Connection Formulae for Asymptotics of Solutions of the Degenerate Third Painleve Equation: II
The degenerate third Painleve' equation, $u"(t)=(u'(t))^2/u(t)-u'(t)/t+1/t(-8c u^2(t)+2ab)+b^2/u(t)$, where $c=+/-1$, $b>0$, and $a$ is a complex parameter, is studied via the Isomonodromy Deformation Method. Asymptotics of general regular and singul…
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Connection F orm ulae for Asymptotics of Solutions of the Degenerate Third P ainlev´ e Equatio n: I I A. V. Kitaev ∗ Steklo v Mathematical Institute F on tank a 27 St. P etersburg 191023 Russia A. V artanian Departmen t of Mathematics College of Charleston Charleston, South Caro lina 2942 4 U. S. A. Ma y 25, 20 22 Abstract The degenerate third P ainlev ´ e equation, u ′′ ( τ ) = ( u ′ ( τ )) 2 u ( τ ) − u ′ ( τ ) τ + 1 τ ( − 8 ǫu 2 ( τ ) + 2 ab ) + b 2 u ( τ ) , where ǫ = ± 1, b ∈ R \ { 0 } , and a ∈ C , is studied v ia the I somonodromy Deformation Metho d. Asymptotics of general regular and singular solutions as τ → ± ∞ and τ → ± i ∞ are derived and parametrized in terms of th e mono dromy data of t h e associated 2 × 2 linear auxiliary problem introduced in [1]. Using these results, th ree-real-parameter families of solutions that hav e infinite sequ ences of zeroes and p oles that are asymptotically lo cated along th e real and imaginary axes are distinguished: asymptotics of t hese zeroes and poles are also obtained. 2000 Mathematics Sub ject Classification. 33E1 7, 34M40, 34M50, 34M55, 34M60 Abbreviated Title. Degenerate Third Pai nlev´ e Equ ation: I I Key W o rds. Asymptotics, P ainlev´ e transcenden ts, isomonodromy deformations, WK B metho d , S t okes phenomena ∗ E-mail: kitaev@p dmi.ras.ru 1 Degenerate Third Painlev ´ e Eq uation: I I 2 1 In tro duction and the Manifold of Mono drom y Data In this pap er we con tinue our study [1] o f the degener ate third Painlev´ e equation, u ′′ ( τ ) = ( u ′ ( τ )) 2 u ( τ ) − u ′ ( τ ) τ + 1 τ ( − 8 ǫu 2 ( τ ) + 2 a b ) + b 2 u ( τ ) , ǫ = ± 1 , (1.1) where the prime denotes differ ent iation with r e spe ct to τ , and b ∈ R \ { 0 } and a ∈ C are para meter s. F or other Painlev´ e equa tions, there exis t families of s olutions that ha ve zero es and p oles a ccumulating at the p oint at infinity (see, for example, [2]). In P art I [1], such s olutions w ere not distinguished; therefore, the main purpo se of this w ork (P art I I) is to determine whether or not suc h solutions exist, and if so , to find their asymptotics. As sho wn in [1], Equation (1.1) c an be presented as a Hamiltonian system, with the Hamiltonian function, H ( τ ), given b y H ( τ ) := a − i 2 b u ( τ ) + 1 2 τ a − i 2 2 + τ 4 u 2 ( τ ) (( u ′ ( τ )) 2 + b 2 ) + 4 ǫu ( τ ) . (1.2) In [1] we int ro duced the auxiliar y function f ( τ ) := τ ( u ′ ( τ ) − i b ) 4 u ( τ ) − 1 2 i a + 1 2 , (1.3) which defines B¨ acklund transformations for Equation (1 .1) : d( τ H ( τ )) d τ = 4 ǫq ( τ ) + i bp ( τ ) , 2 f ( τ ) = q ( τ ) p ( τ ) , − τ 2 d H ( τ ) d τ = 2 f ( τ ) + i a + 1 2 2 − 1 2 i a + 1 2 2 , where q ( τ ) = u ( τ ) and p ( τ ) is a B¨ acklund tra nsformation of u ( τ ). In the Hamiltonian setting, the functions q ( τ ) and p ( τ ) ar e, re sp e c tively , the generalized coordina te and impulse. (More detailed information ab out the functions H ( τ ) and f ( τ ), in particular , the cor resp onding ODEs they satisfy , can b e found in P rop osition 1.3 a nd Rema rk 1.3 of [1 ].) In this work, thes e functions play a n imp ortant role in the study of the zero e s and p ole s of u ( τ ). Section 1 of [1] contains a review o f the literature on the theory and applica tions of Equation (1.1); so her e we mention o nly those pap ers that a re r elated to Equatio n (1.1) and which hav e app eared since the publication of [1]. According to the algebr o-geometr ic classification scheme given in [3], the space of initial c onditions of Equa tio n (1.1) can b e ch ar a cterized by the extended Dynk in diagr am of t yp e D 7 . There is another cas e of the degener ate third P ainlev´ e equation whic h can b e obtained by a similarity reduction of the w ell-known Sine- Go rdon equation: in the classification scheme o f [3], this equation is of t ype D 8 . The la tter equation is k nown to be equiv alent, via a quadratic tra nsformation, to a sp ecial case of the “co mplete” third P ainlev´ e equatio n (t ype D 6 in the classification of [3]); ther e fore, we us e the term “degenerate” to s p ecify only Equation (1.1), or its equiv alen t for ms. Asymptotics of solutions of the D 8 equation were s tudied via the Is omono dromy Deformatio n Me tho d in [2]. Rece n tly , asymptotics of the so-c a lled instan ton solutions of the D 8 equation were o btained in [4] via the exac t WKB a na lysis. W e also ment ion the r ecent w ork [5], where a clas s of semi-flat Calabi-Y au metr ics is obtained in ter ms of r eal solutions of Equation (1.1) with a = 0 . In this work w e apply the Isomono dr o my Deforma tion Metho d: the reader can find so me basic ideas a nd reference s concerning this metho d in Sectio ns 1 and 2 o f [1]. W e als o mention the new mono- graph [6], which reflects some recent developments of this metho d and of a closely rela ted technique based o n a steepest- des cent-t ype analysis of the a sso ciated Riemann- Hilb ert problem [7]. Although Equation (1.1) res em bles one of the standard repr esentativ es of the list of the Painlev ´ e equa tions, its asymptotic study via the Iso mono dromy Deformation Metho d contains, in co nt ra s t to the o ther Painlev ´ e equa tions, a dditional technical difficulties: the problem is that the corresp onding F uc hs- Garnier (or Lax) pair is degener ate (see [1 ] for de ta ils); there fo re, in contrast to the non- degenerate versions of the Painlev´ e equa tio ns, its asso ciated WKB analysis requires a m uc h mor e careful ev alua- tion of the corr ection terms. In fac t, this is one of the r e asons why the present work to ok 6 years to complete since the app eara nce of [1]. In o r der to make the pr esentation as self-co n tained a s p ossible, we now embark on succinctly reminding the r eader ab out s ome of the ba sic facts in tro duced in Sections 1 a nd 2 of [1]. Degenerate Third Painlev ´ e Equation: I I 3 Our study of Equation (1.1) is based on the following F uchs-Garnier (o r Lax) pair (s e e P rop osi- tion 2.1 of [1]): ∂ µ Ψ( µ, τ ) = e U ( µ, τ )Ψ( µ, τ ) , ∂ τ Ψ( µ, τ ) = e V ( µ, τ )Ψ( µ, τ ) , (1.4) where e U ( µ, τ ) = − 2i τ µσ 3 + 2 τ 0 2i A ( τ ) √ − A ( τ ) B ( τ ) − D ( τ ) 0 ! − 1 µ i a + 2 τ A ( τ ) D ( τ ) p − A ( τ ) B ( τ ) + 1 2 ! σ 3 + 1 µ 2 0 e α ( τ ) i τ B ( τ ) 0 , e V ( µ, τ ) = − i µ 2 σ 3 + µ 0 2i A ( τ ) √ − A ( τ ) B ( τ ) − D ( τ ) 0 ! + i a 2 τ − A ( τ ) D ( τ ) p − A ( τ ) B ( τ ) ! σ 3 − 1 2 τ µ 0 e α ( τ ) i τ B ( τ ) 0 , with σ 3 = 1 0 0 − 1 , and e α ( τ ) := − 2 B ( τ ) i a p − A ( τ ) B ( τ ) + τ ( A ( τ ) D ( τ ) + B ( τ ) C ( τ )) . Prop ositio n 1.1. The F r ob enius c omp atibility c ondition of System (1.4) for arbitr ary values of µ ∈ C and for differ entiable, sc alar-value d functions A ( τ ) , B ( τ ) , C ( τ ) , and D ( τ ) is tha t these funct ions satisfy the fol lowing system of isomono dr omy deformatio ns: A ′ ( τ ) = 4 C ( τ ) p − A ( τ ) B ( τ ) , B ′ ( τ ) = − 4 D ( τ ) p − A ( τ ) B ( τ ) , ( τ C ( τ )) ′ = 2i aC ( τ ) − 2 τ A ( τ ) , ( τ D ( τ )) ′ = − 2i aD ( τ ) + 2 τ B ( τ ) , ( p − A ( τ ) B ( τ ) ) ′ = 2( A ( τ ) D ( τ ) − B ( τ ) C ( τ )) . (1.5) Remark 1 .1. Hereafter, all explicit τ dep endencies a r e suppressed, exc ept wher e confusion may a rise . A relation b etw een the F uchs-Garnier pair (1.4) and the degenera te third Painlev ´ e equation (1.1) is given b y Prop ositio n 1.2 ([1]) . L et u = u ( τ ) and ϕ = ϕ ( τ ) solve the system u ′′ = ( u ′ ) 2 u − u ′ τ + 1 τ ( − 8 ǫu 2 + 2 ab ) + b 2 u , ϕ ′ = 2 a τ + b u , (1.6) wher e ǫ = ± 1 , and a, b ∈ C ar e indep endent of τ . Then, A ( τ ) := u ( τ ) τ e i ϕ ( τ ) , B ( τ ) := − u ( τ ) τ e − i ϕ ( τ ) , C ( τ ) := ǫτ A ′ ( τ ) 4 u ( τ ) , D ( τ ) := − ǫτ B ′ ( τ ) 4 u ( τ ) , (1.7) solve System (1.5) . Conversely, let A ( τ ) 6≡ 0 , B ( τ ) 6≡ 0 , C ( τ ) , and D ( τ ) solve System (1.5 ) , and define u ( τ ) := ǫτ p − A ( τ ) B ( τ ) , ϕ ( τ ) := − i 2 ln − A ( τ ) B ( τ ) , and b := u ( τ ) ϕ ′ ( τ ) − 2 a τ . Then b is indep endent of τ , and u ( τ ) and ϕ ( τ ) solve System (1.6) . In this work asymptotics (as τ → ±∞ and τ → ± i ∞ ) of solutions of E quation (1.1) are para metrized in terms o f the mono drom y data of System (1.4). This par ametrization is equiv alent to finding the corres p o nding connection for mulae; indeed, given asymptotics of so me solution as τ → + ∞ , say , one uses it to determine the corr esp onding mono dromy data, and therefore to obtain asy mptotics of the same solution as τ → −∞ or τ → ± i ∞ . F urthermore, emplo ying r esults from [1 ], one arrives a t asymptotics of the same so lution as τ → ± 0 or τ → ± i0. Therefor e , it is impo rtant to remind the reader ab out the definition of the mono dro m y data of System (1.4) given in [1]. Degenerate Third Painlev ´ e Equation: I I 4 On the complex µ -plane, System (1.4) has t wo irreg ular singular p oints, µ = ∞ a nd µ = 0. F or δ ∞ , δ 0 > 0 and k ∈ Z , define the (se ctorial) neigh b orho o ds Ω ∞ k and Ω 0 k , r esp ectively , of these (singular) po int s: Ω ∞ k := µ ∈ C : | µ | > δ − 1 ∞ , − π 2 + π k 2 < arg µ + 1 2 arg τ < π 2 + π k 2 , Ω 0 k := µ ∈ C : | µ | < δ 0 , − π + πk < arg µ − 1 2 arg τ − 1 2 arg( ǫb ) < π + π k . The fo llowing Pro po sition is a direc t consequence of general asymptotic results for linear ODEs [8, 9]. Prop ositio n 1.3 ([1 ]) . F or k ∈ Z , t her e ex ist solutions Y ∞ k ( µ ) = Y ∞ k ( µ, τ ) and X 0 k ( µ ) = X 0 k ( µ, τ ) of System (1.4) which ar e uniquely define d by the fol lowing asymptotic exp ansions: Y ∞ k ( µ ) := µ →∞ µ ∈ Ω ∞ k I + 1 µ Ψ (1) + 1 µ 2 Ψ (2) + · · · exp − i τ µ 2 + a − i 2 ln µ σ 3 , X 0 k ( µ ) := µ → 0 µ ∈ Ω 0 k Ψ 0 (I + Z 1 µ + · · · ) exp − i √ τ ǫb µ σ 3 ! , wher e ln µ := ln | µ | + i a rg µ , Ψ (1) = 0 A √ − AB D 2i 0 , Ψ (2) = ψ (2) 11 0 0 ψ (2) 22 ! , ψ (2) 11 = − i 2 τ √ − AB + τ D C + AD √ − AB , ψ (2) 22 = i τ 2 √ − AB + C D , Ψ 0 = i √ 2 ( ǫb ) 1 / 4 τ 1 / 4 √ B σ 3 ( σ 1 + σ 3 ) , Z 1 = z (11) 1 z (12) 1 − z (12) 1 − z (11) 1 ! , z (11) 1 = i a + 1 2 + 2 τ AD √ − AB 2 2i √ τ ǫb − 2i τ 3 / 2 √ − AB √ ǫb − D √ τ ǫb B , z (12) 1 = i a + 1 2 + 2 τ AD √ − AB 2i √ τ ǫb , with I = ( 1 0 0 1 ) , and σ 1 = ( 0 1 1 0 ) . Remark 1. 2. The canonical solutio ns X 0 k ( µ ) are defined uniquely , pr ovided the branch o f p B ( τ ) is fixed. Hereafter, the br anch of p B ( τ ) is not fixe d; therefore , the s e t of canonica l solutions { X 0 k ( µ ) } k ∈ Z is defined up to a sig n. This ambiguit y do esn’t affect the definition of the Stokes m ultipliers ( see Equations (1 .8 ) below ) , but results in an ambiguit y of s ign in the definition of the connection matrix, G ( see Eq uation (1.11) b elow ) . The c anonic al solutions , Y ∞ k ( µ ) and X 0 k ( µ ), enable one to define the Stokes matric es , S ∞ k and S 0 k : Y ∞ k +1 ( µ ) = Y ∞ k ( µ ) S ∞ k , X 0 k +1 ( µ ) = X 0 k ( µ ) S 0 k . (1.8) The Stokes matrices are independent of the parameters µ a nd τ , and have the following s tructures: S ∞ 2 k = 1 0 s ∞ 2 k 1 , S ∞ 2 k +1 = 1 s ∞ 2 k +1 0 1 , S 0 2 k = 1 s 0 2 k 0 1 , S 0 2 k +1 = 1 0 s 0 2 k +1 1 . The parameter s s ∞ n and s 0 n , n ∈ Z , ar e called the St okes multipliers . One can show that S ∞ k +4 = e − 2 π ( a − i 2 ) σ 3 S ∞ k e 2 π ( a − i 2 ) σ 3 , S 0 k +2 = S 0 k . (1.9) Equations (1.9) s how that th e n umber of independent Stokes mult ipliers does not exceed six; for example, s 0 0 , s 0 1 , s ∞ 0 , s ∞ 1 , s ∞ 2 , and s ∞ 3 . F urthermore, due to the sp ecial structure of System (1.4), that is, the coefficient matrices of o dd (r esp., even) p owers of µ in e U ( µ, τ ) are diago nal (resp., o ff-diagonal) and vic e-versa for e V ( µ, τ ), one can deduce the following rela tions for the Stokes matr ices (multipliers): S ∞ k +2 = σ 3 e − π ( a − i 2 ) σ 3 S ∞ k e π ( a − i 2 ) σ 3 σ 3 , S 0 k = σ 1 S 0 k +1 σ 1 . (1.10) Degenerate Third Painlev ´ e Equation: I I 5 Equations (1 .1 0) reduce the n umber of indep endent Stok es m ultipliers b y a factor of 2 , that is, all Stokes multipliers c a n be expr essed in ter ms of s 0 0 , s ∞ 0 , s ∞ 1 , and the parameter of for mal mono dromy , a . There is one mo re relation b etw een the Stokes multipliers, which follows from the so-ca lled cyclic relation (see below). Define the mono dro my matrice s at infinity , M ∞ , and a t zero , M 0 , by the following relations: Y ∞ 0 ( µ e − 2 π i ) := Y ∞ 0 ( µ ) M ∞ , X 0 0 ( µ e − 2 π i ) := X 0 0 ( µ ) M 0 . Since Y ∞ 0 ( µ ) and X 0 0 ( µ ) are solutions of System (1.4), they differ by a right-hand (matrix) factor G : Y ∞ 0 ( µ ) := X 0 0 ( µ ) G, (1.11) where G is called the c onne ction matrix . As matrices r e lating fundamental solutions of System (1.4), the m ono dr o my , co nnection, a nd Stok es matrices are independent of µ and τ ; furthermore, s ince tr( e U ( µ, τ )) = tr( e V ( µ, τ )) = 0, it follows that det( M 0 ) = det( M ∞ ) = det( G ) = 1 . F rom the definition of the monodro my and co nnec tion matric es, one deduces the following cyclic r elation : GM ∞ = M 0 G. The mono dromy matrices can b e expresse d in terms of the Stokes ma trices: M ∞ = S ∞ 0 S ∞ 1 S ∞ 2 S ∞ 3 e − 2 π ( a − i 2 ) σ 3 , M 0 = S 0 0 S 0 1 . The Stok es multipliers, s 0 0 , s ∞ 0 , and s ∞ 1 , the elements of the connection ma tr ix, ( G ) ij =: g ij , i, j = 1 , 2, and the para meter o f formal monodr omy , a , are ca lle d the mono dr omy data . Consider C 8 with co - ordinates ( a, s 0 0 , s ∞ 0 , s ∞ 1 , g 11 , g 12 , g 21 , g 22 ). The algebra ic v ar iety defined b y det( G ) = 1 and the semi- cyclic r elation G − 1 S 0 0 σ 1 G = S ∞ 0 S ∞ 1 σ 3 e − π ( a − i 2 ) σ 3 is called the manifold of mono dr omy data , M . Since only thre e of the four equations in the se mi-cyclic relation are indep endent, it is clear that dim C ( M ) = 4; more pre c isely , the equations defining M are: s ∞ 0 s ∞ 1 = − 1 − e − 2 π a − i s 0 0 e − π a , g 22 g 21 − g 11 g 12 + s 0 0 g 11 g 22 = ie − π a , g 2 11 − g 2 21 − s 0 0 g 11 g 21 = i s ∞ 0 e − π a , g 2 22 − g 2 12 + s 0 0 g 12 g 22 = i s ∞ 1 e π a , g 11 g 22 − g 12 g 21 = 1 . Remark 1.3. T o a chiev e a one-to-one cor resp ondence b etw een the co e fficie nts of System (1.4) and the p oints on M , one has to fa c torize M by the inv olution G → − G ( cf. Remark 1.2) . W e now describ e the conten ts o f this pap er. In Section 2 the ma in asymptotic results for u ( τ ), H ( τ ), and f ( τ ) as τ → ±∞ a re presented. Solutions u ( τ ) having infinite sequences of zer o es and p oles accumulating at the p oint at infinity of the real and imaginary axes are als o specified, and the asymptotic distribution of these sequences are obtained. In Section 3 (ra ther technical in nature) the asymptotic solutio n o f the direct mo no dromy problem for the µ -pa rt o f System (1.4), under certain restric tio ns on its c o efficien ts, is presented for p os itive real τ . This asy mptotic so lution is based on matching WKB-asymptotics of the fundamental solution of System (1.4) with its approximation in ter ms of parab o lic -cylinder functions near the double-turning po int . In Section 4 the re sults of Section 3 are inverted in or der to solve the in verse mono dromy pr oblem for the µ -part of System (1.4). A t this stag e , explicit asymptotics fo r the co efficients of the µ -pa rt of System (1.4) are par ametrized in terms of the monodr o my data. Under the a ssumption that the mono dromy data are constan t and satisfy certain conditions, one finds that the asymptotics obtained satisfy all of the conditions that were imposed in Section 3. According to the justification scheme presented in [10], it follo ws tha t there exists exact isomono dromy defor mations corresp onding to these mo no dromy data, that is, s olutions of the system of isomono dromy deformations (1.5) , whos e asymptotics co incide with the o ne s obtained in this section. Appendix A contains the Laurent expansio ns at ze r o es and p oles o f the function u ( τ ) together with the corr esp onding expansio ns of the asso cia ted functions H ( τ ) and f ( τ ). These res ults are used in Degenerate Third Painlev ´ e Equation: I I 6 Section 4 to complete the pro of of the a symptotic distribution of zer o es and poles of u ( τ ) for p o s itive real τ . In App endix B the asymptotic results for τ → ∞ obtained in this pap er ar e compared with the corres p o nding as ymptotitcs in [1], a nd misprints from [1] are corrected. This comparison is used in Section 4 to resolve a sign ambiguit y in the solution of the inv er se mono dromy problem. The main b o dy of this pap er is devoted to asymptotics as τ → + ∞ . T o extend the results to negative and pure imaginary τ , one applies the action of the Lie-p oint symmetries τ → − τ and τ → ± i τ on the ma nifold of mono dro m y data that were der ived in Subsection 6.2 o f [1]. Asymptotics for real τ ar e presented in Section 2 (as mentioned ab ove), and asymptotics for imag inary τ are given in App endix C. W e plan to devote the third pa r t o f o ur studies of the degenerate third P ainlev´ e Equation (1.1 ) to the extension of o ur results to complex v alues of the parameter b , and the discuss io n of the b ehavior of some sp ecial solutions on the re al and imaginary axe s tog ether with the co mparison of asymptotic and numerical results. 2 Summary of Results In this work the deta iled a nalysis for asymptotics of u ( τ ) is prese n ted for the case τ → + ∞ and ǫb > 0. The analytic contin uation of the function u ( τ ) from p ositive v alues of the par ameters τ and ǫb to negative v alues of these para meters is not single v alued; therefore, in or der to reflect this fac t, write τ = | τ | e i π ε 1 and ǫ b = | ǫb | e i π ε 2 , wher e ε 1 , ε 2 = 0 , ± 1. The co rresp onding as ymptotics, for bo th po sitive and negative v alues of these parameters , of u ( τ ), and the as so ciated functions H ( τ ) and f ( τ ), are conv enient to expr ess in terms o f the auxiliar y mapping F ε 1 ,ε 2 (see below), which is an isomorphism of the manifold of mono dr omy da ta 1 , M . The following definition of F ε 1 ,ε 2 is based on Section 6 of [1], that is , transfor mation 6.2.1 changing 2 τ → − τ and transforma tion 6.2.2 changing 3 a → − a . Define 4 F ε 1 ,ε 2 : ( a, s 0 0 , s ∞ 0 , s ∞ 1 , g 11 , g 12 , g 21 , g 22 ) → (( − 1) ε 2 a, s 0 0 , s ∞ 0 ( ε 1 , ε 2 ) , s ∞ 1 ( ε 1 , ε 2 ) , g 11 ( ε 1 , ε 2 ) , g 12 ( ε 1 , ε 2 ) , g 21 ( ε 1 , ε 2 ) , g 22 ( ε 1 , ε 2 )), ε 1 , ε 2 = 0 , ± 1: (1) F 0 , 0 is the identit y mapping : s ∞ 0 (0 , 0) = s ∞ 0 , s ∞ 1 (0 , 0) = s ∞ 1 , and g ij (0 , 0) = g ij , i, j = 1 , 2; (2) F 0 , − 1 : s ∞ 0 (0 , − 1 ) = s ∞ 1 e π a , s ∞ 1 (0 , − 1 ) = s ∞ 0 e π a , g 11 (0 , − 1 ) = − g 22 e πa 2 , g 12 (0 , − 1 ) = − ( g 21 + s ∞ 0 g 22 )e − πa 2 , g 21 (0 , − 1 ) = − ( g 12 − s 0 0 g 22 )e πa 2 , and g 22 (0 , − 1 ) = − ( g 11 − s 0 0 g 21 + ( g 12 − s 0 0 g 22 ) s ∞ 0 )e − πa 2 ; (3) F 0 , 1 : s ∞ 0 (0 , 1) = s ∞ 1 e π a , s ∞ 1 (0 , 1) = s ∞ 0 e π a , g 11 (0 , 1) = − i g 12 e πa 2 , g 12 (0 , 1) = − i( g 11 + s ∞ 0 g 12 )e − πa 2 , g 21 (0 , 1) = − i g 22 e πa 2 , and g 22 (0 , 1) = − i( g 21 + s ∞ 0 g 22 )e − πa 2 ; (4) F − 1 , 0 : s ∞ 0 ( − 1 , 0 ) = − s ∞ 0 e − π a , s ∞ 1 ( − 1 , 0 ) = − s ∞ 1 e π a , g 11 ( − 1 , 0 ) = g 21 e − πa 2 , g 12 ( − 1 , 0 ) = − g 22 e πa 2 , g 21 ( − 1 , 0 ) = ( g 11 − s 0 0 g 21 )e − πa 2 , and g 22 ( − 1 , 0 ) = − ( g 12 − s 0 0 g 22 )e πa 2 ; (5) F − 1 , − 1 : s ∞ 0 ( − 1 , − 1) = − s ∞ 1 , s ∞ 1 ( − 1 , − 1) = − s ∞ 0 e 2 π a , g 11 ( − 1 , − 1) = g 12 − s 0 0 g 22 , g 12 ( − 1 , − 1) = − g 11 + s 0 0 g 21 − ( g 12 − s 0 0 g 22 ) s ∞ 0 , g 21 ( − 1 , − 1) = g 22 − ( g 12 − s 0 0 g 22 ) s 0 0 , and g 22 ( − 1 , − 1) = − g 21 + ( g 11 − s 0 0 g 21 ) s 0 0 − ( g 22 − ( g 12 − s 0 0 g 22 ) s 0 0 ) s ∞ 0 ; (6) F − 1 , 1 : s ∞ 0 ( − 1 , 1 ) = − s ∞ 1 , s ∞ 1 ( − 1 , 1 ) = − s ∞ 0 e 2 π a , g 11 ( − 1 , 1 ) = i g 22 , g 12 ( − 1 , 1 ) = − i( g 21 + s ∞ 0 g 22 ), g 21 ( − 1 , 1 ) = i( g 12 − s 0 0 g 22 ), and g 22 ( − 1 , 1 ) = − i( g 11 − s 0 0 g 21 + ( g 12 − s 0 0 g 22 ) s ∞ 0 ); (7) F 1 , 0 : s ∞ 0 (1 , 0) = − s ∞ 0 e π a , s ∞ 1 (1 , 0) = − s ∞ 1 e − π a , g 11 (1 , 0) = ( g 21 + s 0 0 g 11 )e πa 2 , g 12 (1 , 0) = − ( g 22 + s 0 0 g 12 )e − πa 2 , g 21 (1 , 0) = g 11 e πa 2 , and g 22 (1 , 0) = − g 12 e − πa 2 ; (8) F 1 , − 1 : s ∞ 0 (1 , − 1 ) = − s ∞ 1 e 2 π a , s ∞ 1 (1 , − 1 ) = − s ∞ 0 , g 11 (1 , − 1 ) = g 12 e π a , g 12 (1 , − 1 ) = − ( g 11 + s ∞ 0 g 12 )e − π a , g 21 (1 , − 1 ) = g 22 e π a , and g 22 (1 , − 1 ) = − ( g 21 + s ∞ 0 g 22 )e − π a ; (9) F 1 , 1 : s ∞ 0 (1 , 1) = − s ∞ 1 e 2 π a , s ∞ 1 (1 , 1) = − s ∞ 0 , g 11 (1 , 1) = i( g 22 + s 0 0 g 12 )e π a , g 12 (1 , 1) = − i( g 21 + s 0 0 g 11 + ( g 22 + s 0 0 g 12 ) s ∞ 0 )e − π a , and g 22 (1 , 1) = − i( g 11 + s ∞ 0 g 12 )e − π a . 1 There is a misprint on page 1173 (Section 3) of [1]: for items (2), (3) , (5), (6), (8) and (9) in the definition of th e auxiliary mapping F ε 1 ,ε 2 , the chang e a → − a should b e made everywhere. 2 In transformation 6.2.1, ǫb → ǫb and a → a , th at is, ǫ n b n = ǫ o b o and a n = a o . 3 In transformation 6.2.2, τ → τ , that is, τ n = τ o . 4 s 0 0 ( ε 1 , ε 2 ) = s 0 0 . Degenerate Third Painlev ´ e Equation: I I 7 Remark 2. 1. The roots of p ositive quan tities are assumed p ositive, whilst the branches of the ro o ts of complex qua nt ities can be taken arbitra rily , unless sta ted o therwise. F urthermore , it is as sumed that, for ne g ative r eal z , the following branches are alw ays tak en: z 1 / 3 := −| z | 1 / 3 and z 2 / 3 := ( z 1 / 3 ) 2 . Theorem 2.1. L et ε 1 , ε 2 = 0 , ± 1 , ǫ b = | ǫb | e i π ε 2 , and u ( τ ) b e a solution of Equation (1.1) c orr esp onding to the mono dr omy data ( a, s 0 0 , s ∞ 0 , s ∞ 1 , g 11 , g 12 , g 21 , g 22 ) . Supp ose that g 11 ( ε 1 , ε 2 ) g 12 ( ε 1 , ε 2 ) g 21 ( ε 1 , ε 2 ) g 22 ( ε 1 , ε 2 ) 6 = 0 , Re( e ν ( ε 1 , ε 2 ) + 1) ∈ (0 , 1) \ 1 2 , (2.1) wher e e ν ( ε 1 , ε 2 ) + 1 := i 2 π ln( g 11 ( ε 1 , ε 2 ) g 22 ( ε 1 , ε 2 )) . (2.2) Then ther e ex ist δ G satisfying, for 0 < Re( e ν ( ε 1 , ε 2 ) + 1) < 1 2 , the ine qu ality 0 < δ G < 1 3 1 + 2Re( e ν ( ε 1 , ε 2 ) + 1) 7 + 6Re( e ν ( ε 1 , ε 2 ) + 1) , and, for 1 2 < Re( e ν ( ε 1 , ε 2 ) + 1) < 1 , the ine quality 0 < δ G < 1 3 3 − 2Re( e ν ( ε 1 , ε 2 ) + 1) 9 + 2Re( e ν ( ε 1 , ε 2 ) + 1) , such that u ( τ ) has the asymptotic ex p ansion u ( τ ) = τ →∞ e i πε 1 ǫ ( ǫb ) 2 / 3 2 τ 1 / 3 1 − 3 2 sin 2 ( 1 2 ϑ ( ε 1 , ε 2 , τ )) ! (2.3) = τ →∞ e i πε 1 ǫ ( ǫb ) 2 / 3 2 τ 1 / 3 sin( 1 2 ϑ ( ε 1 , ε 2 , τ ) − ϑ 0 ) sin( 1 2 ϑ ( ε 1 , ε 2 , τ ) + ϑ 0 ) sin 2 ( 1 2 ϑ ( ε 1 , ε 2 , τ )) , (2.4) wher e ϑ ( ε 1 , ε 2 , τ ) := φ ( τ ) − i ( e ν ( ε 1 , ε 2 ) + 1) − 1 2 ln φ ( τ ) − i ( e ν ( ε 1 , ε 2 ) + 1) − 1 2 ln 12 + ( − 1 ) ε 2 a ln(2 + √ 3) + π 4 − 3 π 2 ( e ν ( ε 1 , ε 2 ) + 1) + i ln g 11 ( ε 1 , ε 2 ) g 12 ( ε 1 , ε 2 )Γ( e ν ( ε 1 , ε 2 ) + 1) √ 2 π + O ( τ − δ G ln τ ) , (2.5) with φ ( τ ) = 3 √ 3 ( − 1) ε 2 ( ǫb ) 1 / 3 τ 2 / 3 , (2.6) ϑ 0 = − π 2 + i 2 ln(2 + √ 3) , (2.7) and Γ( · · · ) is the Euler gamma function [11] . L et H ( τ ) b e t he H amiltonian function define d by Equation (1.2) c orr esp onding to the function u ( τ ) given ab ove. Then H ( τ ) has t he asymptotic exp ans ion H ( τ ) = τ →∞ e i πε 1 3( ǫb ) 2 / 3 τ 1 / 3 − i( − 1) ε 2 4 √ 3 ( ǫb ) 1 / 3 τ − 1 / 3 ( e ν ( ε 1 , ε 2 ) + 1) − 1 2 + 1 2 √ 3 i( − 1) ε 2 a + 1 2 + i 4 cot( 1 2 ϑ ( ε 1 , ε 2 , τ )) + i 4 cot( 1 2 ϑ ( ε 1 , ε 2 , τ ) − ϑ 0 ) + O ( τ − δ G ) . (2.8) The function f ( τ ) define d by Equation (1.3) has the fol lowing asymptotics: f ( τ ) = τ →∞ e i πε 1 − ( − 1) ε 2 ( ǫb ) 1 / 3 2 τ 2 / 3 i + 3 √ 2 sin( 1 2 ϑ ( ε 1 , ε 2 , τ )) sin ( 1 2 ϑ ( ε 1 , ε 2 , τ ) − ϑ 0 ) ! . (2.9) Remark 2.2. Define the s trip ( in the φ -plane ) D := { τ ∈ C : Re( φ ( τ )) > c 1 , | Im( φ ( τ )) | < c 2 } , (2.10) where φ ( τ ) is given in Equation (2.6) , a nd c 1 , c 2 > 0 are par ameters. In terms o f the origina l v a riable τ , the strip D is a simply-connected domain with conv ex b oundar y of increasing width pro po rtional to | τ | 1 / 3 . The asymptotics of u ( τ ) , H ( τ ) , and f ( τ ) pr esented in Theorem 2.1 are actually v alid in the strip domain D . Degenerate Third Painlev ´ e Equation: I I 8 Theorem 2.2. L et ε 1 , ε 2 = 0 , ± 1 , ǫ b = | ǫb | e i π ε 2 , and u ( τ ) b e a solution of Equation (1.1) c orr esp onding to the mono dr omy data ( a, s 0 0 , s ∞ 0 , s ∞ 1 , g 11 , g 12 , g 21 , g 22 ) . Supp ose that g 11 ( ε 1 , ε 2 ) g 12 ( ε 1 , ε 2 ) g 21 ( ε 1 , ε 2 ) g 22 ( ε 1 , ε 2 ) 6 = 0 , R e i 2 π ln( g 11 ( ε 1 , ε 2 ) g 22 ( ε 1 , ε 2 )) = 1 2 . (2.11) L et the br anch of the fun ction ln( · · · ) b e chosen 5 such that Im(ln( − g 11 ( ε 1 , ε 2 ) g 22 ( ε 1 , ε 2 ))) = 0 . Define 1 ( ε 1 , ε 2 ) := 1 2 π ln( − g 11 ( ε 1 , ε 2 ) g 22 ( ε 1 , ε 2 )) ( ∈ R ) . (2.12) Then ∃ δ ∈ (0 , 1 / 39) such that the function u ( τ ) has, for al l lar ge en ough m ∈ N , se c ond-or der p oles, τ ∞ m , ac cumu lating at t he p oint at infinity, τ ∞ m = m →∞ e i π ε 1 2 π ( − 1) ε 2 m 3 √ 3 ( ǫb ) 1 / 3 3 / 2 1 − 3 1 ( ε 1 , ε 2 ) 4 π ln m m − 3 2 ( ε 1 , ε 2 ) 4 π 1 m + O m 1 2 − 3 δ 2 , (2.13) wher e 2 ( ε 1 , ε 2 ) := 1 ( ε 1 , ε 2 ) ln(24 π ) + ( − 1 ) ε 2 Re( a ) ln(2 + √ 3) + π 2 − 1 2 arg g 11 ( ε 1 , ε 2 ) g 12 ( ε 1 , ε 2 ) g 21 ( ε 1 , ε 2 ) g 22 ( ε 1 , ε 2 ) − arg Γ 1 2 + i 1 ( ε 1 , ε 2 ) + i ( − 1) ε 2 Im( a ) ln(2 + √ 3) + 1 2 ln g 11 ( ε 1 , ε 2 ) g 12 ( ε 1 , ε 2 ) g 21 ( ε 1 , ε 2 ) g 22 ( ε 1 , ε 2 ) ; (2.14) furthermor e, t he function u ( τ ) has, for al l lar ge enou gh m ∈ N , a p air of first -or der zer o es, τ ± m , ac cumulating at t he p oint at infi n ity, τ ± m = m →∞ e i π ε 1 2 π ( − 1) ε 2 m 3 √ 3 ( ǫb ) 1 / 3 3 / 2 1 − 3 1 ( ε 1 , ε 2 ) 4 π ln m m − 3 4 π ( 2 ( ε 1 , ε 2 ) ± 2 ϑ 0 ) 1 m + O m 1 2 − 3 δ 2 , (2.15) wher e ϑ 0 is given in Equation (2 .7) . Remark 2. 3. F urther information concerning the La urent expansio ns of the functions u ( τ ) , H ( τ ) , and f ( τ ) at p o le s and zer o es can b e found in App endix A . Remark 2.4. T o present asymptotics of u ( τ ) , H ( τ ) , and f ( τ ) outside of neigh bo rho o ds of poles and zero es, introduce the technical notion of the cheese-lik e domain, D u , for a solution u ( τ ) : D u := τ ∈ D : | φ ( τ ) − φ ( τ κ m ) | > C | τ κ m | − δ , where the strip do main D is defined by Equation (2.1 0) , φ ( τ ) is given in Eq uation (2.6) , C > 0 is a parameter, κ = ∞ , ± ( τ κ m are the p oles and zero es introduced in Theo rem 2.2) , and 0 < δ < 1 / 39 . In terms of the v ariable φ , the cheese-like domain D u is a m ultiply-connec ted domain which resembles D with circula r “ cheese-holes” cen tered at φ ( τ κ m ) of shr inking radii, whilst in ter ms of the or iginal v aria ble τ the diameter of the ch eese- holes are increa sing, that is, | τ − τ κ m | ∼ | τ κ m | 1 3 − δ . Theorem 2.3. L et ε 1 , ε 2 = 0 , ± 1 , ǫ b = | ǫb | e i π ε 2 , and u ( τ ) b e a solution of Equation (1.1) c orr esp onding to the mono dr omy data ( a, s 0 0 , s ∞ 0 , s ∞ 1 , g 11 , g 12 , g 21 , g 22 ) . Supp ose that c onditions (2.11) ar e valid, t he br anch of ln( · · · ) is chosen as in The or em 2.2 , and 1 ( ε 1 , ε 2 ) is define d by Equation (2.12) . Then ther e exist δ, δ G ∈ R + satisfying the ine qualities 0 < δ < 1 39 , 0 < δ < δ G < 1 15 − 8 δ 5 , such that u ( τ ) has the asymptotic ex p ansion u ( τ ) = τ →∞ e i πε 1 τ ∈ D u ǫ ( ǫb ) 2 / 3 2 τ 1 / 3 sin( 1 2 θ ( ε 1 , ε 2 , τ ) − ϑ 0 ) sin( 1 2 θ ( ε 1 , ε 2 , τ ) + ϑ 0 ) sin 2 ( 1 2 θ ( ε 1 , ε 2 , τ )) , (2.16) 5 The second condition of Equations (2.11) sugge sts that this branch of ln( · · · ) exists. Degenerate Third Painlev ´ e Equation: I I 9 wher e θ ( ε 1 , ε 2 , τ ) := φ ( τ ) + 1 ( ε 1 , ε 2 ) ln φ ( τ ) + 1 ( ε 1 , ε 2 ) ln 1 2 + ( − 1) ε 2 Re( a ) ln(2 + √ 3) + π 2 − 1 2 arg g 11 ( ε 1 , ε 2 ) g 12 ( ε 1 , ε 2 ) g 21 ( ε 1 , ε 2 ) g 22 ( ε 1 , ε 2 ) − arg Γ 1 2 + i 1 ( ε 1 , ε 2 ) + i ( − 1) ε 2 Im( a ) ln(2 + √ 3) + 1 2 ln g 11 ( ε 1 , ε 2 ) g 12 ( ε 1 , ε 2 ) g 21 ( ε 1 , ε 2 ) g 22 ( ε 1 , ε 2 ) + O ( τ − δ G ln τ ) , (2.17) with φ ( τ ) and ϑ 0 given, r esp e ctively, in Equations (2 .6) and (2.7) . L et H ( τ ) b e t he H amiltonian function define d by Equation (1.2) c orr esp onding to the function u ( τ ) given ab ove. Then H ( τ ) has t he asymptotic exp ans ion H ( τ ) = τ →∞ e i πε 1 τ ∈ D u 3( ǫb ) 2 / 3 τ 1 / 3 + ( − 1) ε 2 4 √ 3 ( ǫb ) 1 / 3 τ − 1 / 3 1 ( ε 1 , ε 2 ) + 1 2 √ 3 ( − 1) ε 2 a − i 2 + 1 4 cot( 1 2 θ ( ε 1 , ε 2 , τ )) + 1 4 cot( 1 2 θ ( ε 1 , ε 2 , τ ) − ϑ 0 ) + O ( τ − δ G ) . (2.18) The function f ( τ ) define d by Equation (1.3) has the fol lowing asymptotics: f ( τ ) = τ →∞ e i πε 1 τ ∈ D u − ( − 1) ε 2 ( ǫb ) 1 / 3 2 τ 2 / 3 i + 3 √ 2 sin( 1 2 θ ( ε 1 , ε 2 , τ )) sin ( 1 2 θ ( ε 1 , ε 2 , τ ) − ϑ 0 ) ! . (2.19) Remark 2 .5. F or rea l, non-zero v a lues of b , s ing ular rea l solutions u ( τ ) of E q uation (1.1) are sp ecified by the following “singular real reductio n” for the mono dromy data 6 : s 0 0 = − s 0 0 , s ∞ 0 ( ε 1 , ε 2 ) = − s ∞ 1 ( ε 1 , ε 2 ) e 2 π a , g 11 ( ε 1 , ε 2 ) = − g 22 ( ε 1 , ε 2 ) , g 12 ( ε 1 , ε 2 ) = − g 21 ( ε 1 , ε 2 ) , Im( a ) = 0 . (2.20) In this case, asymptotics of τ ∞ m , τ ± m , u ( τ ) , H ( τ ) , and f ( τ ) a re as given in Eq uations (2.13) , (2.1 5) , (2.16) , (2.18) , and (2 .19) , respectively , but with the c ha nges 1 ( ε 1 , ε 2 ) → 0 ( ε 1 , ε 2 ) , 2 ( ε 1 , ε 2 ) → ♯ 0 ( ε 1 , ε 2 ) , and θ ( ε 1 , ε 2 , τ ) → Θ 0 ( ε 1 , ε 2 , τ ) , where 0 ( ε 1 , ε 2 ) := 1 π ln | g 11 ( ε 1 , ε 2 ) | , (2.21) ♯ 0 ( ε 1 , ε 2 ) := 0 ( ε 1 , ε 2 ) ln(2 4 π ) + ( − 1) ε 2 Re( a ) ln(2 + √ 3) − π 2 − arg g 11 ( ε 1 , ε 2 ) g 12 ( ε 1 , ε 2 )Γ 1 2 + i 0 ( ε 1 , ε 2 ) , (2.22) Θ 0 ( ε 1 , ε 2 , τ ) := φ ( τ ) + 0 ( ε 1 , ε 2 ) ln φ ( τ ) + 0 ( ε 1 , ε 2 ) ln 1 2 + ( − 1) ε 2 Re( a ) ln(2 + √ 3) − π 2 − arg g 11 ( ε 1 , ε 2 ) g 12 ( ε 1 , ε 2 )Γ 1 2 + i 0 ( ε 1 , ε 2 ) + O ( τ − δ G ln τ ) . (2.23) 3 Asymptotic Solution of the Direct Mono drom y Problem 3.1 Notation In this sec tion the mono dr omy data intro duced in Section 1 is ca lculated as τ → + ∞ fo r ǫb > 0 (corresp onding to ε 1 = ε 2 = 0): this c o nstitutes the first step tow a rds the pro of of the results stated in Section 2. This calcula tion cons ists of three comp onents: (i) the WKB analysis of the µ -part of System (1.4), that is, ∂ µ Ψ( µ ) = e U ( µ, τ )Ψ( µ ) , (3.1) where Ψ( µ ) = Ψ( µ, τ ); (ii) the a pproximation of Ψ( µ ) in the neig hborho o ds of the turning p oints; and (iii) the ma tching of these asymptotics. 6 There exist regular real solutions (cf. Part I [ 1]) which are sp ecified by a different r eal r eduction: we plan to discuss this i n a subsequent work. Degenerate Thir d Painlev ´ e Equation: II 10 Some so lutions u ( τ ) of Eq ua tion (1.1) might (and ac tually do) hav e p oles and ze r o es lo cated on the po sitive re a l semi-axis. In or der to be able to study such solutions, one must consider a slightly mo re general co mplex do main e D u : this coincides with the c heese-like domain D u int ro duced in Remar k 2.4; how e ver, since, a priori , one do es not know the so lutions u ( τ ) that hav e such p oles a nd ze ro es, no r the exact lo ca tions of these p ole s a nd zero es (cf. E quations (2.13) and (2.1 5)), it is ne c essary to int ro duce a formal definition for e D u . Denote b y P u and Z u , res p ectively , the countable sets of p oles a nd zero es of u ( τ ): as follows fro m the Painlev ´ e prop erty , these sets migh t hav e a ccumu lation points at 0 and ∞ . Define neighbor ho o ds of P u and Z u , res p ectively: for δ > 0, P δ u := τ ∈ C : | φ ( τ ) − φ ( τ p ) | > C | τ p | − δ , τ p ∈ P u , (3.2) Z δ u := τ ∈ C : | φ ( τ ) − φ ( τ z ) | > C | τ z | − δ , τ z ∈ Z u . ( 3.3) Now, the c heese-like domain e D u is defined: e D u := D \ ( P δ u ∪ Z δ u ) , (3.4) where D is g iven in Equation (2.1 0). Remark 3. 1 .1. Thr oughout this s ection, and for br evity of no tation, the following conv ent ion is adopted: in a symptotics of a ll express ions a nd formulae dep ending on u ( τ ) , the “notatio n” τ → + ∞ means Re( τ ) → + ∞ and τ ∈ e D u . F urther nota tion used thro ughout this s ection is now summarized: (1) I = ( 1 0 0 1 ) is the 2 × 2 ident ity matrix, σ 1 = ( 0 1 1 0 ), σ 2 = 0 − i i 0 , and σ 3 = 1 0 0 − 1 are the Pauli matrices, σ ± := 1 2 ( σ 1 ± i σ 2 ) a re the raising (+) and low ering ( − ) matr ices, and R ± := { x ∈ R : ± x > 0 } ; (2) for ( ς 1 , ς 2 ) ∈ R × R , the function ( z − ς 1 ) i ς 2 : C \ ( −∞ , ς 1 ) → C , z 7→ exp(i ς 2 ln( z − ς 1 )), with the branch cut taken along ( −∞ , ς 1 ) and the principal branch of the log arithm chosen; (3) for a sc alar ω o and a 2 × 2 matrix ˆ Υ, ω ad( σ 3 ) o ˆ Υ := ω σ 3 o ˆ Υ ω − σ 3 o ; (4) for a 2 × 2 matrix-v alued function Y ( z ), Y ( z ) = z → z 0 O ( ∗ ∗ ∗ ) (resp., o ( ∗ ∗ ∗ )) means Y ij ( z ) = z → z 0 O ( ∗ ∗ ∗ ij ) (resp., o ( ∗ ∗ ∗ ij )), i , j = 1 , 2; (5) for B ( · · · ) ∈ M 2 ( C ), || B ( · · · ) || := ( P 2 i,j =1 B ij ( · · · ) B ij ( · · · )) 1 / 2 denotes the Hilber t-Schmidt norm, where ⋆ ⋆ ⋆ denotes complex conjugatio n of ⋆ ⋆ ⋆ ; (6) for some δ ∗ > 0 a nd sufficiently small, O δ ∗ ( p ) denotes the δ ∗ -neighborho o d of the p o int p , that is, O δ ∗ ( p ) := { z ∈ C : | z − p | < δ ∗ } . 3.2 WKB Analysis This subse ction is devoted to the WKB analysis of Equation (3.1) as τ → + ∞ (and ǫb > 0). In order to tr ansform Equation (3.1) int o a for m a mena ble to WKB analysis, one uses the result of Prop ositio n 4.1.1 in [1], which is summariz ed here for the reader’s co n venience. Prop ositio n 3.2. 1 ([1]) . L et A ( τ ) = a ( τ ) τ − 2 / 3 , B ( τ ) = b ( τ ) τ − 2 / 3 , C ( τ ) = c ( τ ) τ − 1 / 3 , D ( τ ) = d ( τ ) τ − 1 / 3 , e µ = µτ 1 / 6 , e Ψ( e µ ) := τ − (1 / 12) σ 3 Ψ( e µτ − 1 / 6 ) , (3.5) wher e e Ψ( e µ ) = e Ψ( e µ, τ ) . Then ∂ e µ e Ψ( e µ ) = τ 2 / 3 A ( e µ, τ ) e Ψ( e µ ) , (3.6) wher e A ( e µ, τ ) := − 2i e µσ 3 + 0 − 4i √ − a ( τ ) b ( τ ) b ( τ ) − 2 d ( τ ) 0 ! − i r ( τ )( ǫb ) 1 / 3 2 e µ σ 3 + 1 e µ 2 0 i ǫb b ( τ ) i b ( τ ) 0 , (3.7) with i r ( τ )( ǫb ) 1 / 3 2 = i a + 1 2 τ − 2 / 3 + 2 a ( τ ) d ( τ ) p − a ( τ ) b ( τ ) . (3.8) Degenerate Thir d Painlev ´ e Equation: II 11 It is now imp orta n t to state under what co nditions the WKB analysis of E quation (3.6 ) is under- taken. Define the functions ˆ r 0 ( τ ), ˆ u 0 ( τ ), and h 0 ( τ ) via the following equations 7 : p − a ( τ ) b ( τ ) + c ( τ ) d ( τ ) + a ( τ ) d ( τ ) τ − 2 / 3 2 p − a ( τ ) b ( τ ) − 1 4 a − i 2 2 τ − 4 / 3 = 3 4 ( ǫb ) 2 / 3 − h 0 ( τ ) τ − 2 / 3 , (3.9) r ( τ ) = − 2 + ˆ r 0 ( τ ) , (3.10) p − a ( τ ) b ( τ ) = ( ǫb ) 2 / 3 2 (1 + ˆ u 0 ( τ )) , (3.11) and assume that the functions ˆ r 0 ( τ ), ˆ u 0 ( τ ), and h 0 ( τ ), which are holomo rphic in the domain e D u , satisfy the c o nditions O ( τ − 1 3 + δ 1 ) 6 τ → + ∞ | ˆ r 0 ( τ ) | 6 τ → + ∞ O ( τ δ ) , O ( τ − 1 3 + δ 1 ) 6 τ → + ∞ | ˆ u 0 ( τ ) | 6 τ → + ∞ O ( τ 2 δ ) , 1 | 1 + ˆ u 0 ( τ ) | 6 τ → + ∞ O ( τ δ ) , | h 0 ( τ ) | 6 τ → + ∞ O ( τ δ ) , 0 < δ 1 < 1 3 , 0 6 δ < 1 39 , (3.12) where the parameter δ is the one that was introduced in the definition of the cheese-like domain e D u (cf. Equatio ns (3.2), (3.3), a nd (3.4)). Remark 3.2.1 . E ven though, at this juncture, the upp er bound 1 / 39 for the growth expo nent , δ , given in c o nditions (3.12) might seem artificial, it must b e noted that it a rises na turally during the course of the ensuing as y mptotic analysis. It is also assumed that the functions a ( τ ), b ( τ ), c ( τ ), and d ( τ ) are rela ted v ia the “integral of motion” asso ciated with the underlying Hamiltonian structure of E quation (1 .1) (see [1], Lemma 2.1): a ( τ ) d ( τ ) + b ( τ ) c ( τ ) + i a p − a ( τ ) b ( τ ) τ − 2 / 3 = − i ǫb 2 , ǫ = ± 1 . (3.13) Remark 3.2. 2. I t is worth noting that E quations (3.9) – (3 .11) and (3.1 3) are self-consistent; in fact, they ar e equiv alent to a ( τ ) d ( τ ) = ( ǫb ) 2 / 3 2 (1 + ˆ u 0 ( τ )) − i( ǫb ) 1 / 3 2 + i( ǫb ) 1 / 3 ˆ r 0 ( τ ) 4 − i 2 a − i 2 τ − 2 / 3 , (3.14) b ( τ ) c ( τ ) = ( ǫb ) 2 / 3 2 (1 + ˆ u 0 ( τ )) − i( ǫb ) 1 / 3 2 + i( ǫb ) 1 / 3 ˆ u 0 ( τ ) 1 + ˆ u 0 ( τ ) − ˆ r 0 ( τ ) 4 − i 2 a + i 2 τ − 2 / 3 , (3.15) − h 0 ( τ ) τ − 2 / 3 = ( ǫb ) 2 / 3 2 ( ˆ u 0 ( τ )) 2 + 1 2 ˆ u 0 ( τ ) ˆ r 0 ( τ ) 1 + ˆ u 0 ( τ ) − ( ˆ r 0 ( τ )) 2 8 + ( ǫb ) 1 / 3 2 a − i 2 τ − 2 / 3 (1 + ˆ u 0 ( τ )) ; (3.16) furthermore, via Equations (3.11) , (3.14) , and (3.15), c ( τ ) d ( τ ) = − i( ǫb ) 1 / 3 2 − i( ǫb ) 1 / 3 ˆ r 0 ( τ ) 4 + i 2 a − i 2 τ − 2 / 3 × i( ǫb ) 1 / 3 2 − i( ǫb ) 1 / 3 ˆ u 0 ( τ ) 1 + ˆ u 0 ( τ ) − ˆ r 0 ( τ ) 4 + i 2 a + i 2 τ − 2 / 3 . (3.17) In cer tain domains of the complex e µ -plane (see the discussion below), the leading term of a s ymp- totics (as τ → + ∞ ) of a fundamental so lution, e Ψ( e µ ), of Equation (3.6) is given by the follo wing WKB formula 8 (see, for exa mple, Chapter 5 of [9]), e Ψ WKB ( e µ ) = T ( e µ ) exp − σ 3 i τ 2 / 3 Z e µ l ( ξ ) d ξ − Z e µ diag(( T ( ξ )) − 1 ∂ ξ T ( ξ )) d ξ ! , (3.18) 7 Equations (63)–(66) in [1] are expressed in terms of the functions r 0 ( τ ) and u 0 ( τ ), whereas Equations (3.9) –(3.11) and conditions (3.12) are expressed in terms of th e functions ˆ r 0 ( τ ) := r 0 ( τ ) τ − 1 / 3 and ˆ u 0 ( τ ) := u 0 ( τ ) τ − 1 / 3 . 8 F or si mplicity of notation, the τ dependencies of e Ψ WKB ( e µ ), l ( e µ ), and T ( e µ ) are suppressed. Degenerate Thir d Painlev ´ e Equation: II 12 where l ( e µ ) := p det( A ( e µ )) , (3.19) and T ( e µ ), which dia g onalizes A ( e µ ), that is, ( T ( e µ )) − 1 A ( e µ ) T ( e µ ) = − i l ( e µ ) σ 3 , is g iven by T ( e µ ) = i p 2i l ( e µ )( A 11 ( e µ ) − i l ( e µ )) ( A ( e µ ) − i l ( e µ ) σ 3 ) σ 3 , (3.20) where A 11 ( e µ ) is the (1 1)-ele ment of the matrix A ( e µ ) (cf. Equa tio n (3.7)). Prop ositio n 3.2.2. L et T ( e µ ) b e given in Equation (3.20) , with A ( e µ ) and l ( e µ ) define d by Equa- tions (3.7) and (3.19 ) , re sp e ctively. Then det( T ( e µ )) = 1 , and tr(( T ( e µ )) − 1 ∂ e µ T ( e µ )) = 0; furthermor e, diag(( T ( e µ )) − 1 ∂ e µ T ( e µ )) = − 1 2 A 12 ( e µ ) ∂ e µ A 21 ( e µ ) − A 21 ( e µ ) ∂ e µ A 12 ( e µ ) 2i l ( e µ ) A 11 ( e µ ) + 2 l 2 ( e µ ) σ 3 . (3.21) Pr o of . Set T ( e µ ) = T 11 ( e µ ) T 12 ( e µ ) T 21 ( e µ ) T 22 ( e µ ) . F rom the form ula for T ( e µ ) giv en in Equation (3.20), with A ( e µ ) and l ( e µ ) defined by Equations (3.7) and (3.19), res pec tively , one s hows at T 11 ( e µ ) = T 22 ( e µ ) = i( A 11 ( e µ ) − i l ( e µ )) p 2i l ( µ )( A 11 ( e µ ) − i l ( e µ )) , T 12 ( e µ ) = − i A 12 ( e µ ) p 2i l ( µ )( A 11 ( e µ ) − i l ( e µ )) , T 21 ( e µ ) = i A 21 ( e µ ) p 2i l ( µ )( A 11 ( e µ ) − i l ( e µ )) , (3.22) whence (as det( A ( e µ )) = l 2 ( e µ ) and tr ( A ( e µ )) = 0 ) det( T ( e µ )) = ( T 11 ( e µ )) 2 − T 12 ( e µ ) T 21 ( e µ ) = 1. Since det( T ( e µ )) = 1, it follows that ( T ( e µ )) − 1 ∂ e µ T ( e µ ) = T 11 ( e µ ) ∂ e µ T 11 ( e µ ) − T 12 ( e µ ) ∂ e µ T 21 ( e µ ) T 11 ( e µ ) ∂ e µ T 12 ( e µ ) − T 12 ( e µ ) ∂ e µ T 11 ( e µ ) T 11 ( e µ ) ∂ e µ T 21 ( e µ ) − T 21 ( e µ ) ∂ e µ T 11 ( e µ ) T 11 ( e µ ) ∂ e µ T 11 ( e µ ) − T 21 ( e µ ) ∂ e µ T 12 ( e µ ) , whence diag(( T ( e µ )) − 1 ∂ e µ T ( e µ )) = 1 2 ∂ e µ ( T 11 ( e µ )) 2 − T 12 ( e µ ) ∂ e µ T 21 ( e µ ) 0 0 1 2 ∂ e µ ( T 11 ( e µ )) 2 − T 21 ( e µ ) ∂ e µ T 12 ( e µ ) , which implies that tr(( T ( e µ )) − 1 ∂ e µ T ( e µ )) = ∂ e µ (( T 11 ( e µ )) 2 − T 12 ( e µ ) T 21 ( e µ )) = ∂ e µ (1) = 0. W riting out explic- itly diag (( T ( e µ )) − 1 ∂ e µ T ( e µ )), o ne shows that (( T ( e µ )) − 1 ∂ e µ T ( e µ )) 11 = i 4 ∂ e µ A 11 ( e µ ) l ( e µ ) − A 11 ( e µ ) ∂ e µ l ( e µ ) l 2 ( e µ ) − A 12 ( e µ ) ∂ e µ A 21 ( e µ ) (2i l ( e µ ) A 11 ( e µ ) + 2 l 2 ( e µ )) + A 12 ( e µ ) A 21 ( e µ ) (i( l ( e µ ) ∂ e µ A 11 ( e µ ) + A 11 ( e µ ) ∂ e µ l ( e µ )) + 2 l ( e µ ) ∂ e µ l ( e µ )) (2i l ( e µ ) A 11 ( e µ ) + 2 l 2 ( e µ )) 2 , (( T ( e µ )) − 1 ∂ e µ T ( e µ )) 22 = i 4 ∂ e µ A 11 ( e µ ) l ( e µ ) − A 11 ( e µ ) ∂ e µ l ( e µ ) l 2 ( e µ ) − A 21 ( e µ ) ∂ e µ A 12 ( e µ ) (2i l ( e µ ) A 11 ( e µ ) + 2 l 2 ( e µ )) + A 12 ( e µ ) A 21 ( e µ ) (i( l ( e µ ) ∂ e µ A 11 ( e µ ) + A 11 ( e µ ) ∂ e µ l ( e µ )) + 2 l ( e µ ) ∂ e µ l ( e µ )) (2i l ( e µ ) A 11 ( e µ ) + 2 l 2 ( e µ )) 2 : now, using the fact that tr(( T ( e µ )) − 1 ∂ e µ T ( e µ )) = 0, one ar rives at Equation (3.21). Corollary 3.2.1. L et e Ψ WKB ( e µ ) b e given in Equation (3 .18) , with l ( e µ ) define d by Equation (3.19) and T ( e µ ) given in Equation (3 .20) . Then det( e Ψ WKB ( e µ )) = 1 . The domains in the complex e µ -plane where E quation (3.18) gives the a symptotic a pproximation of so lutio ns to Equation (3.6 ) ar e defined in ter ms of the S tokes gr aph (see, for example, [8, 9]). The vertices of the Sto kes gr aph are the sing ular p oints of Equa tion (3.6), that is, e µ = 0 a nd e µ = ∞ , and the turning p oints , which ar e the ro o ts of the equation l 2 ( e µ ) = 0. The edges of the Sto kes g raph are the Stokes curves , defined as Re( ∫ e µ e µ TP l ( ξ ) d ξ ) = 0, where e µ TP denotes a turning po int . Canonic al domains ar e thos e domains in the complex e µ -pla ne containing one, and only o ne, Stokes c urve and bo unded b y t wo adjacen t Stok es cur ves. Note that the restric tio n of any br anch of l ( e µ ) to a canonic a l Degenerate Thir d Painlev ´ e Equation: II 13 domain is a single-v alued function. In each ca nonical domain, for any c hoice of the branch o f l ( e µ ), there exists a fundamental solution of Equation (3.6) which has a symptotics whose leading term is given b y Equation (3.18) . F rom the definition of l ( e µ ) given b y Equation (3.19), o ne ar rives at l 2 ( e µ ) = 4 e µ 4 e µ 2 − α 2 2 e µ 2 + 2 α 2 + e µ 4 a − i 2 + e µ 2 h 0 ( τ ) τ − 2 / 3 , (3.23) where α := ( ǫb ) 1 / 6 √ 2 > 0 . It follows from E quation (3.23) that there a re six turning p oints: tw o turning p oints c oalesce (as τ → + ∞ ) at α + O ( τ − 1 3 + δ 2 ), another pair c o alesce at − α + O ( τ − 1 3 + δ 2 ), a nd the remaining t wo turning po int s approach (as τ → + ∞ ) ± i √ 2 α + O ( τ − 2 3 + δ ), respec tively . Deno te by e µ 1 the turning p oint in the first quadrant of the c o mplex e µ -plane whic h approaches α + O ( τ − 1 3 + δ 2 ), and by e µ 2 the pure imag inary turning p oint which approaches i √ 2 α + O ( τ − 2 3 + δ ). Deno te by G 1 the part of the Stokes graph in the first quadrant of the complex e µ -plane which consists of the vertices 0, ∞ , e µ 1 , and e µ 2 , and the edges (+i ∞ , e µ 2 ), (0 , e µ 1 ), ( e µ 2 , e µ 1 ), and ( e µ 1 , + ∞ ): the complete Stokes graph is the union of the mirr or images of G 1 with r esp ect to the r eal and imag inary axes in the complex e µ -plane. Prop ositio n 3.2. 3 . L et l 2 ( e µ ) b e given in Equation (3.23) . Then Z e µ e µ 0 l ( ξ ) d ξ = τ → + ∞ Υ τ ( e µ ) − Υ τ ( e µ 0 ) + O ( E l ( e µ )) , (3.24) wher e e µ 0 ∈ C \ ( O τ − 1 3 + δ 2 ( ± α ) ∪ O τ − 2 3 + δ ( ± i √ 2 α ) ∪ { 0 , ∞} ) and the p ath of inte gr ation lie s in the c orr esp onding 9 c anonic al domain, Υ τ ( ξ ) := ξ + 2 α 2 ξ p ξ 2 + 2 α 2 + τ − 2 / 3 a − i 2 ln ξ + p ξ 2 + 2 α 2 + τ − 2 / 3 2 √ 3 a − i 2 + h 0 ( τ ) α 2 ln √ 3 p ξ 2 + 2 α 2 − ξ + 2 α √ 3 p ξ 2 + 2 α 2 + ξ + 2 α ! ξ − α ξ + α ! , (3.25) and E l ( e µ ) = E ♮ l ( e µ ) − E ♮ l ( e µ 0 ) , with E ♮ l ( ξ ) := ( c 1 ( δ 0 )( h 0 ( τ )) 2 + c 2 ( δ 0 )) τ − 4 / 3 ( ξ ∓ α ) 2 , ξ ∈ O δ 0 ( ± α ) , ( c 3 ( δ 0 )( h 0 ( τ )) 2 + c 4 ( δ 0 )) τ − 4 / 3 ( ξ ∓ i √ 2 α ) 1 / 2 , ξ ∈ O δ 0 ( ± i √ 2 α ) , ( c 5 ( δ 0 )( h 0 ( τ )) 2 + c 6 ( δ 0 )) τ − 4 / 3 , ξ ∈ O δ 0 (0) , ( c 7 ( δ 0 )( h 0 ( τ )) 2 + c 8 ( δ 0 )) τ − 4 / 3 , ξ ∈ O δ 0 ( ∞ ) , (3.26) wher e c k ( z ) , k = 1 , . . . , 8 , ar e holomorph ic functions of z in a neighb orho o d of z = 0 with c k (0) 6 = 0 , and δ 0 > 0 and sufficiently smal l. Pr o of . Reca ll the express ion for l 2 ( e µ ) (= l 2 ( e µ, τ )) given in Equation (3 .23). Set l 2 ∞ ( e µ ) := l 2 ( e µ, + ∞ ) = 4 e µ − 4 ( e µ 2 − α 2 ) 2 ( e µ 2 + 2 α 2 ) . (3.27) Define ∆ τ ( e µ ) := l 2 ( e µ ) − l 2 ∞ ( e µ ) l 2 ∞ ( e µ ) = e µ 2 ( h 0 ( τ ) + e µ 2 ( a − i 2 )) τ − 2 / 3 ( e µ 2 − α 2 ) 2 ( e µ 2 + 2 α 2 ) . (3.28) Now, via conditions (3.12), a nd writing l ( e µ ) = l ∞ ( e µ )(1 + ∆ τ ( e µ )) 1 / 2 = τ → + ∞ l ∞ ( e µ ) 1 + 1 2 ∆ τ ( e µ ) + O ((∆ τ ( e µ )) 2 ) , (3.29) 9 The same canonical domain to whic h e µ 0 belongs. Degenerate Thir d Painlev ´ e Equation: II 14 one arr ives at l ( e µ ) = τ → + ∞ 2 1 − α 2 e µ 2 p e µ 2 + 2 α 2 + ( h 0 ( τ ) + e µ 2 ( a − i 2 )) τ − 2 / 3 ( e µ 2 − α 2 ) p e µ 2 + 2 α 2 + O e µ 2 ( h 0 ( τ ) + e µ 2 ( a − i 2 )) 2 τ − 4 / 3 ( e µ 2 − α 2 ) 3 ( e µ 2 + 2 α 2 ) 3 / 2 ! . (3.30) In or de r to o btain the leading term in Equation (3 .24), one in tegrates the fir st tw o terms in E qua- tion (3.30). Analogously , integrating the err o r ter m in Equatio n (3 .30), one finds an explicit expressio n for the function E ♮ l ( e µ ): since this la tter ex pression is quite cumber some, only its asymptotics at the singular and turning p oints are presented in Equation (3.2 6). Corollary 3. 2.2. Set e µ 0 = α + τ − 1 / 3 e Λ , wher e e Λ = τ → + ∞ O ( τ ε ) , 0 < δ < ε < 1 / 9 . Then Z e µ e µ 0 l ( ξ ) d ξ = τ → + ∞ Υ τ ( e µ ) + Υ ♯ τ + O ( E ♮ l ( e µ )) + O ( τ − 2 3 − 2( ε − δ ) ) + O ( τ − 1+3 ε ) , (3.31) wher e Υ τ ( e µ ) and E ♮ l ( e µ ) ar e define d in Pr op osition 3.2.3 , Υ ♯ τ := ∓ 3 √ 3 α 2 ∓ 2 √ 3 τ − 2 / 3 e Λ 2 − τ − 2 / 3 a − i 2 ln ( √ 3 ± 1) α e i π s ( ± ) ∓ τ − 2 / 3 2 √ 3 a − i 2 + h 0 ( τ ) α 2 ln e Λ − 1 3 ln τ − ln(3 α ) , (3 .32) and s ( ± ) = (1 ∓ 1 ) / 2 , with the upp er (r esp., lower) signs taken if the p ositive (+) (r esp., ne gative ( − )) br anch of t he squar e-ro ot function p ξ 2 + 2 α 2 is chosen. Pr o of . Substituting e µ 0 as the arg ument of the functions Υ τ ( ξ ) and E ♮ l ( ξ ) (cf. Equation (3.25) and the first line of Equation (3.26), r esp ectively) and e x panding with resp ect to (the sma ll par ameter) τ − 1 / 3 e Λ, one ar rives a t the following estimates: Υ τ ( e µ 0 ) = τ → + ∞ Υ ♯ τ + O ( τ − 1 e Λ 3 ) and E ♮ l ( e µ 0 ) = τ → + ∞ O τ − 2 / 3 ( c 1 ( δ 0 )( h 0 ( τ )) 2 + c 2 ( δ 0 )) e Λ − 2 , where Υ ♯ τ is defined by Equation (3.32). The inequality δ < ε < 1 / 9 (see conditions (3.12) fo r the function h 0 ( τ )) is introduced in or der to guara nt ee that the last tw o error estimates in Equation (3 .3 1) a r e decaying after m ultiplication by the lar ge para meter τ 2 / 3 (cf. E quation (3.18)). Prop ositio n 3.2.4. L et T ( e µ ) b e given in Equation (3.20) , with A ( e µ ) define d by Equation (3.7) , and l 2 ( e µ ) given in Equation (3 .23) . Then Z e µ e µ 0 diag(( T ( ξ )) − 1 ∂ ξ T ( ξ )) d ξ = τ → + ∞ ( I τ ( e µ ) + O ( E T ( e µ ))) σ 3 , ( 3.33 ) wher e e µ 0 ∈ C \ ( O τ − 1 3 + δ 2 ( ± α ) ∪ O τ − 2 3 + δ ( ± i √ 2 α ) ∪ { 0 , ∞} ) and the p ath of inte gr ation lie s in the c orr esp onding c anonic al domain, I τ ( e µ ) = p τ 8i ( τ ( e µ ) − τ ( e µ 0 )) , (3.34) with p τ := 4i ˆ r 0 ( τ ) 2(1 + ˆ u 0 ( τ )) + ( − 2 + ˆ r 0 ( τ )) 1 + ˆ u 0 ( τ ) , (3.35) τ ( ξ ) := 1 √ 3 ln √ 3 p ξ 2 + 2 α 2 − ξ + 2 α √ 3 p ξ 2 + 2 α 2 + ξ + 2 α ! ξ − α ξ + α ! + 4 p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 × arctanh 4 p ξ 2 + 2 α 2 ξ ( ˆ r 0 ( τ ) + p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ) ! − arctanh 4 p ξ 2 + 2 α 2 ξ ( ˆ r 0 ( τ ) − p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ) ! + arcta nh 8( − 2 + ˆ r 0 ( τ )) ξ 2 + α 2 ( − 2 + ˆ r 0 ( τ )) 2 + 12 α 2 α 2 ˆ r 0 ( τ ) p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ! ! , (3.36) Degenerate Thir d Painlev ´ e Equation: II 15 and E T ( e µ ) = E ♮ T ( e µ ) − E ♮ T ( e µ 0 ) , wher e E ♮ T ( ξ ) := τ − 2 / 3 ln( ξ ∓ α ) (1+ ˆ u 0 ( τ ))( C 1 ( τ )+ˆ r 0 ( τ )) + ( ˆ u 0 ( τ )+ ˆ u 0 ( τ )+ 1 2 ˆ r 0 ( τ ) 1+ ˆ u 0 ( τ ) )( C 2 ( τ )+ h 0 ( τ )) ˆ r 0 ( τ )( ξ ∓ α ) 2 τ 2 / 3 , | ξ ∓ α | < τ − 1 3 + δ 1 < | ˆ r 0 ( τ ) | , τ − 2 / 3 (1+ ˆ u 0 ( τ ))ˆ r 0 ( τ ) + ( ˆ u 0 ( τ )+ ˆ u 0 ( τ )+ 1 2 ˆ r 0 ( τ ) 1+ ˆ u 0 ( τ ) )( C 3 ( τ )+ h 0 ( τ )) ( C 4 ( τ )+ˆ r 0 ( τ ))( ξ ∓ i √ 2 α ) 1 / 2 τ 2 / 3 , ˆ r 0 ( τ ) 6 = 6 , ξ ∈ O δ 0 ( ± i √ 2 α ) , τ − 2 / 3 (1+ ˆ u 0 ( τ ))ˆ r 0 ( τ ) + ( ˆ u 0 ( τ ) + ˆ u 0 ( τ )+ 1 2 ˆ r 0 ( τ ) 1+ ˆ u 0 ( τ ) ) ( C 5 ( τ ) ˆ r 0 ( τ ) + C 6 ( τ ) ( ˆ r 0 ( τ )) 3 )( C 7 ( τ )+ h 0 ( τ )) τ 2 / 3 , ξ ∈ O δ 0 (0) , τ − 2 / 3 (1+ ˆ u 0 ( τ ))ˆ r 0 ( τ ) + ( ˆ u 0 ( τ ) + ˆ u 0 ( τ )+ 1 2 ˆ r 0 ( τ ) 1+ ˆ u 0 ( τ ) ) ( C 8 ( τ ) ˆ r 0 ( τ ) + C 9 ( τ ) ( ˆ r 0 ( τ )) 3 )( C 10 ( τ )+ h 0 ( τ )) τ 2 / 3 , ξ ∈ O δ 0 ( ∞ ) , (3.37) with 10 C j ( τ ) = τ → + ∞ O (1 ) , j = 1 , . . . , 10 . Pr o of . F rom Equation (3.7) a nd E quations (3.27)–(3.29), one shows that, via E quation (3.1 0) (cf. Equation (3.21)), 2i l ( ξ ) A 11 ( ξ ) + 2 l 2 ( ξ ) = τ → + ∞ P ∞ ( ξ ) + P 1 ( ξ )∆ τ ( ξ ) + O ξ − 1 l ∞ ( ξ ) ˆ r 0 ( τ )(∆ τ ( ξ )) 2 , (3 .38) where P ∞ ( ξ ) := 2 l 2 ∞ ( ξ ) + 2 l ∞ ( ξ ) 2 ξ + α 2 ( − 2 + ˆ r 0 ( τ )) ξ , (3.39) P 1 ( ξ ) := 2 l 2 ∞ ( ξ ) + l ∞ ( ξ ) 2 ξ + α 2 ( − 2 + ˆ r 0 ( τ )) ξ , (3.40) and, via Eq uations (3.7), (3.1 1), and (3.14) (cf. Eq uation (3.21)), A 12 ( ξ ) ∂ ξ A 21 ( ξ ) − A 21 ( ξ ) ∂ ξ A 12 ( ξ ) = τ → + ∞ 2i α 4 ˆ r 0 ( τ ) p τ ξ 3 + O τ − 2 / 3 ξ 3 (1 + ˆ u 0 ( τ )) , (3.41) where p τ is defined by Eq uation (3.3 5). Substitut ing Equatio ns (3.38) and (3.4 1) into Equation (3.21) and expanding (2i l ( ξ ) A 11 ( ξ ) + 2 l 2 ( ξ )) − 1 int o a series of powers of ∆ τ ( ξ ), one ar rives at Z e µ e µ 0 diag(( T ( ξ )) − 1 ∂ ξ T ( ξ )) d ξ = τ → + ∞ ( I τ ( e µ ) + O ( E T ( e µ ))) σ 3 , where I τ ( e µ ) := − i α 4 ˆ r 0 ( τ ) p τ Z e µ e µ 0 1 ξ 3 P ∞ ( ξ ) d ξ , (3.42) and E T ( e µ ) := i α 4 ˆ r 0 ( τ ) p τ Z e µ e µ 0 P 1 ( ξ )∆ τ ( ξ ) ξ 3 ( P ∞ ( ξ )) 2 d ξ + τ − 2 / 3 (1 + ˆ u 0 ( τ )) Z e µ e µ 0 1 ξ 3 P ∞ ( ξ ) d ξ . (3.43) Via Equation (3 .27), it follows from E quation (3.39) that 1 ξ 3 P ∞ ( ξ ) = ξ ( ξ (4 ξ 2 + 2 α 2 ( − 2 + ˆ r 0 ( τ ))) − 4( ξ 2 − α 2 )( ξ 2 + 2 α 2 ) 1 / 2 ) 2( ξ 2 − α 2 )( ξ 2 + 2 α 2 ) 1 / 2 (( ξ (4 ξ 2 + 2 α 2 ( − 2 + ˆ r 0 ( τ )))) 2 − 16( ξ 2 − α 2 ) 2 ( ξ 2 + 2 α 2 )) : (3 .44) writing the fa ctorization ( ξ (4 ξ 2 + 2 α 2 ( − 2 + ˆ r 0 ( τ )))) 2 − 16( ξ 2 − α 2 ) 2 ( ξ 2 + 2 α 2 ) = 16 α 2 ( − 2 + ˆ r 0 ( τ )) ξ 4 + ξ 2 α 2 ( − 2 + ˆ r 0 ( τ )) 3 + − 2 + ˆ r 0 ( τ ) 2 2 ! − 2 α 4 ( − 2 + ˆ r 0 ( τ )) ! = 16 α 2 ( − 2 + ˆ r 0 ( τ ))( ξ 2 + ˆ z + )( ξ 2 + ˆ z − ) , where ˆ z ± := α 2 2( − 2 + ˆ r 0 ( τ )) 3 + − 2 + ˆ r 0 ( τ ) 2 2 ∓ ˆ r 0 ( τ ) 4 p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ! , (3.45) 10 In case ˆ r 0 ( τ ) → 6, the factor ( ξ ∓ i √ 2 α ) 1 / 2 which app ears in the s econd line of E quation (3.37) should be changed to ξ ∓ i √ 2 α . Degenerate Thir d Painlev ´ e Equation: II 16 one shows that, for ˆ z + 6≡ ˆ z − , I τ ( e µ ) = − i α 2 ˆ r 0 ( τ ) p τ 32( − 2 + ˆ r 0 ( τ )) Z e µ e µ 0 ξ 2 (4 ξ 2 + 2 α 2 ( − 2 + ˆ r 0 ( τ ))) p ξ 2 + 2 α 2 ( ξ 2 − α 2 )( ξ 2 + 2 α 2 )( ξ 2 + ˆ z + )( ξ 2 + ˆ z − ) d ξ − 4 Z e µ e µ 0 ξ ( ξ 2 + ˆ z + )( ξ 2 + ˆ z − ) d ξ ! . (3.46) W rite the partial fraction de c omp o sition ξ 2 (4 ξ 2 + 2 α 2 ( − 2 + ˆ r 0 ( τ ))) ( ξ 2 − α 2 )( ξ 2 + 2 α 2 )( ξ 2 + ˆ z + )( ξ 2 + ˆ z − ) = A 0 ( τ ) ξ − α + B 0 ( τ ) ξ + α + C 0 ( τ ) ξ + D 0 ( τ ) ξ 2 + 2 α 2 + E 0 ( τ ) ξ + F 0 ( τ ) ξ 2 + ˆ z + + G 0 ( τ ) ξ + H 0 ( τ ) ξ 2 + ˆ z − : (3.47) to determine A 0 ( τ ) , . . . , H 0 ( τ ), one notes that the left -hand side of Equation (3 .47) is symmetric ( ξ → − ξ ), w he nce it follows that A 0 ( τ ) = − B 0 ( τ ) , C 0 ( τ ) = E 0 ( τ ) = G 0 ( τ ) = 0 , (3.48) and to determine the remaining co efficients, that is, A 0 ( τ ), D 0 ( τ ), F 0 ( τ ), and H 0 ( τ ), one compares residues (at ξ = α , ξ 2 = − 2 α 2 , and ξ 2 = − ˆ z ± ) o n the left- and right-hand sides of Eq uation (3.47) a nd uses the r elations (cf. Equation (3.45)) ˆ z + + ˆ z − = α 2 ( − 2 + ˆ r 0 ( τ )) 3 + − 2 + ˆ r 0 ( τ ) 2 2 ! and ˆ z + ˆ z − = − 2 α 4 ( − 2 + ˆ r 0 ( τ )) to show that A 0 ( τ ) = α ˆ r 0 ( τ ) 3( α 2 + ˆ z + )( α 2 + ˆ z − ) = 4( − 2 + ˆ r 0 ( τ )) 3 α 3 ˆ r 0 ( τ ) , D 0 ( τ ) = 4 α 2 ˆ r 0 ( τ ) − 2 4 α 2 3( − 2 α 2 + ˆ z − )( − 2 α 2 + ˆ z + ) = 8( − 2 + ˆ r 0 ( τ )) 3 α 2 (6 − ˆ r 0 ( τ )) , F 0 ( τ ) = 2 α 2 ˆ z + ( − 2 + ˆ r 0 ( τ )) − 4( ˆ z + ) 2 ( α 2 + ˆ z + )( − 2 α 2 + ˆ z + )( ˆ z + − ˆ z − ) , H 0 ( τ ) = 4( ˆ z − ) 2 − 2 α 2 ˆ z − ( − 2 + ˆ r 0 ( τ )) ( α 2 + ˆ z − )( − 2 α 2 + ˆ z − )( ˆ z + − ˆ z − ) ⇒ F 0 ( τ ) + H 0 ( τ ) = − 16( − 2 + ˆ r 0 ( τ )) α 2 ˆ r 0 ( τ )(6 − ˆ r 0 ( τ )) . (3.49) (Note: it is the quantit y F 0 ( τ ) + H 0 ( τ ), and no t F 0 ( τ ) a nd H 0 ( τ ) indiv idua lly , that is r equisite for the ensuing calculation.) Substituting Equatio ns (3.49), (3.4 8), and (3.4 7) into E quation (3.4 6), one arrives at, after an integration argument and neglecting ξ - independent terms (that is, with abuse of nomenclature, “c o nstants of integration”), I τ ( e µ ) = − i α 2 ˆ r 0 ( τ ) p τ 32( − 2 + ˆ r 0 ( τ )) 2 αA 0 ( τ ) + D 0 ( τ ) + 8( − 2 + ˆ r 0 ( τ )) α 2 ˆ r 0 ( τ )( ˆ r 0 ( τ ) − 6 ) ln( p ξ 2 + 2 α 2 + ξ ) − 8( − 2 + ˆ r 0 ( τ )) α 2 ˆ r 0 ( τ )( ˆ r 0 ( τ ) − 6 ) ln( p ξ 2 + 2 α 2 − ξ ) + √ 3 αA 0 ( τ ) ln √ 3 p ξ 2 + 2 α 2 − ξ + 2 α √ 3 p ξ 2 + 2 α 2 + ξ + 2 α ! ξ − α ξ + α ! + 16( − 2 + ˆ r 0 ( τ )) α 2 ˆ r 0 ( τ ) p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 arctanh 8( − 2 + ˆ r 0 ( τ )) ξ 2 + α 2 ( − 2 + ˆ r 0 ( τ )) 2 + 12 α 2 α 2 ˆ r 0 ( τ ) p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ! + arctanh 4 p ξ 2 + 2 α 2 ξ ( ˆ r 0 ( τ ) + p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ) ! − arctanh 4 p ξ 2 + 2 α 2 ξ ( ˆ r 0 ( τ ) − p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ) ! ! ! e µ e µ 0 . (3.50) F rom E q uations (3.49), one no tes that 2 αA 0 ( τ ) + D 0 ( τ ) + 16( − 2 + ˆ r 0 ( τ )) α 2 ˆ r 0 ( τ )( ˆ r 0 ( τ ) − 6 ) = 0; Degenerate Thir d Painlev ´ e Equation: II 17 whence (cf. Equation (3.50)) 2 αA 0 ( τ ) + D 0 ( τ ) + 8( − 2 + ˆ r 0 ( τ )) α 2 ˆ r 0 ( τ )( ˆ r 0 ( τ ) − 6 ) ln( p ξ 2 + 2 α 2 + ξ ) − 8( − 2 + ˆ r 0 ( τ )) α 2 ˆ r 0 ( τ )( ˆ r 0 ( τ ) − 6 ) ln( p ξ 2 + 2 α 2 − ξ ) = − 8( − 2 + ˆ r 0 ( τ )) α 2 ˆ r 0 ( τ )( ˆ r 0 ( τ ) − 6 ) ln(2 α 2 ) , which is a ξ -indep endent term (a “constant of integration”): neglecting this term, o ne arr ives a t (a fter simplification) Equatio ns (3.34)–(3.36). One now studies the er ror term O ( E T ( e µ )) defined by Equatio n (3.43). Recall the expressio n for ( ξ 3 P ∞ ( ξ )) − 1 given in Equation (3.44): fr o m this latter expressio n, and those for ∆ τ ( ξ ) (cf. E q ua- tion (3.28)) and P 1 ( ξ ) (cf. Eq ua tion (3.40)), one shows that P 1 ( ξ )∆ τ ( ξ ) ξ 3 ( P ∞ ( ξ )) 2 = ξ 2 ( h 0 ( τ ) + ξ 2 ( a − i 2 ))(2 ξ l ∞ ( ξ ) + 2 ξ 2 + α 2 ( − 2 + ˆ r 0 ( τ ))) τ − 2 / 3 8( ξ 2 − α 2 ) 3 ( ξ 2 + 2 α 2 ) 3 / 2 ( ξ l ∞ ( ξ ) + 2 ξ 2 + α 2 ( − 2 + ˆ r 0 ( τ ))) 2 . (3.51) Using Equation (3.51), one ev a luates the first integral in Equation (3.43) explicitly (as do ne ab ov e, for the second in tegra l in Equatio n (3.43)): the resulting express io n for E T ( e µ ) is quite cumberso me; therefore, only its a symptotics a t the singular and turning points a re defined b y Equation (3.37). Corollary 3.2.3. L et e µ 0 b e define d as in Cor ol lary 3.2.2 , that is, e µ 0 = α + τ − 1 / 3 e Λ , wher e e Λ = τ → + ∞ O ( τ ε ) , and δ, δ 1 b e define d in c onditions (3.1 2) . Then, for 0 < δ < ε < 1 / 9 and ε < δ 1 < 1 / 3 , Z e µ e µ 0 diag(( T ( ξ )) − 1 ∂ ξ T ( ξ )) d ξ = τ → + ∞ p τ 8i ( τ ( e µ ) − ♯ τ ) + O ( E ♮ T ( e µ )) + O ( τ − 2( ε − δ ) ) + O ( τ − 1 3 + ε + δ ) σ 3 , (3.52) wher e p τ , τ ( e µ ) , and E ♮ T ( e µ ) ar e define d by Equations (3.3 5) , (3.36) , and (3.3 7) , re sp e ctively, and ♯ τ := ∓ 1 √ 3 ln(3 α ) + 2 p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ln ˆ r 0 ( τ ) + 4 + p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 − ˆ r 0 ( τ ) − 4 + p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ! ∓ 2 p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ln ˆ r 0 ( τ ) + 4 √ 3 − p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ˆ r 0 ( τ ) + 4 √ 3 + p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ! × ˆ r 0 ( τ ) − 4 √ 3 + p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ˆ r 0 ( τ ) − 4 √ 3 − p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ! ! ± 1 √ 3 − 1 3 ln τ + ln e Λ , (3.53) with the upp er (r esp., lower) signs taken if the p ositive (+) (r esp., ne gative ( − )) br anch of the squar e- r o ot function p ξ 2 + 2 α 2 is chosen. Pr o of . Substitut ing e µ 0 as the argument o f the functions τ ( ξ ) and E ♮ T ( ξ ) (cf. E q uation (3.3 6) and the first line of Equation (3.37), r esp ectively) and e x panding with resp ect to (the sma ll par ameter) τ − 1 / 3 e Λ, one ar rives a t the following estimates: τ ( e µ 0 ) = τ → + ∞ ♯ τ + O ( τ − 1 / 3 e Λ) , and E ♮ T ( e µ 0 ) = τ → + ∞ O τ − 2 / 3 ln τ (1 + ˆ u 0 ( τ ))( C 1 ( τ ) + ˆ r 0 ( τ )) + O (2 ˆ u 0 ( τ ) + 1 2 ˆ r 0 ( τ ))( C 2 ( τ ) + h 0 ( τ )) ˆ r 0 ( τ ) e Λ 2 ! , where ♯ τ is defined by Eq ua tion (3.53), and C j ( τ ) = τ → + ∞ O (1 ), j = 1 , 2. Demanding that the a b ove error estimates are decaying, and using the definition of the par ameters δ, δ 1 given in conditions (3.1 2) for the functions ˆ r 0 ( τ ), ˆ u 0 ( τ ), and h 0 ( τ ), one a rrives at the inequa lities stated in the Coro llary . Corollary 3.2.4. Un der t he c onditions of Cor ol lary 3.2.2 , for the br anch of l ( ξ ) that is p ositive for lar ge and smal l p ositive ξ , − i τ 2 / 3 Z e µ e µ 0 l ( ξ ) d ξ = τ → + ∞ Re( e µ ) → + ∞ − i τ 2 / 3 e µ 2 + a − i 2 ln e µ + i τ 2 / 3 3( √ 3 − 1) α 2 + i2 √ 3 e Λ 2 Degenerate Thir d Painlev ´ e Equation: II 18 − i 2 √ 3 a − i 2 + h 0 ( τ ) α 2 1 3 ln τ − ln e Λ + C WKB ∞ + O ( τ − 2( ε − δ ) ) + O ( τ − 1 3 +3 ε ) , (3.54) wher e C WKB ∞ := i a − i 2 ln ( √ 3 + 1) α 2 ! − i 2 √ 3 a − i 2 + h 0 ( τ ) α 2 ln 6 α ( √ 3 + 1) 2 , (3.55) and − i τ 2 / 3 Z e µ e µ 0 l ( ξ ) d ξ = τ → + ∞ Re( e µ ) → +0 i τ 2 / 3 2 √ 2 α 3 e µ − i τ 2 / 3 3 √ 3 α 2 − i2 √ 3 e Λ 2 + i 2 √ 3 a − i 2 + h 0 ( τ ) α 2 × 1 3 ln τ − ln e Λ + C WKB 0 + O ( τ − 2( ε − δ ) ) + O ( τ − 1 3 +3 ε ) , (3.56) wher e C WKB 0 := − i a − i 2 ln √ 3 + 1 √ 2 ! + i 2 √ 3 a − i 2 + h 0 ( τ ) α 2 ln(3 α e − i π ) . (3.57) Pr o of . F ollows fr o m Corollar y 3.2 .2, E quation (3.31), by choo sing the corresp o nding branches in Equations (3.25) and (3.32) and taking the limits Re( e µ ) → + ∞ and Re( e µ ) → +0: the erro r estimate O ( E ♮ l ( ξ )) in Equation (3 .31) is defined b y Equation (3.26). Corollary 3.2.5. Un der t he c onditions of Cor ol lary 3.2.3 , for the br anch of l ( ξ ) that is p ositive for lar ge and smal l p ositive ξ , Z e µ e µ 0 diag(( T ( ξ )) − 1 ∂ ξ T ( ξ )) d ξ = τ → + ∞ Re( e µ ) → + ∞ − p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 8 √ 3 − 1 3 ln τ + ln e Λ + I ♯ ∞ ( τ ) + O ˆ r 0 ( τ ) e µ 2 + O ( τ δ − 2 δ 1 ln τ ) + O ( τ − 2( ε − δ ) ) + O ( τ − 2 3 +2 δ ) + O ( τ − 1 3 + ε + δ ) σ 3 , (3.58) wher e I ♯ ∞ ( τ ) := 1 4 ln ˆ r 0 ( τ ) + 4 √ 3 − p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ˆ r 0 ( τ ) + 4 √ 3 + p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ! ˆ r 0 ( τ ) − 4 √ 3 + p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ˆ r 0 ( τ ) − 4 √ 3 − p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ! ! + p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 8 √ 3 ln 6 α ( √ 3 + 1) 2 − 1 2 ln ˆ u 0 ( τ ) ˆ r 0 ( τ ) − 3 4 ln 2 − i π 4 , (3.59) and Z e µ e µ 0 diag(( T ( ξ )) − 1 ∂ ξ T ( ξ )) d ξ = τ → + ∞ Re( e µ ) → +0 p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 8 √ 3 − 1 3 ln τ + ln e Λ + I ♯ 0 ( τ ) + O ( ˆ r 0 ( τ ) e µ ) + O ( τ δ − 2 δ 1 ln τ ) + O ( τ − 2( ε − δ ) ) + O ( τ − 2 3 +2 δ ) + O ( τ − 1 3 + ε + δ ) + O ( τ − 1 3 + δ − δ 1 ) σ 3 , (3.6 0 ) wher e I ♯ 0 ( τ ) := − 1 4 ln ˆ r 0 ( τ ) + 4 √ 3 − p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ˆ r 0 ( τ ) + 4 √ 3 + p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ! ˆ r 0 ( τ ) − 4 √ 3 + p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ˆ r 0 ( τ ) − 4 √ 3 − p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ! ! − p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 8 √ 3 ln(3 α e − i π ) + 1 2 ln ˆ u 0 ( τ ) ˆ r 0 ( τ ) + 3 4 ln 2 + i π 4 . (3.61) Degenerate Thir d Painlev ´ e Equation: II 19 Pr o of . T aking the limit Re( e µ ) → + ∞ in Equation (3.52), with τ ( ξ ) and E ♮ T ( ξ ) defined by Equa - tions (3.36) a nd (3.37), res pec tively , and using conditions (3.12), one ar rives at τ ( e µ ) = τ → + ∞ Re( e µ ) → + ∞ 1 √ 3 ln √ 3 − 1 √ 3 + 1 ! − 2 p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ln ˆ r 0 ( τ ) − 4 + p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ˆ r 0 ( τ ) − 4 − p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ! × − ˆ r 0 ( τ ) − 4 + p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ˆ r 0 ( τ ) + 4 + p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ! ! + O ( e µ − 2 ) , (3.62) and E ♮ T ( e µ ) = τ → + ∞ Re( e µ ) → + ∞ O ( τ − 2 δ 1 ) + O ( τ − 2 3 +2 δ ) . (3.63) One now computes as ymptotics of p τ (cf. Equation (3.35)). Using conditions (3.1 2) and E quation (3.16), one s hows that − 2 + ˆ r 0 ( τ ) 1 + ˆ u 0 ( τ ) = τ → + ∞ ˆ r 0 ( τ )(4 − ˆ r 0 ( τ )) 4 + 2(1 + ˆ u 0 ( τ )) − 4 + O ( τ − 2 / 3 κ 2 0 ( τ )) , (3.64) where κ 2 0 ( τ ) := 4 α 2 h 0 ( τ ) + α 2 ( a − i 2 ) 1 + ˆ u 0 ( τ ) ! , (3.65) whence, substituting Equation (3.64) into Equation (3 .35), one obta ins p τ = τ → + ∞ 4i ˆ r 0 ( τ ) 4 ˆ u 0 ( τ ) + ˆ r 0 ( τ )(4 − ˆ r 0 ( τ )) 4 + O ( τ − 2 / 3 κ 2 0 ( τ )) . (3.66) Solving Equa tion (3.16) for ˆ u 0 ( τ ) and tak ing in to account co nditions (3.12), o ne deduces that 16 ˆ u 0 ( τ ) = τ → + ∞ − ˆ r 0 ( τ )(4 − ˆ r 0 ( τ )) + ˆ r 0 ( τ ) p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 1 + O τ − 2 / 3 κ 2 0 ( τ ) ( ˆ r 0 ( τ )) 2 . (3.67) Now, substituting Equation (3.6 7) in to Eq uation (3.66), one arrives at p τ = τ → + ∞ i p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 1 + O τ − 2 / 3 κ 2 0 ( τ ) ( ˆ r 0 ( τ )) 2 . (3.68) Substituting the expansions (3.62), (3.63), and (3.68) in to Equa tion (3.52) (with the upp er signs in the form ula for ♯ τ taken (cf. Equa tion (3.53))), a nd taking in to account conditions (3.12), one shows that Z e µ e µ 0 diag(( T ( ξ )) − 1 ∂ ξ T ( ξ )) d ξ = τ → + ∞ Re( e µ ) → + ∞ I ∞ ( τ ) + O ( τ δ − 2 δ 1 ln τ ) + O ( τ − 2( ε − δ ) ) + O ( τ − 1 3 + ε + δ ) + O ( τ − 2 3 +2 δ ) + O ( ˆ r 0 ( τ ) e µ − 2 ) σ 3 , (3.69) where I ∞ ( τ ) = − p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 8 √ 3 − 1 3 ln τ + ln e Λ + b I ∞ ( τ ) , (3.70) with b I ∞ ( τ ) := p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 8 √ 3 ln 6 α ( √ 3 + 1) 2 − 1 4 ln ˆ r 0 ( τ ) − 4 + p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ˆ r 0 ( τ ) − 4 − p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ! + 1 4 ln ˆ r 0 ( τ ) + 4 √ 3 − p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ˆ r 0 ( τ ) + 4 √ 3 + p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ! ˆ r 0 ( τ ) − 4 √ 3 + p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ˆ r 0 ( τ ) − 4 √ 3 − p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ! ! . (3 .71) Degenerate Thir d Painlev ´ e Equation: II 20 T aking the limit Re( e µ ) → +0 in E q uation (3.52), with τ ( ξ ) and E ♮ T ( ξ ) defined by E q uations (3.3 6) and (3.37), r esp ectively , and using co nditions (3.12), o ne obtains τ ( e µ ) = τ → + ∞ Re( e µ ) → +0 2 p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ln ˆ r 0 ( τ ) p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 + (( ˆ r 0 ( τ )) 2 − 4 ˆ r 0 ( τ ) + 1 6 ) ˆ r 0 ( τ ) p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 − (( ˆ r 0 ( τ )) 2 − 4 ˆ r 0 ( τ ) + 1 6 ) ! + 1 √ 3 ln(e i π ) + O ( e µ ) , (3.72) and E ♮ T ( e µ ) = τ → + ∞ Re( e µ ) → +0 O ( τ − 2 δ 1 ) + O ( τ − 2 3 +2 δ ) , (3.73) whence, substituting the expansions (3.72), (3.73), and (3.68) into Equa tio n (3.52) (with the low e r signs in the formula for ♯ τ taken (cf. E quation (3.53))), and taking in to account conditions (3 .12), one arr ives at Z e µ e µ 0 diag(( T ( ξ )) − 1 ∂ ξ T ( ξ )) d ξ = τ → + ∞ Re( e µ ) → +0 I 0 ( τ ) + O ( τ δ − 2 δ 1 ln τ ) + O ( τ − 2( ε − δ ) ) + O ( τ − 1 3 + ε + δ ) + O ( τ − 2 3 +2 δ ) + O ( ˆ r 0 ( τ ) e µ ) σ 3 , (3.74) where I 0 ( τ ) = p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 8 √ 3 − 1 3 ln τ + ln e Λ + b I 0 ( τ ) , (3.75) with b I 0 ( τ ) := − p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 8 √ 3 ln(3 α e − i π ) − 1 4 ln ˆ r 0 ( τ ) + 4 + p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 − ˆ r 0 ( τ ) − 4 + p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ! − 1 4 ln ˆ r 0 ( τ ) + 4 √ 3 − p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ˆ r 0 ( τ ) + 4 √ 3 + p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ! ˆ r 0 ( τ ) − 4 √ 3 + p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ˆ r 0 ( τ ) − 4 √ 3 − p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ! ! + 1 4 ln ˆ r 0 ( τ ) p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 + (( ˆ r 0 ( τ )) 2 − 4 ˆ r 0 ( τ ) + 1 6 ) ˆ r 0 ( τ ) p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 − (( ˆ r 0 ( τ )) 2 − 4 ˆ r 0 ( τ ) + 1 6 ) ! . (3.76) One now simplifies Equatio ns (3.71) and (3.76); in or der to do s o , howev er, several es timates a re necessary . Rewrite Equatio n (3.16) as follows: ( ˆ u 0 ( τ )) 2 1 + ˆ u 0 ( τ ) + ˆ r 0 ( τ ) ˆ u 0 ( τ ) 2(1 + ˆ u 0 ( τ )) − ( ˆ r 0 ( τ )) 2 8 = − τ − 2 / 3 κ 2 0 ( τ ) 8 α 2 . (3 .77) Via E quations (3.35) and (3.6 8), using Equation (3.77) to eliminate ( ˆ u 0 ( τ )) 2 , a nd tak ing into acco unt conditions (3.12), one arrives at, after simplification, ˆ r 0 ( τ ) p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 − (( ˆ r 0 ( τ )) 2 − 4 ˆ r 0 ( τ ) + 1 6 ) = τ → + ∞ 8( − 2 + ˆ r 0 ( τ )) 1 + ˆ u 0 ( τ ) 1 + O ( τ − 2 3 + δ ) + O ( τ − 1 3 + δ − δ 1 ) . (3.78) The estimates (3.79)–(3.83) b elow are derived analogously : ˆ r 0 ( τ ) p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 + (( ˆ r 0 ( τ )) 2 − 4 ˆ r 0 ( τ ) + 1 6 ) = τ → + ∞ 16(1 + ˆ u 0 ( τ )) 1 + O ( τ − 2 3 + δ ) + O ( τ − 1 3 + δ − δ 1 ) , (3.79) ˆ r 0 ( τ ) + 4 + p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 = τ → + ∞ 2 ˆ r 0 ( τ )(1 + ˆ u 0 ( τ )) ˆ u 0 ( τ ) 1 + O ( τ − 2 3 − δ ) + O ( τ δ − 2 δ 1 ) , (3.80) Degenerate Thir d Painlev ´ e Equation: II 21 ˆ r 0 ( τ ) + 4 − p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 = τ → + ∞ 8 ˆ u 0 ( τ )( − 2 + ˆ r 0 ( τ )) ˆ r 0 ( τ )(1 + ˆ u 0 ( τ )) 1 + O ( τ − 2 3 − δ ) + O ( τ δ − 2 δ 1 ) , (3.81) ˆ r 0 ( τ ) − 4 + p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 = τ → + ∞ 16 ˆ u 0 ( τ ) ˆ r 0 ( τ ) 1 + O ( τ − 2 3 − δ ) + O ( τ δ − 2 δ 1 ) , (3.82) ˆ r 0 ( τ ) − 4 − p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 = τ → + ∞ − 2 ˆ r 0 ( τ ) ˆ u 0 ( τ ) 1 + O ( τ − 2 3 − δ ) + O ( τ δ − 2 δ 1 ) . (3.83) Substituting the e s timates (3.8 2) and (3.83) in to Equa tion (3.71), o ne prov es that b I ∞ ( τ ) = τ → + ∞ I ♯ ∞ ( τ ) + O ( τ − 2 3 − δ ) + O ( τ δ − 2 δ 1 ) , (3.84) where I ♯ ∞ ( τ ) is defined by Equation (3.59), and, s ubstituting the es timates (3.78)–(3.81) into Equa- tion (3.76), o ne prov es that b I 0 ( τ ) = τ → + ∞ I ♯ 0 ( τ ) + O ( τ − 2 3 + δ ) + O ( τ − 1 3 + δ − δ 1 ) + O ( τ δ − 2 δ 1 ) , (3.85) where I ♯ 0 ( τ ) is defined by Eq ua tion (3.61). Hence, via expansions (3.8 4) and (3.85), and Equations (3.69), (3.70), (3.74), and (3.7 5), one obtains the results s ta ted in the Prop osition. Prop ositio n 3.2.5. L et T ( e µ ) b e given in Equation (3.20) , with A ( e µ ) define d by Equation (3.7) , and l ( e µ ) given in Equation (3.23) with the br anches define d as in Cor ol lary 3.2 .4 . Then T ( e µ ) = τ → + ∞ Re( e µ ) → + ∞ ( b ( τ )) − 1 2 ad( σ 3 ) I + 1 e µ 0 − 2 α 4 (1 + ˆ u 0 ( τ )) (2 − ˆ r 0 ( τ )) α 2 +( a − i 2 ) τ − 2 / 3 8 α 4 (1+ ˆ u 0 ( τ )) 0 ! + O ( c 1 ( τ )( ˆ r 0 ( τ )) 2 + c 2 ( τ )) e µ 2 , (3.86) and T ( e µ ) = τ → + ∞ Re( e µ ) → +0 1 √ 2 b ( τ ) 2 √ 2 α 3 − 1 2 ad( σ 3 ) 1 1 − 1 1 + e µ ( − 2 + ˆ r 0 ( τ )) 4 √ 2 α 1 − 1 1 1 + O e µ 2 ( c 3 ( τ )( ˆ r 0 ( τ )) 2 + c 4 ( τ ) ˆ u 0 ( τ ) + c 5 ( τ )) , (3.87) wher e c j ( τ ) = τ → + ∞ O (1 ) , j = 1 , . . . , 5 . Pr o of . Here, the pro of of e stimate (3.86) is presented: the estimate (3.8 7) is proven analog ously . Via Equations (3.10), (3.11), and (3.14), and co nditions (3.12), o ne shows that l ( e µ ) = τ → + ∞ Re( e µ ) → + ∞ 2 e µ + 1 e µ a − i 2 τ − 2 / 3 + O ( e µ − 3 ) , i( A ( e µ ) − i l ( e µ ) σ 3 ) σ 3 = τ → + ∞ Re( e µ ) → + ∞ 4I e µ + 0 − 4 √ − a ( τ ) b ( τ ) b ( τ ) − 2i d ( τ ) 0 ! + 1 e µ ( − 2 + ˆ r 0 ( τ )) α 2 + a − i 2 τ − 2 / 3 I + 1 e µ 2 0 8 α 6 b ( τ ) − b ( τ ) 0 ! + O ( e µ − 3 I) , 1 p 2i l ( e µ )( A 11 ( e µ ) − i l ( e µ )) = τ → + ∞ Re( e µ ) → + ∞ 1 4 e µ 1 − 1 8 e µ 2 ( − 2 + ˆ r 0 ( τ )) α 2 + 3 a − i 2 τ − 2 / 3 + O ( C 1 ( τ )( ˆ r 0 ( τ )) 2 + C 2 ( τ )) e µ 4 , where C j ( τ ) = τ → + ∞ O (1 ), j = 1 , 2 ; hence, v ia Equation (3.20) and co nditions (3.12), one ar rives at the estimate (3.8 6). Degenerate Thir d Painlev ´ e Equation: II 22 Prop ositio n 3.2.6. L et T ( e µ ) b e given in Equation (3.20) , with A ( e µ ) define d by Equation (3.7) , and l 2 ( e µ ) given in Equation (3 .23) . Under the c onditions of Cor ol lary 3.2.3: T 11 ( e µ ) = T 22 ( e µ ) = e µ = e µ 0 τ → + ∞ α ˆ r 0 ( τ ) τ 1 / 6 q 8 √ 3 α ˆ r 0 ( τ ) e Λ 1 − h 0 ( τ ) 48 α 2 e Λ 2 − ( a − i 2 ) 48 e Λ 2 + 1 2 α 4 − ˆ r 0 ( τ ) + 4 √ 3 ˆ r 0 ( τ ) + 7 6 ! × τ − 1 / 3 e Λ + O ( τ 2 δ − 4 ε ) + O ( τ − ε − δ 1 ) + O ( τ 2 ε − 2 δ 1 ) ! , (3.88) T 12 ( e µ ) = e µ = e µ 0 τ → + ∞ − 8 α 4 ˆ u 0 ( τ ) τ 1 / 6 b ( τ ) q 8 √ 3 α ˆ r 0 ( τ ) e Λ 1 − h 0 ( τ ) 48 α 2 e Λ 2 − ( a − i 2 ) 48 e Λ 2 − 1 2 α 4 − ˆ r 0 ( τ ) + 4 √ 3 ˆ r 0 ( τ ) − 7 6 ! − 2 α ˆ u 0 ( τ ) τ − 1 / 3 e Λ + O ( τ 2 δ − 4 ε ) + O ( τ − ε − δ 1 ) , (3.89) T 21 ( e µ ) = e µ = e µ 0 τ → + ∞ − b ( τ ) τ 1 / 6 2 α 2 q 8 √ 3 α ˆ r 0 ( τ ) e Λ ˆ r 0 ( τ ) + 2 ˆ u 0 ( τ ) − α − 2 ( a − i 2 ) τ − 2 / 3 1 + ˆ u 0 ( τ ) ! 1 − h 0 ( τ ) 48 α 2 e Λ 2 − ( a − i 2 ) 48 e Λ 2 − 1 2 α 4 − ˆ r 0 ( τ ) + 4 √ 3 ˆ r 0 ( τ ) − 7 6 ! + 4 α ˆ r 0 ( τ ) + 2 ˆ u 0 ( τ ) − α − 2 ( a − i 2 ) τ − 2 / 3 1 + ˆ u 0 ( τ ) ! − 1 τ − 1 / 3 e Λ + O ( τ 2 δ − 4 ε ) + O ( τ − ε − δ 1 ) ! , (3.90) wher e := ( +1 , arg( e Λ) = 0 , − 1 , arg( e Λ) = π . Pr o of . Using the definition of A ( e µ ) (cf. Eq ua tion (3 .7) ), together with Equations (3.10), (3.11), and (3.14), a nd conditions (3 .12) and those in Coro llary 3.2.3, one der ives the following expansions: 1 p 2i l ( e µ )( A 11 ( e µ ) − i l ( e µ ) ) = e µ = e µ 0 τ → + ∞ τ 1 / 6 q 8 √ 3 α ˆ r 0 ( τ ) e Λ 1 − h 0 ( τ ) 48 α 2 e Λ 2 − ( a − i 2 ) 48 e Λ 2 − 1 2 α 4 − ˆ r 0 ( τ ) + 4 √ 3 ˆ r 0 ( τ ) − 7 6 τ − 1 / 3 e Λ + O ( h 0 ( τ )) 2 e Λ 4 + O τ − 1 / 3 h 0 ( τ ) ˆ r 0 ( τ ) e Λ ! ! , i A 11 ( e µ ) + l ( e µ ) = e µ = e µ 0 τ → + ∞ α ˆ r 0 ( τ ) 1 + 4 − ˆ r 0 ( τ ) + 4 √ 3 α ˆ r 0 ( τ ) + 4 √ 3 α ˆ r 0 ( τ ) h 0 ( τ ) 24 α 2 e Λ 2 + ( a − i 2 ) 24 e Λ 2 !! τ − 1 / 3 e Λ + − 2 + ˆ r 0 ( τ ) − 14 √ 3 α 2 ˆ r 0 ( τ ) ! τ − 2 / 3 e Λ 2 + O τ − 1 / 3 ( h 0 ( τ )) 2 ˆ r 0 ( τ ) e Λ 3 !! , − i A 12 ( e µ ) = e µ = e µ 0 τ → + ∞ 1 b ( τ ) − 8 α 4 ˆ u 0 ( τ ) − 1 6 α 3 τ − 1 / 3 e Λ + 24 α 2 τ − 2 / 3 e Λ 2 + O ( τ − 1 e Λ 3 ) , i A 21 ( e µ ) = e µ = e µ 0 τ → + ∞ b ( τ ) − ( − 2 + ˆ r 0 ( τ )) 2 α 2 (1 + ˆ u 0 ( τ )) − 1 α 2 + ( a − i 2 ) τ − 2 / 3 2 α 4 (1 + ˆ u 0 ( τ )) + 2 α 3 τ − 1 / 3 e Λ − 3 α 4 τ − 2 / 3 e Λ 2 + O ( τ − 1 e Λ 3 ) . According to the choice o f the bra nch of l ( ξ ) in Coro lla ry 3.2.4, l ( ξ ) > 0 for p ositive ξ o utside the neighborho o d of ξ = α containing the double-tur ning points; therefore, one arrives at the definition of stated in the Pr o p o sition. The asympto tics (3.88)–(3.90) a r e obtained b y substituting the ab ove expansions into Equations (3.22). Degenerate Thir d Painlev ´ e Equation: II 23 3.3 The Mo del Pr oblem and Asymptotics Nea r the T urning Poin ts F or the calculation of the mono dromy data, one needs an approximation that is mor e accur ate than that given by the WKB for m ula (cf. Equation (3.18)) for the solution of Equation (3.6) in prop er neighborho o ds of the turning p oints. There ar e t wo simple turning p o ints appro aching ± i √ 2 α : the approximate solution of E quation (3.6) in neighbor ho ods o f thes e turning p oints is given in terms of Airy functions. There ar e also t wo pair s of turning p oints, one pair coalescing at − α and ano ther pair coales c ing at + α (double-turning p oints): it is well kno wn that, in neig hborho o ds of ± α , the approximate solution of Equatio n (3.6) is ex pr essed in terms of parab olic-cylinder functions (see, for example, [8, 9]). I n order to o btain asymptotics of u ( τ ), and the a sso ciated functions H ( τ ) and f ( τ ), it is sufficient to study a subset of the complete set of the mono dr omy data , which can be calculated via the appr oximation of the g eneral solution of E quation (3.6) in a neig hbo rho o d of the double- turning po in t + α . F or the asymptotic conditions (3.12) on the functions ˆ r 0 ( τ ), ˆ u 0 ( τ ), and h 0 ( τ ), this approximation is not straightforw ard, and is given in Lemma 3 .3.1 be low. Lemma 3 .3.1. L et e µ = α + τ − 1 / 3 e Λ , wher e e Λ = τ → + ∞ O ( τ ε ) , 0 < ε < 1 / 9 , and ν + 1 := − i 2 √ 3 h 0 ( τ ) α 2 + a − i 2 − p τ 4 + 1 2 , (3.91) wher e p τ is define d by Equation (3.35) . In c onjunction with Equations (3.9) – (3.11) and (3.13) , and c onditions (3.12) , imp ose t he fol lowing r estrictions: 0 < τ → + ∞ Re( ν + 1) < τ → + ∞ 1 , Im( ν + 1) = τ → + ∞ O (1 ) , (3.92) 0 < δ < ε < 1 9 , 6 ε + 2 ε Re( ν + 1 ) + δ 2 < δ 1 < 1 3 . ( 3.93 ) Then ther e ex ist s a fundamental solution of Equation (3.6) with asymptotic re pr esentation e Ψ( e µ, τ ) = τ → + ∞ F τ ( e Λ) I + O τ 6 ε +2 ε Re( ν +1)+ δ 2 − δ 1 ψ 0 ( e Λ) , (3.94) wher e F τ ( e Λ) := i τ 1 / 6 √ − 8 α 4 ˆ u 0 ( τ ) √ b ( τ ) √ κ 0 ( τ ) 0 i √ b ( τ ) ( α ˆ r 0 ( τ ) τ 1 / 6 − i τ − 1 / 6 ℓ τ e Λ ) √ κ 0 ( τ ) √ − 8 α 4 ˆ u 0 ( τ ) − i √ κ 0 ( τ ) √ b ( τ ) τ − 1 / 6 √ − 8 α 4 ˆ u 0 ( τ ) , (3.95) κ 2 0 ( τ ) is define d by Equation (3.65) , ℓ τ = − p τ + p p 2 τ − ˆ r 0 ( τ )(8 − ˆ r 0 ( τ )) = τ → + ∞ ℓ ∞ τ + O τ − 2 / 3 κ 2 0 ( τ )( O (1) + ( ˆ r 0 ( τ )) 2 ) ( ˆ r 0 ( τ )) 2 , (3.96) with ℓ ∞ τ := − p τ + i4 √ 3 , (3.97) and ψ 0 ( e Λ) is a fundamental solution of ∂ ψ 0 ( e Λ) ∂ e Λ = i4 √ 3 e Λ σ 3 + e q ( τ ) σ − + e p ( τ ) σ + ψ 0 ( e Λ) , (3.98) wher e e p ( τ ) = − i κ 0 ( τ ) , e q ( τ ) = − i κ 0 ( τ ) κ 2 0 ( τ ) + ℓ ∞ τ + 4( a − i 2 ) ˆ u 0 ( τ ) 1 + ˆ u 0 ( τ ) ! . (3.99) The function ψ 0 ( e Λ) c an b e explicitly pr esente d in the fol lowing form: ψ 0 ( e Λ) = D − 1 − ν (ie i π 4 2 3 / 2 3 1 / 4 e Λ) D ν (e i π 4 2 3 / 2 3 1 / 4 e Λ) ˆ ∂ e Λ D − 1 − ν (ie i π 4 2 3 / 2 3 1 / 4 e Λ) ˆ ∂ e Λ D ν (e i π 4 2 3 / 2 3 1 / 4 e Λ) ! , (3.100) wher e D ∗ ∗ ∗ ( · · · ) is the p ar ab olic-cylinder function [1 1] , and ˆ ∂ e Λ := ( e p ( τ )) − 1 ( ∂ ∂ e Λ − i4 √ 3 e Λ) . Degenerate Thir d Painlev ´ e Equation: II 24 Pr o of . The deriv a tion of approximation (3.94) cons ists of the following sequence of inv ertible linear transformatio ns, F j : SL 2 ( C ) → SL 2 ( C ), j = 1 , . . . , 7: (i) (i) (i) F 1 : e Ψ( e µ ) 7→ Φ( e Λ) := e Ψ( α + τ − 1 / 3 e Λ) , (ii) (ii) (ii) F 2 : Φ( e Λ) 7→ b Φ( e Λ) := ( b ( τ )) 1 2 σ 3 Φ( e Λ) , (iii) (iii) (iii) F 3 : b Φ( e Λ) 7→ φ ( e Λ ) := ( N ( τ )) − 1 b Φ( e Λ) , (iv) (iv) (iv) F 4 : φ ( e Λ) 7→ e Φ( e Λ) := τ − 1 6 σ 3 φ ( e Λ) , (v) (v) (v) F 5 : e Φ( e Λ) 7→ b φ ( e Λ) := 1 0 − ℓ τ e Λ 1 e Φ( e Λ) , (vi) (vi) (vi) F 6 : b φ ( e Λ) 7→ ψ ( e Λ) := ( G ( τ )) − 1 b φ ( e Λ) , (vii) (vii) (vii) F 7 : ψ ( e Λ) 7→ ψ 0 ( e Λ) := ( χ 0 ( e Λ)) − 1 ψ ( e Λ) , where the unimo dular, ma trix-v a lue d functions N ( τ ), G ( τ ), and χ 0 ( e Λ), res pec tiv ely , are describ ed in steps (iii) (iii) (iii), (vi) (vi) (vi), and (vii) (vii) (vii) b elow. (i) (i) (i) Let e Ψ( e µ ) solve Equation (3.6). Then applying the transfor ma tion F 1 , using Equations (3.10), (3.11), and (3.1 4), and conditions (3.12), one shows tha t ∂ Φ( e Λ) ∂ e Λ = τ → + ∞ ( τ 1 / 3 Q + + τ − 1 / 3 Q − ) + e Λ Q 1 + τ − 1 / 3 e Λ 2 Q 2 + O ( τ − 2 / 3 e Λ 3 Q 3 ) Φ( e Λ) , (3.10 1) where Q ± = ( b ( τ )) − 1 2 ad( σ 3 ) ˆ P ± , Q j = ( b ( τ )) − 1 2 ad( σ 3 ) ˆ P j , j = 1 , 2 , 3 , with ˆ P + := − i α ˆ r 0 ( τ ) − 8i α 4 ˆ u 0 ( τ ) i( ˆ r 0 ( τ )+2 ˆ u 0 ( τ )) 2 α 2 (1+ ˆ u 0 ( τ )) i α ˆ r 0 ( τ ) ! , ˆ P − := 0 0 − i( a − i 2 ) 2 α 4 (1+ ˆ u 0 ( τ )) 0 ! , ˆ P 1 := i( − 4 + ˆ r 0 ( τ )) − 16i α 3 − 2i α 3 − i( − 4 + ˆ r 0 ( τ )) , ˆ P 2 := − i( − 2+ ˆ r 0 ( τ )) α 24i α 2 3i α 4 i( − 2+ ˆ r 0 ( τ )) α ! , ˆ P 3 := O (1 ) + O ( ˆ r 0 ( τ )) O (1 ) O (1 ) O (1) + O ( ˆ r 0 ( τ )) . Note that tr( Q ± ) = tr( ˆ P ± ) = tr( Q j ) = tr( ˆ P j ) = 0, j = 1 , 2 , 3; furthermore, Eq uation (3.77) implies det( ˆ P + ) = − 8 α 2 ( ˆ u 0 ( τ )) 2 1 + ˆ u 0 ( τ ) + ˆ r 0 ( τ ) ˆ u 0 ( τ ) 2(1 + ˆ u 0 ( τ )) − ( ˆ r 0 ( τ )) 2 8 = κ 2 0 ( τ ) τ − 2 / 3 , (3.102) where κ 2 0 ( τ ) is defined by E quation (3.6 5). (ii) (ii) (ii) Let Φ( e Λ) so lve Equation (3.10 1). Then applying the transfor ma tion F 2 , one obtains ∂ b Φ( e Λ) ∂ e Λ = τ → + ∞ ( τ 1 / 3 ˆ P + + τ − 1 / 3 ˆ P − ) + e Λ ˆ P 1 + τ − 1 / 3 e Λ 2 ˆ P 2 + O ( τ − 2 / 3 e Λ 3 ˆ P 3 ) b Φ( e Λ) . (3.103) (iii) (iii) (iii) The idea b ehind the following tr ansformation for E quation (3.10 3) is to put the ma trix ˆ P + int o Jo rdan canonical fo rm, that is, to find a function N ( τ ) suc h that ( N ( τ )) − 1 ˆ P + N ( τ ) = i κ 0 ( τ ) τ − 1 / 3 σ 3 + σ + . The following solution for N ( τ ) is chosen: N ( τ ) = p − 8i α 4 ˆ u 0 ( τ ) 0 i ( α ˆ r 0 ( τ )+ κ 0 ( τ ) τ − 1 / 3 ) √ − 8i α 4 ˆ u 0 ( τ ) 1 √ − 8i α 4 ˆ u 0 ( τ ) . Degenerate Thir d Painlev ´ e Equation: II 25 Let b Φ( e Λ) solve Equation (3.103). Then applying the tra nsformation F 3 , using E quations (3.10) and (3.11), co nditions (3.12), a nd Equa tions (3.77), (3.65), a nd (3.102), o ne shows that ∂ φ ( e Λ) ∂ e Λ = τ → + ∞ ( τ 1 / 3 P ♯ + + τ − 1 / 3 P ♯ − ) + e Λ P ♯ 1 + τ − 1 / 3 e Λ 2 P ♯ 2 + O ( τ − 2 / 3 e Λ 3 ( N ( τ )) − 1 ˆ P 3 N ( τ )) φ ( e Λ) , (3.104) where P ♯ ± := ( N ( τ )) − 1 P ± N ( τ ), P ♯ j := ( N ( τ )) − 1 ˆ P j N ( τ ), j = 1 , 2, with P ♯ + = i κ 0 ( τ ) τ − 1 / 3 σ 3 + σ + , P ♯ − = − 4( a − i 2 ) ˆ u 0 ( τ ) (1 + ˆ u 0 ( τ )) σ − , ( P ♯ 1 ) 11 = − ( P ♯ 1 ) 22 = τ → + ∞ p τ + 2i κ 0 ( τ ) τ − 1 / 3 α ˆ u 0 ( τ ) + O κ 2 0 ( τ ) τ − 2 / 3 ˆ r 0 ( τ ) ˆ u 0 ( τ ) + O κ 2 0 ( τ ) τ − 2 / 3 ˆ r 0 ( τ ) , ( P ♯ 1 ) 12 = 2 α ˆ u 0 ( τ ) , ( P ♯ 1 ) 21 = τ → + ∞ − 2i κ 0 ( τ ) p τ τ − 1 / 3 + O κ 2 0 ( τ ) τ − 2 / 3 ˆ u 0 ( τ ) + O ( κ 2 0 ( τ ) τ − 2 / 3 ) , ( P ♯ 2 ) 11 = − ( P ♯ 2 ) 22 = − i(6 + ( − 2 + ˆ r 0 ( τ ))(3 + ˆ u 0 ( τ ))) α ˆ u 0 ( τ ) − 3i κ 0 ( τ ) τ − 1 / 3 α 2 ˆ u 0 ( τ ) , ( P ♯ 2 ) 12 = − 3 α 2 ˆ u 0 ( τ ) , ( P ♯ 2 ) 21 = τ → + ∞ − ˆ r 0 ( τ )(8 − ˆ r 0 ( τ )) − 2 κ 0 ( τ ) α ˆ u 0 ( τ ) (6 + ( − 2 + ˆ r 0 ( τ ))(3 + ˆ u 0 ( τ ))) τ − 1 / 3 + O κ 2 0 ( τ ) τ − 2 / 3 ˆ u 0 ( τ ) + O ( κ 2 0 ( τ ) τ − 2 / 3 ) , where p τ is defined by Equation (3.3 5). (iv) (iv) (iv) Let φ ( e Λ) so lve Equation (3.10 4). Then applying the transfor ma tion F 4 , one pr ov es that ∂ e Φ( e Λ) ∂ e Λ = τ → + ∞ i κ 0 ( τ ) 1 − 4( a − i 2 ) ˆ u 0 ( τ ) 1+ ˆ u 0 ( τ ) − i κ 0 ( τ ) ! + e Λ p τ 0 − 2i κ 0 ( τ ) p τ − p τ + e Λ 2 0 0 − ˆ r 0 ( τ )(8 − ˆ r 0 ( τ )) 0 + O ( τ − 1 / 3 E τ ( e Λ)) e Φ( e Λ) , (3.105) where, with the help of co nditions (3.1 2), E τ ( e Λ) = τ → + ∞ e Λ O κ 0 ( τ ) ˆ u 0 ( τ ) + O κ 2 0 ( τ ) τ − 1 / 3 ˆ r 0 ( τ ) ˆ u 0 ( τ ) O (( ˆ u 0 ( τ )) − 1 ) O κ 2 0 ( τ ) ˆ u 0 ( τ ) + O ( κ 2 0 ( τ )) O κ 0 ( τ ) ˆ u 0 ( τ ) + O κ 2 0 ( τ ) τ − 1 / 3 ˆ r 0 ( τ ) ˆ u 0 ( τ ) + e Λ 2 O (1 ) + O ( ˆ r 0 ( τ )) O ( τ − 1 / 3 ( ˆ u 0 ( τ )) − 1 ) O ( κ 0 ( τ )) + O ( κ 0 ( τ ) ˆ r 0 ( τ )) O (1 ) + O ( ˆ r 0 ( τ )) + e Λ 3 O ( τ − 1 / 3 ) + O ( τ − 1 / 3 ˆ r 0 ( τ )) O ( τ − 2 / 3 ( ˆ u 0 ( τ )) − 1 ) O ( ˆ r 0 ( τ )) + O (( ˆ r 0 ( τ )) 2 ) O ( τ − 1 / 3 ) + O ( τ − 1 / 3 ˆ r 0 ( τ )) = τ → + ∞ O e Λ κ 2 0 ( τ ) τ − 1 / 3 ˆ r 0 ( τ ) ˆ u 0 ( τ ) + O ( e Λ 2 ) + O ( e Λ 2 ˆ r 0 ( τ )) O ( e Λ( ˆ u 0 ( τ )) − 1 ) O e Λ κ 2 0 ( τ ) ˆ u 0 ( τ ) + O ( e Λ 2 κ 0 ( τ )) + O ( e Λ 3 ( ˆ r 0 ( τ )) 2 ) O e Λ κ 2 0 ( τ ) τ − 1 / 3 ˆ r 0 ( τ ) ˆ u 0 ( τ ) + O ( e Λ 2 ) + O ( e Λ 2 ˆ r 0 ( τ )) . (3.106) (v) (v) (v) O ne now pro cee ds to eliminate the e Λ 2 term in Equation (3.105). Applying the tra ns formation F 5 , with a τ -dependent pa rameter ℓ τ , one o btains ∂ b φ ( e Λ) ∂ e Λ = τ → + ∞ i κ 0 ( τ ) 1 − ℓ τ − 4( a − i 2 ) ˆ u 0 ( τ ) 1+ ˆ u 0 ( τ ) − i κ 0 ( τ ) ! + e Λ p τ + ℓ τ 0 − 2i κ 0 ( τ )( ℓ τ + p τ ) − ( p τ + ℓ τ ) + e Λ 2 0 0 − ˆ r 0 ( τ )(8 − ˆ r 0 ( τ )) − 2 p τ ℓ τ − ℓ 2 τ 0 + O ( τ − 1 / 3 b E τ ( e Λ)) b φ ( e Λ) , (3.107) where b E τ ( e Λ) := 1 0 − ℓ τ e Λ 1 E τ ( e Λ) 1 0 ℓ τ e Λ 1 . (3 .108) Degenerate Thir d Painlev ´ e Equation: II 26 One now c ho oses ℓ τ so that the quadra tic term in Equa tion (3.107) is annihila ted: ℓ 2 τ + 2 p τ ℓ τ + ˆ r 0 ( τ )(8 − ˆ r 0 ( τ )) = 0; th us, ℓ τ = − p τ + p p 2 τ − ˆ r 0 ( τ )(8 − ˆ r 0 ( τ )) . (3.109) A straig ht forward ca lculation shows that (cf. Equation (3.68)) p 2 τ − ˆ r 0 ( τ )(8 − ˆ r 0 ( τ )) = τ → + ∞ − 48 + O τ − 2 / 3 κ 2 0 ( τ )( O (1) + ( ˆ r 0 ( τ )) 2 ) ( ˆ r 0 ( τ )) 2 ; hence, making the choice √ − 1 = i, ℓ τ = τ → + ∞ ℓ ∞ τ + O τ − 2 / 3 κ 2 0 ( τ )( O (1) + ( ˆ r 0 ( τ )) 2 ) ( ˆ r 0 ( τ )) 2 , (3.110) where ℓ ∞ τ is defined by Equation (3.97). Now, with the help of Equations (3.106), (3.10 8), and (3.110), and conditions (3.1 2), one re- writes Equa tion (3.107) as ∂ b φ ( e Λ) ∂ b Λ = τ → + ∞ i κ 0 ( τ ) 1 − ℓ ∞ τ − 4( a − i 2 ) ˆ u 0 ( τ ) 1+ ˆ u 0 ( τ ) − i κ 0 ( τ ) ! + e Λ i4 √ 3 0 8 √ 3 κ 0 ( τ ) − i4 √ 3 + O ( e E τ ( e Λ)) ! b φ ( e Λ) , (3.111) where e E τ ( e Λ) := ( e E τ ( e Λ)) 11 ( e E τ ( e Λ)) 12 ( e E τ ( e Λ)) 21 ( e E τ ( e Λ)) 22 ! , (3.112) with ( e E τ ( e Λ)) 11 = − ( e E τ ( e Λ)) 22 = τ → + ∞ O e Λ τ − 2 / 3 κ 2 0 ( τ ) ( ˆ r 0 ( τ )) 2 ! + O e Λ τ − 2 / 3 κ 2 0 ( τ ) + O e Λ 2 τ − 1 / 3 ˆ r 0 ( τ ) + O e Λ 2 τ − 1 / 3 ℓ ∞ τ ˆ u 0 ( τ ) ! , ( e E τ ( e Λ)) 12 = τ → + ∞ O e Λ τ − 1 / 3 ˆ u 0 ( τ ) ! , ( e E τ ( e Λ)) 21 = τ → + ∞ O τ − 2 / 3 κ 2 0 ( τ ) ( ˆ r 0 ( τ )) 2 + O τ − 2 / 3 κ 2 0 ( τ ) + O e Λ τ − 2 / 3 κ 3 0 ( τ ) ( ˆ r 0 ( τ )) 2 ! + O e Λ τ − 2 / 3 κ 3 0 ( τ ) + O e Λ τ − 1 / 3 κ 2 0 ( τ ) ˆ u 0 ( τ ) ! + O e Λ 2 τ − 1 / 3 κ 0 ( τ ) + O e Λ 2 τ − 2 / 3 ℓ ∞ τ κ 2 0 ( τ ) ˆ r 0 ( τ ) ˆ u 0 ( τ ) ! + O e Λ 3 τ − 1 / 3 ( ˆ r 0 ( τ )) 2 + O e Λ 3 τ − 1 / 3 ℓ ∞ τ + O e Λ 3 τ − 1 / 3 ℓ ∞ τ ˆ r 0 ( τ ) + O e Λ 3 τ − 1 / 3 ( ℓ ∞ τ ) 2 ˆ u 0 ( τ ) ! . (vi) (vi) (vi) One now pro ceeds to diagonalize the e Λ term in Equation (3.111). Set G ( τ ) := e − i π 4 p κ 0 ( τ ) i( κ 0 ( τ )) − 1 0 1 1 . (3.113) Then applying the transfor mation F 6 , one s hows that ∂ ψ ( e Λ) ∂ e Λ = B 0 ( e Λ) + R 0 ( e Λ) ψ ( e Λ) , (3.114) Degenerate Thir d Painlev ´ e Equation: II 27 where B 0 ( e Λ) := i4 √ 3 e Λ σ 3 + e q ( τ ) σ − + e p ( τ ) σ + , (3.115) with e p ( τ ), e q ( τ ) given in Equations (3.9 9), and R 0 ( e Λ) := τ → + ∞ O ( G ( τ )) − 1 e E τ ( e Λ) G ( τ ) , (3.116) where e E τ ( e Λ) is defined by Equation (3.112). (vii) (vii) (vii) Let ψ 0 ( e Λ) b e a fundamental solution of ∂ ψ 0 ( e Λ) ∂ e Λ = B 0 ( e Λ) ψ 0 ( e Λ), which co incides with E qua- tion (3.9 8). Changing v ar iables in Eq uation (3.98) according to D ( x ) := ψ 0 ( e Λ), wher e e Λ = x 0 x , with x 0 = e − i π 4 2 − 3 / 2 3 − 1 / 4 , one proves that D ( x ) solves the standard eq uation ∂ x D ( x ) = x 2 σ 3 + q ∗ ( τ ) σ − + p ∗ ( τ ) σ + D ( x ) , where p ∗ ( τ ) := x 0 e p ( τ ) and q ∗ ( τ ) := x 0 e q ( τ ), with fundamental solution giv en in terms of the par ab olic- cylinder function, D ∗ ∗ ∗ ( · · · ) (see, for example, [2 , 6, 1 2]): D ( x ) = D − 1 − ν (i x ) D ν ( x ) ˙ D − 1 − ν (i x ) ˙ D ν ( x ) , (3.117) where ˙ D ∗ ∗ ∗ ( z ) := ( p ∗ ( τ )) − 1 ( ∂ z D ∗ ∗ ∗ ( z ) − z 2 D ∗ ∗ ∗ ( z )), a nd ν + 1 := − p ∗ ( τ ) q ∗ ( τ ) = − i 8 √ 3 κ 2 0 ( τ ) + ℓ ∞ τ + 4( a − i 2 ) ˆ u 0 ( τ ) 1 + ˆ u 0 ( τ ) ! . (3.118) Now, applying definitions (3.65) and (3 .97), one r e - writes Equation (3.1 18) as definition (3.9 1) ; th us, the repr e sentation for ψ 0 ( e Λ) given in Eq uations (3.91) and (3.9 8)–(3.100) is obtained. Finally , in order to prov e the erro r es timate in Equation (3.94), one has to e stimate the func- tion χ 0 ( e Λ) defined in the transfor mation F 7 . Applying the transfo rmation F 7 , one re-writes Equa- tion (3.114) as follows: ∂ χ 0 ( e Λ) ∂ e Λ = τ → + ∞ R 0 ( e Λ) χ 0 ( e Λ) + h B 0 ( e Λ) , χ 0 ( e Λ) i , (3.119) where B 0 ( e Λ) is defined by E q uation (3.115), [ B 0 ( e Λ) , χ 0 ( e Λ)] := B 0 ( e Λ) χ 0 ( e Λ) − χ 0 ( e Λ) B 0 ( e Λ) is the ma- trix commutator, and R 0 ( e Λ) is defined by Equation (3.116). The nor malized solution ( χ 0 (0) = I) of Equation (3.119) is given b y χ 0 ( e Λ) = I + Z e Λ 0 ψ 0 ( e Λ)( ψ 0 ( ξ )) − 1 R 0 ( ξ ) χ 0 ( ξ ) ψ 0 ( ξ )( ψ 0 ( e Λ)) − 1 d ξ . (3.120) T o prove the required estima te for χ 0 ( e Λ), one uses the method of successive approximations, χ ( n ) 0 ( e Λ) := I + Z e Λ 0 ψ 0 ( e Λ)( ψ 0 ( ξ )) − 1 R 0 ( ξ ) χ ( n − 1) 0 ( ξ ) ψ 0 ( ξ )( ψ 0 ( e Λ)) − 1 d ξ , n ∈ N , with χ (0) 0 ( e Λ) = I, to construct a Neumann ser ies solution for χ 0 ( e Λ), that is, χ 0 ( e Λ) := lim n →∞ χ ( n ) 0 ( e Λ). In this case, howev er, it suffices to estimate the norm of the asso ciated resolvent kernel. Via the ab ov e iteration ar gument, it fo llows that || χ 0 ( e Λ) − I || 6 exp Z e Λ 0 || ψ 0 ( e Λ) |||| ( ψ 0 ( ξ )) − 1 |||| R 0 ( ξ ) |||| ψ 0 ( ξ ) |||| ( ψ 0 ( e Λ)) − 1 || | d ξ | ! − 1 , (3.121) where | d ξ | denotes in tegra tion w ith r esp ect to arc length. One now estimates the norms appea ring in Equation (3.12 1). Using Equations (3.1 12), (3.113), and (3.116), conditions (3.12), and the conditions of Corolla ry 3.2.3, one shows that R 0 ( ξ ) = τ → + ∞ O ( τ 2 ε + δ − δ 1 ) O ( τ ε + δ 2 − δ 1 ) O ( τ 3 ε + 3 δ 2 − δ 1 ) O ( τ 2 ε + δ − δ 1 ) ! ; Degenerate Thir d Painlev ´ e Equation: II 28 recalling the definitio n of the matrix nor m (cf. Subsection 3.1), one ar rives at || R 0 ( ξ ) || = τ → + ∞ O τ 3 ε + 3 δ 2 − δ 1 . (3.122) F or the function ψ 0 ( ξ ), one has to find a uniform approximation for ξ ∈ R ∪ i R . T o do so, o ne uses the following integral re presentation for the parab olic- cylinder function [13]: D ν ( x ) = 2 ν 2 e − x 2 4 Γ( − ν 2 ) Z + ∞ 0 e − tx 2 2 t − ν 2 − 1 (1 + t ) ν − 1 2 d t, Re( ν ) < 0 , | a rg( x ) | 6 π / 4 . (3.123) As this int egr al r e presentation will be applied to the entries of the matr ix-v alued function ψ 0 ( ξ ) (cf. Equation (3.100)), it implies the following restriction on ν : 0 < τ → + ∞ Re( ν + 1) < τ → + ∞ 1 . Since, in the sector | ar g( x ) | 6 π / 4, the exp onents in the integral representation (3.1 23) are less than or equal to one, the following estimate for D ν ( x ) can b e deduced: | D ν ( x ) | 6 √ π 2 Re( ν ) / 2 Γ( 1 − Re( ν ) 2 ) , | ar g( x ) | 6 π/ 4 . F rom the ab ov e inequalit y , one derives, for the elements of the sec o nd column of ψ 0 ( ξ ), the following estimates: | ( ψ 0 ( ξ )) 12 | = τ → + ∞ O (1 ) , | ( ψ 0 ( ξ )) 22 | = τ → + ∞ O (1 ) + O ( ξ | κ 0 ( τ ) | − 1 ) , ar g ( ξ ) ∈ ( − π / 2 , 0) . In order to a pply , simultaneously , the same integral represe ntation (3.123) for the elements of the first column of ψ 0 ( ξ ), one has to r estrict ar g ( ξ ) to − π/ 2; hence, one ar rives at the following, analo gous estimates: | ( ψ 0 ( ξ )) 11 | = τ → + ∞ O (1 ) , | ( ψ 0 ( ξ )) 21 | = τ → + ∞ O (1 ) + O ( ξ | κ 0 ( τ ) | − 1 ) , ar g ( ξ ) = − π / 2 . Thu s, for arg( ξ ) = − π / 2, one a rrives at the following estimate for || ψ 0 ( ξ ) || : || ψ 0 ( ξ ) || = τ → + ∞ O (1 ) + O ( ξ | κ 0 ( τ ) | − 1 ) . (3.124 ) In or der to find estimates for ψ 0 ( ξ ) on the other St okes rays arg( ξ ) = 0 , π / 2 , ± π , . . . , o ne has to use the linear relations for the para b o lic-cylinder functions re lating any three of the four func- tions D ν ( ± x ) and D − ν − 1 ( ± i x ) (see, for example, [11], pg . 1094 , Eq ua tions 9.2 48 9.248 9.248 1.– 3.), and impo se the additional restrictio n Im( ν + 1) = τ → + ∞ O (1 ), in which case, for all the Stokes r ays a rg( ξ ) = 0 , ± π / 2 , ± π , . . . , 0 6 | ξ | < + ∞ , one verifies an estimate similar to Equation (3.124). Noting that det( ψ 0 ( ξ )) = − ( p ∗ ( τ )) − 1 exp( − i π 2 ( ν + 1)), o ne obtains || ( ψ 0 ( ξ )) − 1 || = τ → + ∞ | κ 0 ( τ ) | e − π 2 Im( ν + 1) || ψ 0 ( ξ ) || . (3 .125) Using the asymptotic expansions for the par a b o lic-cylinder functions (see Remark 3 .3.1 b elow), one shows tha t || ψ 0 ( e Λ) || = τ → + ∞ O ( | κ 0 ( τ ) | − 1 e Λ Re( ν +1) ) , || ( ψ 0 ( e Λ)) − 1 || = τ → + ∞ O ( e Λ Re( ν +1) ) . (3.126) Combining the estimates (3.122), (3.1 24), (3.12 5), and (3.126), and assuming that 6 ε + 2 ε Re( ν + 1 ) + δ 2 < δ 1 , one deduces fro m Equation (3.12 1) that || χ 0 ( e Λ) − I || 6 τ → + ∞ O τ 6 ε +2 ε Re( ν +1)+ δ 2 − δ 1 . Degenerate Thir d Painlev ´ e Equation: II 29 Hence, forming the comp osition of the inv ertible linear transformations F 1 , . . . , F 7 , that is (the “sym- bo l” ◦ denotes “comp osition”), e Ψ( e µ ) = F − 1 1 ◦ F − 1 2 ◦ F − 1 3 ◦ F − 1 4 ◦ F − 1 5 ◦ F − 1 6 ◦ F − 1 7 ψ 0 ( e Λ) = ( b ( τ )) − 1 2 σ 3 N ( τ ) τ 1 6 σ 3 1 0 ℓ τ e Λ 1 G ( τ ) | {z } =: F τ ( e Λ) χ 0 ( e Λ) ψ 0 ( e Λ) , one arr ives at the asymptotic re presentation for e Ψ( e µ ) given in Equation (3.94). Remark 3.3.1. In Lemma 3 .3.1 and herea fter, the matrix-v a lued function ψ 0 ( e Λ) ( cf. E q uation (3.100)) plays a pivotal ro le; therefore , for the reader’s co nv enience , its as y mptotics are presented here: ψ 0 ( e Λ) = e Λ →∞ arg( e Λ)= kπ 2 I + ∞ X j =1 ψ j ( τ ) e Λ − j exp i2 √ 3 e Λ 2 − ( ν + 1) ln(e i π 4 2 3 2 3 1 4 e Λ) σ 3 R k , k = − 1 , 0 , 1 , 2 , where ψ j ( τ ) ar e off-dia gonal ( resp., diagonal ) matrices for j o dd ( resp., j even ) , R − 1 := e − π i 2 ( ν +1) 0 0 − ( p ∗ ( τ )) − 1 , R 0 := e − π i 2 ( ν +1) 0 − i √ 2 π p ∗ ( τ )Γ( ν +1) e − π i 2 ( ν +1) − ( p ∗ ( τ )) − 1 ! , R 1 := e 3 π i 2 ( ν +1) √ 2 π Γ( − ν ) e π i( ν +1) − i √ 2 π p ∗ ( τ )Γ( ν +1) e − π i 2 ( ν +1) − ( p ∗ ( τ )) − 1 ! , R 2 := e 3 π i 2 ( ν +1) √ 2 π Γ( − ν ) e π i( ν +1) 0 − ( p ∗ ( τ )) − 1 e − 2 π i( ν +1) ! , and Γ( · · · ) is the (Euler) g amma function [11] . The ab ove as y mptotic expansion can b e deduced from the a symptotics of the par ab olic-cylinder functions ( see, for example, [13]) . The dia gonal/o ff-diagonal structure of the matrices ψ j ( τ ) is a consequence of the “ σ 3 -reduction” for Eq uation (3.9 8): ψ 0 ( e Λ) → σ 3 ψ 0 ( − e Λ) . In Lemmata 3.4.1 a nd 3.4.2 ( see Subsec tion 3 .4 b elow ) , explicit knowledge of the following matrices is es sential: ψ 1 ( τ ) = 0 κ 0 ( τ ) 8 √ 3 − i( ν +1) κ 0 ( τ ) 0 ! , ψ 2 ( τ ) = − i( ν +1)( ν +2) 16 √ 3 0 0 i ν ( ν +1) 16 √ 3 ! , ψ 3 ( τ ) = 0 i ν ( ν − 1) κ 0 ( τ ) 384 − ( ν +1)( ν +2)( ν +3) 16 √ 3 κ 0 ( τ ) 0 ! . 3.4 Matc hing of Asymptotics In this subsection the connec tio n matrix (cf. Equa tion (1.11)), G , is c a lculated asymptotically in terms of the matrix elemen ts of the function A ( e µ , τ ) defined by Equation (3.7), that is, the functions ˆ r 0 ( τ ), ˆ u 0 ( τ ), h 0 ( τ ), and b ( τ ); thus, under conditions (3.12), the direct mo no dromy problem for Equation (3.6) is solved asymptotically . Prop ositio n 3.4.1. L et e µ = e µ 0 = α + τ − 1 / 3 e Λ , wher e e Λ = τ → + ∞ O ( τ ε ) , 0 < ε < 1 / 9 . Th en under c onditions (3.12) , with 0 < δ < ε < 1 / 9 and ε < δ 1 < 1 / 3 , ( F τ ( e Λ)) − 1 T ( e µ ) = e µ = e µ 0 τ → + ∞ 1 3 1 / 4 s κ 0 ( τ ) e Λ ! σ 3 − 1 8 (1 + O ( τ δ − 2 ε )) 1 + O ( τ δ − 2 ε ) − 1 2 ( + 1 )(1 + O ( τ 2 δ − 2 ε )) 4( − 1)(1 + O ( τ 2 δ − 2 ε )) × 1 α 3 / 2 s b ( τ ) ˆ r 0 ( τ ) ˆ u 0 ( τ ) ! σ 3 , (3.127) wher e is define d in Pr op osition 3.2.6 . Degenerate Thir d Painlev ´ e Equation: II 30 Pr o of . The pro of is a str aightforw ard, though algebraica lly tedious, consequence of Prop osi- tion 3.2.6 (cf. E quations (3.88)–(3.90)) and Lemma 3.3.1 (cf. E q uations (3 .9 5)–(3.97)). Mo re pre c is ely , using conditions (3.1 2), the conditio ns of Cor ollary 3 .2.3, Equations (3 .3 5) and (3.65), and rep eated application of E quation (3.77), one shows that ( F τ ( e Λ)) − 1 T ( α + τ − 1 / 3 e Λ) 11 = τ → + ∞ − 1 8 · · · 3 1 / 4 α √ α s ˆ r 0 ( τ ) b ( τ ) ˆ u 0 ( τ ) s κ 0 ( τ ) e Λ 1 + O ( h 0 ( τ ) e Λ − 2 ) , ( F τ ( e Λ)) − 1 T ( α + τ − 1 / 3 e Λ) 12 = τ → + ∞ α √ α 3 1 / 4 s ˆ u 0 ( τ ) ˆ r 0 ( τ ) b ( τ ) s κ 0 ( τ ) e Λ 1 + O ( h 0 ( τ ) e Λ − 2 ) , ( F τ ( e Λ)) − 1 T ( α + τ − 1 / 3 e Λ) 21 = τ → + ∞ − √ 3 ( + 1) 2 · · · 3 1 / 4 α √ α s b ( τ ) ˆ r 0 ( τ ) ˆ u 0 ( τ ) s e Λ κ 0 ( τ ) × 1 + O (4 ˆ u 0 ( τ ) + ˆ r 0 ( τ ) + 2 ( ˆ u 0 ( τ )) 2 ) h 0 ( τ ) ˆ r 0 ( τ )(1 + ˆ u 0 ( τ )) e Λ 2 !! , ( F τ ( e Λ)) − 1 T ( α + τ − 1 / 3 e Λ) 22 = τ → + ∞ 4 √ 3 ( − 1) α √ α 3 1 / 4 s e Λ κ 0 ( τ ) s ˆ u 0 ( τ ) b ( τ ) ˆ r 0 ( τ ) × 1 + O (4 ˆ u 0 ( τ ) + ˆ r 0 ( τ ) + 2 ( ˆ u 0 ( τ )) 2 ) h 0 ( τ ) ˆ r 0 ( τ )(1 + ˆ u 0 ( τ )) e Λ 2 !! ; hence, via conditions (3.12) and the conditions of Coro llary 3.2.3, one a rrives a t, after simplifica tion, Equation (3.127). Lemma 3. 4.1. L et e Ψ( e µ, τ ) b e the fundamental solution of Equation (3.6) with asymptotics given in L emma 3 .3.1 , and Y ∞ 0 ( e µ, τ ) b e t he c anonic al solution of Equation (3.1) . Define 11 L ∞ ( τ ) := e Ψ( e µ, τ ) − 1 τ − 1 12 σ 3 Y ∞ 0 ( τ − 1 6 e µ, τ ) . Assume that the p ar ameters ε , δ , δ 1 , and ν + 1 satisfy t he c onditions (3.92) and (3.9 3); furthermor e, let the function b ( τ ) satisfy the fol lowing c onditions: τ 1 3 Im( a ) ( ˆ u 0 ( τ )) 2 b ( τ )( ˆ r 0 ( τ )) 2 ! τ 6 ε + δ − δ 1 = τ → + ∞ O ( τ − ˆ δ 1 ) , b ( τ )( ˆ r 0 ( τ )) 2 τ 1 3 Im( a ) ( ˆ u 0 ( τ )) 2 τ 6 ε +4 ε Re( ν +1) − δ 1 = τ → + ∞ O ( τ − ˆ δ 2 ) , (3.128) τ 1 3 Im( a ) ( ˆ u 0 ( τ )) 2 b ( τ )( ˆ r 0 ( τ )) 2 ! τ − ε − 2 ε Re( ν +1) = τ → + ∞ O ( τ − ˆ δ 3 ) , b ( τ )( ˆ r 0 ( τ )) 2 τ 1 3 Im( a ) ( ˆ u 0 ( τ )) 2 τ 2 δ +2 ε Re ( ν +1) − 3 ε = τ → + ∞ O ( τ − ˆ δ 4 ) , (3.129) wher e ˆ δ k > 0 , k = 1 , 2 , 3 , 4 . Then L ∞ ( τ ) = τ → + ∞ i R − 1 0 exp t ∞ + c ∞ − I ♯ ∞ ( τ ) σ 3 s κ 0 ( τ ) ˆ u 0 ( τ ) b ( τ ) ˆ r 0 ( τ ) ! σ 3 σ 2 × I + O ( τ − ∞ δ 11 ) O ( τ − ∞ δ 12 ) O ( τ − ∞ δ 21 ) O ( τ − ∞ δ 22 ) !! , (3.130) wher e R 0 is define d in R emark 3.3.1 , I ♯ ∞ ( τ ) is define d by Equation (3.59) , t ∞ := i3( √ 3 − 1) α 2 τ 2 / 3 + 1 3 ν + 1 2 − i a 2 ln τ , (3.131) 11 Note that τ − 1 12 σ 3 Y ∞ 0 ( τ − 1 6 e µ , τ ) is also a fundamen tal s olution of Equat ion (3.6); therefore, L ∞ ( τ ) is indep enden t of e µ . Degenerate Thir d Painlev ´ e Equation: II 31 c ∞ := 3 2 ln α − 1 4 ln 3 + ( ν + 1 ) ln(e i π 4 2 3 / 2 3 1 / 4 ) + C WKB ∞ , (3.132) with C WKB ∞ define d by Equation (3.55) , and ∞ δ 11 = ∞ δ 22 := min { δ 1 − δ 2 − 6 ε − 2 ε Re( ν + 1) , 2( ε − δ ) , 1 3 − 3 ε } , ∞ δ 12 := min { ˆ δ 1 , ˆ δ 3 } , and ∞ δ 21 := min { ˆ δ 2 , ˆ δ 4 } . Pr o of . Denote by e Ψ WKB ( e µ , τ ) the s olution of E q uation (3.6) which has WK B asymptotics given by Equations (3.1 8)–(3.20) in the canonical domain con taining the Stokes curve approa ching the pos itive real e µ -a x is from ab ove as e µ → + ∞ . Now, re-write L ∞ ( τ ) in the following, equiv alent form: L ∞ ( τ ) = ( e Ψ( e µ , τ )) − 1 e Ψ WKB ( e µ, τ ) ( e Ψ WKB ( e µ, τ )) − 1 τ − 1 12 σ 3 Y ∞ 0 ( τ − 1 6 e µ, τ ) . Noting that e Ψ( e µ, τ ), e Ψ WKB ( e µ, τ ), and τ − 1 12 σ 3 Y ∞ 0 ( τ − 1 6 e µ, τ ) are all solutions of Equation (3.6) , it follows that they differ by right-hand, e µ -indep endent ma tr ix factors. Using this fact, one ev alua tes asymptotically (as τ → + ∞ ) each pair b y taking separate limits, that is, e µ → α and e µ → + ∞ , resp ectively; more precisely , ( e Ψ( e µ, τ )) − 1 e Ψ WKB ( e µ , τ ) = τ → + ∞ ( F τ ( e Λ)(I + O ( τ 6 ε +2 ε Re( ν +1)+ δ 2 − δ 1 )) ψ 0 ( e Λ)) − 1 T ( e µ )e WKB( e µ,τ ) | {z } e µ = e µ 0 = α + τ − 1 / 3 e Λ , e Λ ∼ τ ε , 0 <ε< 1 / 9 , arg( e Λ)=0 , ( e Ψ WKB ( e µ , τ )) − 1 τ − 1 12 σ 3 Y ∞ 0 ( τ − 1 6 e µ, τ ) = τ → + ∞ ( T ( e µ )e WKB( e µ,τ ) ) − 1 τ − 1 12 σ 3 Y ∞ 0 ( τ − 1 6 e µ, τ ) | {z } e µ →∞ , arg( e µ )=0 , where WKB( e µ, τ ) := − σ 3 i τ 2 / 3 R e µ e µ 0 l ( ξ ) d ξ − R e µ e µ 0 diag(( T ( ξ )) − 1 ∂ ξ T ( ξ )) d ξ . T o c alculate a symptotics of ( e Ψ( e µ , τ )) − 1 e Ψ WKB ( e µ, τ ), one us es Equa tion (3.12 7), with = +1 , a nd Equation (3.100) (in conjunction with Remark 3.3.1) to show that ( e Ψ( e µ , τ )) − 1 e Ψ WKB ( e µ, τ ) = τ → + ∞ I + ( ψ 0 ( e Λ)) − 1 O ( τ 6 ε +2 ε Re( ν +1)+ δ 2 − δ 1 ) ψ 0 ( e Λ) ( ψ 0 ( e Λ)) − 1 ( F τ ( e Λ)) − 1 T ( e µ ) = τ → + ∞ I + ( ψ 0 ( e Λ)) − 1 O ( τ 6 ε +2 ε Re( ν +1)+ δ 2 − δ 1 ) ψ 0 ( e Λ) R − 1 0 × exp − i2 √ 3 e Λ 2 − ( ν + 1) ln (e i π 4 2 3 / 2 3 1 / 4 e Λ) σ 3 × 1 − κ 0 ( τ ) 8 √ 3 e Λ 0 1 ! + 1 e Λ 0 0 i( ν +1) κ 0 ( τ ) 0 ! + 1 e Λ 2 i ν ( ν +1) 16 √ 3 0 0 − i( ν +1)( ν +2) 16 √ 3 ! + 1 e Λ 3 0 − i ν ( ν − 1) κ 0 ( τ ) 384 ( ν +1)( ν +2)( ν +3) 16 √ 3 κ 0 ( τ ) 0 ! + O 1 e Λ 4 O (1 ) 0 0 O (1 ) ! × 1 3 1 / 4 s κ 0 ( τ ) e Λ ! σ 3 − 1 8 1 − 1 0 + O ( τ δ − 2 ε ) O ( τ δ − 2 ε ) O ( τ 2 δ − 2 ε ) 0 × 1 α 3 / 2 s b ( τ ) ˆ r 0 ( τ ) ˆ u 0 ( τ ) ! σ 3 = τ → + ∞ I + ( ψ 0 ( e Λ)) − 1 O ( τ 6 ε +2 ε Re( ν +1)+ δ 2 − δ 1 ) ψ 0 ( e Λ) R − 1 0 × α 3 / 2 3 1 / 4 s κ 0 ( τ ) ˆ u 0 ( τ ) b ( τ ) ˆ r 0 ( τ ) e Λ ! σ 3 exp − i2 √ 3 e Λ 2 − ( ν + 1) ln(e i π 4 2 3 / 2 3 1 / 4 e Λ) σ 3 i σ 2 × I + τ − 2 ε O ( τ 2 δ ) O ˆ u 0 ( τ ) b ( τ ) ˆ r 0 ( τ ) O b ( τ ) ˆ r 0 ( τ ) τ 2 δ ˆ u 0 ( τ ) O ( τ δ ) . (3.133 ) F or the calculatio n of asymptotics for ( e Ψ WKB ( e µ, τ )) − 1 τ − 1 12 σ 3 Y ∞ 0 ( τ − 1 6 e µ, τ ), one uses the asymptotics of Y ∞ 0 ( τ − 1 6 e µ, τ ) given in Prop osition 1 .3, Equation (3.86), Eq uations (3.54) and (3 .5 5), Equations (3.58) and (3.59), a nd conditions (3 .93), to show that ( e Ψ WKB ( e µ , τ )) − 1 τ − 1 12 σ 3 Y ∞ 0 ( τ − 1 6 e µ, τ ) = τ → + ∞ e f W ∞ ( τ ) σ 3 I + O ( τ − 2( ε − δ ) ) + O ( τ − 1 3 +3 ε ) , (3.134) Degenerate Thir d Painlev ´ e Equation: II 32 where f W ∞ ( τ ) := τ → + ∞ − i3( √ 3 − 1 ) α 2 τ 2 / 3 − i2 √ 3 e Λ 2 + i a 6 ln τ + I ♯ ∞ ( τ ) − C WKB ∞ + i 2 √ 3 a − i 2 + h 0 ( τ ) α 2 − p τ 4 1 3 ln τ − ln e Λ . (3 .135) Hence, via expansions (3.1 33) and (3 .134), up on taking in to acco unt definition (3.91), and co ndi- tions (3.92) a nd (3.93), one arrives at L ∞ ( τ ) = τ → + ∞ i R − 1 0 exp t ∞ + c ∞ − I ♯ ∞ ( τ ) σ 3 s κ 0 ( τ ) ˆ u 0 ( τ ) b ( τ ) ˆ r 0 ( τ ) ! σ 3 σ 2 (I + E ∞ ( τ )) , where t ∞ and c ∞ are defined b y Eq uations (3.131) a nd (3.132), r esp ectively , and E ∞ ( τ ) = τ → + ∞ τ 6 ε +2 ε Re( ν +1)+ δ 2 − δ 1 O (1 ) O κ 0 ( τ )( ˆ u 0 ( τ )) 2 e η 1 ( τ ) b ( τ )( ˆ r 0 ( τ )) 2 O b ( τ )( ˆ r 0 ( τ )) 2 e − η 1 ( τ ) κ 0 ( τ )( ˆ u 0 ( τ )) 2 O (1 ) + τ − 2 ε O ( τ 2 δ ) O ( ˆ u 0 ( τ )) 2 τ ε e η 1 ( τ ) b ( τ )( ˆ r 0 ( τ )) 2 O ( ˆ r 0 ( τ )) 2 b ( τ ) τ 2 δ τ − ε e − η 1 ( τ ) ( ˆ u 0 ( τ )) 2 O ( τ δ ) + O ( τ − 2( ε − δ ) ) + O ( τ − 1 3 +3 ε ) O (1 ) o (1) o (1) O (1 ) , where η 1 ( τ ) := 1 3 Im( a ) ln τ − 2 ε Re( ν + 1) ln τ . Inv o king conditions ( 3.12 8) and (3.12 9), and condi- tions (3.92) a nd (3.93), one arrives at Equation (3.13 0). Lemma 3. 4.2. L et e Ψ( e µ, τ ) b e the fundamental solution of Equation (3.6) with asymptotics given in L emma 3 .3.1 , and X 0 0 ( e µ , τ ) b e the c anonic al solution of Equation (3.1) . Defin e 12 L 0 ( τ ) := e Ψ( e µ, τ ) − 1 τ − 1 12 σ 3 X 0 0 ( τ − 1 6 e µ, τ ) . Assume that the p ar ameters ε , δ , δ 1 , and ν + 1 satisfy t he c onditions (3.92) and (3.9 3); furthermor e, let 6 ε + δ − δ 1 < 0 , 6 ε + 4 ε Re( ν + 1) − δ 1 < 0 , − 3 ε + 2 ε Re( ν + 1 ) + 2 δ < 0 . (3.136) Then L 0 ( τ ) = τ → + ∞ − i R − 1 2 exp t 0 + c 0 + I ♯ 0 ( τ ) σ 3 s κ 0 ( τ ) ˆ r 0 ( τ ) ˆ u 0 ( τ ) ! σ 3 σ 2 × I + O ( τ − 0 δ 11 ) O ( τ − 0 δ 12 ) O ( τ − 0 δ 21 ) O ( τ − 0 δ 22 ) , (3.137) wher e R 2 is define d in R emark 3.3.1 , I ♯ 0 ( τ ) is define d by Equation (3.6 1) , t 0 := i3 √ 3 α 2 τ 2 / 3 + 1 3 ν + 1 2 ln τ , (3.138) c 0 := − 9 4 ln 2 − 1 4 ln 3 + ( ν + 1 ) ln(e i π 4 2 3 / 2 3 1 / 4 ) − C WKB 0 , (3.139) with C WKB 0 define d by Equation (3.57) , and 0 δ 11 = 0 δ 22 := min { δ 1 − δ 2 − 6 ε − 2 ε Re( ν + 1) , 2( ε − δ ) , 1 3 − 3 ε } , 0 δ 12 := min { δ 1 − δ − 6 ε, ε + 2 ε Re ( ν + 1) } , and 0 δ 21 := min { δ 1 − 6 ε − 4 ε Re ( ν + 1) , 3 ε − 2 δ − 2 ε Re ( ν + 1 ) } . 12 Note that τ − 1 12 σ 3 X 0 0 ( τ − 1 6 e µ, τ ) is also a fundamen tal solution of Equation (3.6); therefore, L 0 ( τ ) is indep enden t of e µ . Degenerate Thir d Painlev ´ e Equation: II 33 Pr o of . Denote by e Ψ WKB ( e µ , τ ) the s olution of E q uation (3.6) which has WK B asymptotics given by Equations (3.1 8)–(3.20) in the canonical domain con taining the Stokes curve approa ching the pos itive real e µ -a x is from ab ove as e µ → +0. Now, re-wr ite L 0 ( τ ) in the following, equiv alent form: L 0 ( τ ) = ( e Ψ( e µ, τ )) − 1 e Ψ WKB ( e µ , τ ) ( e Ψ WKB ( e µ, τ )) − 1 τ − 1 12 σ 3 X 0 0 ( τ − 1 6 e µ, τ ) . Noting that e Ψ( e µ, τ ), e Ψ WKB ( e µ , τ ), and τ − 1 12 σ 3 X 0 0 ( τ − 1 6 e µ, τ ) are all solutions of Equatio n (3.6), it follows that they differ by rig h t-hand, e µ -indep endent matr ix factor s. Using this fact, one ev aluates asymptot- ically (as τ → + ∞ ) ea ch pa ir by taking separate limits, that is, e µ → α and e µ → +0, respectively; more precisely , ( e Ψ( e µ, τ )) − 1 e Ψ WKB ( e µ , τ ) = τ → + ∞ ( F τ ( e Λ)(I + O ( τ 6 ε +2 ε Re( ν +1)+ δ 2 − δ 1 )) ψ 0 ( e Λ)) − 1 T ( e µ )e WKB( e µ,τ ) | {z } e µ = e µ 0 = α + τ − 1 / 3 e Λ , e Λ ∼ τ ε , 0 <ε< 1 / 9 , arg( e Λ)= π , ( e Ψ WKB ( e µ, τ )) − 1 τ − 1 12 σ 3 X 0 0 ( τ − 1 6 e µ, τ ) = τ → + ∞ ( T ( e µ )e WKB( e µ,τ ) ) − 1 τ − 1 12 σ 3 X 0 0 ( τ − 1 6 e µ, τ ) | {z } e µ → 0 , arg( e µ )=0 , where WKB( e µ, τ ) := − σ 3 i τ 2 / 3 R e µ e µ 0 l ( ξ ) d ξ − R e µ e µ 0 diag(( T ( ξ )) − 1 ∂ ξ T ( ξ )) d ξ . T o c alculate a symptotics of ( e Ψ( e µ , τ )) − 1 e Ψ WKB ( e µ, τ ), one us es Equa tion (3.12 7), with = − 1 , a nd Equation (3.100) (in conjunction with Remark 3.3.1) to show that ( e Ψ( e µ, τ )) − 1 e Ψ WKB ( e µ , τ ) = τ → + ∞ I + ( ψ 0 ( e Λ)) − 1 O ( τ 6 ε +2 ε Re( ν +1)+ δ 2 − δ 1 ) ψ 0 ( e Λ) ( ψ 0 ( e Λ)) − 1 ( F τ ( e Λ)) − 1 T ( e µ ) = τ → + ∞ I + ( ψ 0 ( e Λ)) − 1 O ( τ 6 ε +2 ε Re( ν +1)+ δ 2 − δ 1 ) ψ 0 ( e Λ) R − 1 2 × exp − i2 √ 3 e Λ 2 − ( ν + 1) ln (e i π 4 2 3 / 2 3 1 / 4 e Λ) σ 3 × 1 − κ 0 ( τ ) 8 √ 3 e Λ 0 1 ! + 1 e Λ 0 0 i( ν +1) κ 0 ( τ ) 0 ! + 1 e Λ 2 i ν ( ν +1) 16 √ 3 0 0 − i( ν +1)( ν +2) 16 √ 3 ! + 1 e Λ 3 0 − i ν ( ν − 1) κ 0 ( τ ) 384 ( ν +1)( ν +2)( ν +3) 16 √ 3 κ 0 ( τ ) 0 ! + O 1 e Λ 4 O (1 ) 0 0 O (1 ) ! × e − i π 2 3 1 / 4 s κ 0 ( τ ) e Λ ! σ 3 − 1 8 1 0 − 8 + O ( τ δ − 2 ε ) O ( τ δ − 2 ε ) 0 O ( τ 2 δ − 2 ε ) × 1 α 3 / 2 s b ( τ ) ˆ r 0 ( τ ) ˆ u 0 ( τ ) ! σ 3 = τ → + ∞ − I + ( ψ 0 ( e Λ)) − 1 O ( τ 6 ε +2 ε Re( ν +1)+ δ 2 − δ 1 ) ψ 0 ( e Λ) × R − 1 2 − i 8 · · · 3 1 / 4 α 3 / 2 s b ( τ ) κ 0 ( τ ) ˆ r 0 ( τ ) ˆ u 0 ( τ ) e Λ ! σ 3 × exp − i2 √ 3 e Λ 2 − ( ν + 1) ln (e i π 4 2 3 / 2 3 1 / 4 e Λ) σ 3 × I + τ − 2 ε O ( τ δ ) O ˆ u 0 ( τ ) τ 2 δ b ( τ ) ˆ r 0 ( τ ) O b ( τ ) ˆ r 0 ( τ ) ˆ u 0 ( τ ) O ( τ 2 δ ) . (3.140) F or the calculation o f asymptotics for ( e Ψ WKB ( e µ , τ )) − 1 τ − 1 12 σ 3 X 0 0 ( τ − 1 6 e µ, τ ), one uses the asymptotics of X 0 0 ( τ − 1 6 e µ, τ ) given in Pr o p o sition 1 .3, Equatio n (3.87), E quations (3.56) and (3.57), Equations (3.60) and (3.61), a nd conditions (3 .93), to show that ( e Ψ WKB ( e µ , τ )) − 1 τ − 1 12 σ 3 X 0 0 ( τ − 1 6 e µ, τ ) = τ → + ∞ e f W 0 ( τ ) σ 3 0 i2 3 / 4 α 3 / 2 √ b ( τ ) i √ b ( τ ) 2 3 / 4 α 3 / 2 0 × I + O ( τ − 2( ε − δ ) ) + O ( τ − 1 3 +3 ε ) , (3 .141) Degenerate Thir d Painlev ´ e Equation: II 34 where f W 0 ( τ ) := τ → + ∞ i3 √ 3 α 2 τ 2 / 3 + i2 √ 3 e Λ 2 + I ♯ 0 ( τ ) − C WKB 0 − i 2 √ 3 a − i 2 + h 0 ( τ ) α 2 − p τ 4 1 3 ln τ − ln e Λ . (3.142) Hence, via expansions (3.1 40) and (3 .141), up on taking in to acco unt definition (3.91), and co ndi- tions (3.92) a nd (3.93), one arrives at L 0 ( τ ) = τ → + ∞ − i R − 1 2 exp t 0 + c 0 + I ♯ 0 ( τ ) σ 3 s κ 0 ( τ ) ˆ r 0 ( τ ) ˆ u 0 ( τ ) ! σ 3 σ 2 (I + E 0 ( τ )) , where t 0 and c 0 are defined b y Eq uations (3.138) a nd (3.139), r esp ectively , and E 0 ( τ ) = τ → + ∞ τ 6 ε +2 ε Re( ν +1)+ δ 2 − δ 1 O (1 ) O ( κ 0 ( τ ) τ − 2 ε Re ( ν +1) ) O (( κ 0 ( τ )) − 1 τ 2 ε Re( ν +1) ) O (1) + τ − 2 ε O ( τ 2 δ ) O ( τ − 2 ε Re( ν +1 / 2) ) O ( τ 2 δ +2 ε Re ( ν +1 / 2) ) O ( τ δ ) + O ( τ − 2( ε − δ ) ) + O ( τ − 1 3 +3 ε ) O (1 ) o (1) o (1) O (1) . Inv o king conditions (3.13 6), and conditions (3.9 2) and (3.93), one arrives at Equation (3 .1 37). Theorem 3.4. 1. Assu ming c onditions (3.12) , (3.92) , (3.9 3) , (3.1 2 8) , (3.1 29) , and (3.136) , define δ 0 := min { 0 δ ij } i,j =1 , 2 . F u rthermor e, supp ose that τ 1 3 Im( a ) ( ˆ u 0 ( τ )) 2 b ( τ )( ˆ r 0 ( τ )) 2 ! τ − δ 0 = τ → + ∞ O ( τ − δ G 12 ) , b ( τ )( ˆ r 0 ( τ )) 2 τ 1 3 Im( a ) ( ˆ u 0 ( τ )) 2 τ − δ 0 = τ → + ∞ O ( τ − δ G 21 ) , (3.143 ) wher e δ G 12 , δ G 21 > 0 . L et δ G := min { δ 0 , δ G 12 , δ G 21 , ∞ δ 12 , ∞ δ 21 } . (3.144) Then the c onne ction matrix ( cf. Equation (1.11)) has t he fol lowing asymptotics: G = τ → + ∞ G ∞ (I + O ( τ − δ G )) , (3.145) wher e G ∞ := − ˆ r 0 ( τ ) √ b ( τ ) ˆ u 0 ( τ ) e ( z 0 ( τ ) − z ∞ ( τ )) e − 2 π i( ν +1) − e i π 4 2 3 / 2 3 1 / 4 √ 2 π √ b ( τ ) Γ( ν +1) e ( z 0 ( τ )+ z ∞ ( τ )) e − 2 π i( ν +1) e i π 4 √ 2 π √ b ( τ ) 2 3 / 2 3 1 / 4 Γ( − ν ) e − ( z 0 ( τ )+ z ∞ ( τ )) e i π ( ν +1) − ˆ u 0 ( τ ) ˆ r 0 ( τ ) √ b ( τ ) e − ( z 0 ( τ ) − z ∞ ( τ )) , (3.146) with z ∞ ( τ ) = t ∞ + c ∞ − I ♯ ∞ ( τ ) , (3.147) z 0 ( τ ) = t 0 + c 0 + I ♯ 0 ( τ ) , (3.148) wher e t ∞ , c ∞ , I ♯ ∞ ( τ ) , t 0 , c 0 , and I ♯ 0 ( τ ) ar e define d by Equations (3.131) , (3.13 2) , (3.5 9) , (3.1 38) , (3.139) , and (3.61) , r esp e ctively. Pr o of . Starting from the definition of the connection matrix (cf. E quation (1 .11)), one has the fol- lowing sequence of formulae: Y ∞ 0 ( µ ) := X 0 0 ( µ ) G ⇒ (cf. Equations (3.5)) Y ∞ 0 ( τ − 1 / 6 e µ ) = X 0 0 ( τ − 1 / 6 e µ ) G ⇒ τ − 1 12 σ 3 Y ∞ 0 ( τ − 1 / 6 e µ ) = τ − 1 12 σ 3 X 0 0 ( τ − 1 / 6 e µ ) G ⇒ (cf. Lemmata 3 .4.1 and 3.4.2) e Ψ( e µ, τ ) L ∞ ( τ ) = e Ψ( e µ, τ ) L 0 ( τ ) G ⇒ L ∞ ( τ ) = L 0 ( τ ) G ⇒ G = ( L 0 ( τ )) − 1 L ∞ ( τ ) . (3.149) Degenerate Thir d Painlev ´ e Equation: II 35 Using E quations (3.130)–(3.1 32), Eq ua tions (3.137)–(3.139), and the formulae for R 0 and R 2 given in Remark 3.3 .1, one ar rives at G = τ → + ∞ G ∞ (I + E G ( τ )) , where G ∞ is defined by E quations (3.146)–(3.1 48), and E G ( τ ) := τ → + ∞ O ( τ − ∞ δ 11 ) O ( τ − ∞ δ 12 ) O ( τ − ∞ δ 21 ) O ( τ − ∞ δ 22 ) ! + ( G ∞ ) − 1 O ( τ − 0 δ 11 ) O ( τ − 0 δ 12 ) O ( τ − 0 δ 21 ) O ( τ − 0 δ 22 ) G ∞ . The estimate for E G ( τ ) (cf. Equations (3.1 4 4) and (3 .1 45)) follows from the definitions of ∞ δ ij and 0 δ ij , i, j = 1 , 2, g iven in Lemmata 3.4 .1 and 3.4.2, r esp ectively , and conditions (3.1 4 3). 4 Asymptotic Solution of the In v erse Mono d rom y Problem In Section 3 the connection matr ix , G , for Equation (3.6) is c alculated asymptotically under cer tain conditions on its c o efficien t functions 13 . Using these conditions, one can derive the τ -depe nden t c la ss of functions to which G b elong s . W e are not, how ever, going to pr o ceed in this, most ge neral, way; rather, we will inv oke the isomono drom y condition on G , that is , its matrix elements, g ij , i, j = 1 , 2, are constants, and then inv ert the formula for G to obtain the coe fficien t functions of Equatio n (3.6 ) and verify tha t they sa tisfy a ll of the imp ose d co nditions for this iso mono dromy ca se. The latter pro cedur e pro duces explicit formulae for the co efficient functions of Equation (3 .6) , which give rise to asymptotics of the s o lution of the system of isomo no dromy defor mations (1.5) (see, als o, Eq uations (3 .5)) and, in tur n, de fine a s ymptotics o f the solution o f the dege ner ate third Painlev´ e equation (1.1) and the related functions H ( τ ) a nd f ( τ ) (cf. Equations (1 .2) a nd (1.3), resp ectively). Z e ro es a nd poles of the leading term of a s ymptotics of u ( τ ) define the c ent ers of the cheese-holes of the domain e D u (cf. definitions (2.10) , (3.2), (3.3), and (3.4)). W e a lso pr ov e that, for larg e enough τ , there is a one-to-o ne corres p o ndenc e b etw een the zero es and p oles of the solution u ( τ ) and the zer o es and p ole s o f its leading term of asymptotics , that is, in ea ch c heese-ho le there is lo c ated one, and only one, zero or po le of u ( τ ). Prop ositio n 4. 1. L et g ij , i, j = 1 , 2 , denote the matrix elements of G . Assu me that g 11 g 12 g 21 g 22 6 = 0 and the c onditions of The or em 3.4 .1 ar e valid . Then the fun ctions ˆ r 0 ( τ ) , ˆ u 0 ( τ ) , h 0 ( τ ) , κ 2 0 ( τ ) , and b ( τ ) have the fol lowing asymptotics: ˆ r 0 ( τ ) = τ → + ∞ 12 1 − i √ 2 co s( e ϑ ( τ )) = τ → + ∞ 6 co s ϑ 0 cos( 1 2 ( e ϑ ( τ ) + ϑ 0 )) cos( 1 2 ( e ϑ ( τ ) − ϑ 0 )) , ( 4.1) ˆ u 0 ( τ ) = τ → + ∞ − 3 2 co s 2 ( 1 2 ( e ϑ ( τ ) + ϑ 0 )) + O ( τ − 1 3 + δ − δ 1 ) = τ → + ∞ − sin 2 ϑ 0 cos 2 ( 1 2 ( e ϑ ( τ ) + ϑ 0 )) + O ( τ − 1 3 + δ − δ 1 ) , (4.2) h 0 ( τ ) = τ → + ∞ i2 √ 3 α 2 ( e ν + 1 ) − 1 2 + i 2 √ 3 a − i 2 − sin( e ϑ ( τ )) cos ϑ 0 2 √ 2 cos( 1 2 ( e ϑ ( τ ) + ϑ 0 )) cos( 1 2 ( e ϑ ( τ ) − ϑ 0 )) + O ( τ − δ G ) + O ( τ δ − 2 δ 1 ) , (4.3) κ 2 0 ( τ ) = τ → + ∞ i8 √ 3 ( e ν + 1 ) − 1 2 + i 2 √ 3 a − i 2 − sin( e ϑ ( τ )) cos ϑ 0 2 √ 2 cos( 1 2 ( e ϑ ( τ ) + ϑ 0 )) cos( 1 2 ( e ϑ ( τ ) − ϑ 0 )) ! + 4( a − i 2 ) cos 2 ( 1 2 ( e ϑ ( τ ) + ϑ 0 )) cos( 1 2 ( e ϑ ( τ ) + ϑ 0 ) + ϑ 0 ) cos( 1 2 ( e ϑ ( τ ) + ϑ 0 ) − ϑ 0 ) + O ( τ − δ G ) + O ( τ δ − 2 δ 1 ) + O ( τ − 2 3 +3 δ ) , (4.4) p b ( τ ) = τ → + ∞ cos( 1 2 ( e ϑ ( τ ) − ϑ 0 )) 4 co s( ϑ 0 ) cos( 1 2 ( e ϑ ( τ ) + ϑ 0 )) + O ( τ − 2 3 ) + O ( τ δ − 2 δ 1 ) ! exp(Φ( τ )) , (4.5) 13 See E quations (3.7)-(3.11) and (3.13), and conditions (3 .12), (3.92) , (3.93), (3.128), (3.129), (3.136), and (3.143). Degenerate Thir d Painlev ´ e Equation: II 36 wher e e ν + 1 := i 2 π ln( g 11 g 22 ) , 0 < Re( e ν + 1 ) < 1 , (4.6) e ϑ ( τ ) := 6 √ 3 α 2 τ 2 / 3 − i ( e ν + 1 ) − 1 2 ln(6 √ 3 α 2 τ 2 / 3 ) − i ( e ν + 1) − 1 2 ln 12 + a − i 2 ln(2 + √ 3) + i ln g 11 g 12 Γ( e ν + 1 ) √ 2 π − 3 π 2 ( e ν + 1 ) + 7 π 4 + O ( τ − δ G ln τ ) , (4.7) ϑ 0 := − π 2 + i 2 ln(2 + √ 3) , cos ϑ 0 = i √ 2 , sin ϑ 0 = − r 3 2 , (4.8) Φ( τ ) := − i3 α 2 τ 2 / 3 − i a 6 ln τ + i π ( e ν + 1 ) − 1 2 + ln g 11 + 2 ln(2 α ) + i a 2 ln α 2 2 − ( e ν + 1 ) − 1 2 ln(2 + √ 3) + O ( τ − δ G ) , (4.9) and δ G > 0 is define d by Equation (3.14 4) . Pr o of . Multiplying the diago na l element s of G (cf. Equations (3.145)–(3.148)) and taking into account co nditions (3.9 2) and definition (4.6), one shows that ν + 1 = τ → + ∞ ( e ν + 1 )(1 + O ( τ − δ G )) , (4.10) where δ G > 0 is defined b y Equation (3.144). Multiplying the element s of the first r ow o f G , one s hows, via Equation (4.10), that g 11 g 12 = τ → + ∞ exp i π 4 + 1 2 ln 2 π + ln(2 3 2 3 1 4 ) − ln Γ( e ν + 1) − 4 π i( e ν + 1) + ln ˆ r 0 ( τ ) ˆ u 0 ( τ ) + 2 z 0 ( τ ) + O ( τ − δ G ) , (4.11) where, via E quations (3.148), (3.139), (3.61), and (4 .10), 2 z 0 ( τ ) = τ → + ∞ 2 t 0 − 3 ln 2 − 1 2 ln 3 + 3 π i 2 + 2( e ν + 1 ) ln(e i π 4 2 3 2 3 1 4 ) + 2i a − i 2 ln √ 3 + 1 √ 2 ! + 2 ( e ν + 1 ) − 1 2 ln 3 α + 2 π i( e ν + 1) − ln ˆ r 0 ( τ ) ˆ u 0 ( τ ) − 1 2 ln Q ( τ ) + O ( τ − δ G ) , (4.1 2) with t 0 defined by Equation (3.13 8), and Q ( τ ) := ˆ r 0 ( τ ) − 4 √ 3 + p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ˆ r 0 ( τ ) − 4 √ 3 − p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ! ˆ r 0 ( τ ) + 4 √ 3 − p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ˆ r 0 ( τ ) + 4 √ 3 + p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 ! . (4.13) Substituting Equa tio ns (4.12) and (4.13) into Equation (4.11) a nd solving for Q ( τ ), one ar r ives a t 12 − ˆ r 0 ( τ ) − √ 3 p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 12 − ˆ r 0 ( τ ) + √ 3 p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 = τ → + ∞ exp(2i e ϑ ( τ )) , (4.14) where e ϑ ( τ ) is defined by Equation (4.7). Ther e ar e t wo solutions of E quation (4.14) for ˆ r 0 ( τ ). Define the pa rameter ϑ 0 by the firs t equation of (4.8); then, one of the solutions for ˆ r 0 ( τ ) is given in Equa- tion (4.1 ), whilst the other is given by the same formula as in Equation (4 .1 ) but with the ch ange e ϑ ( τ ) → e ϑ ( τ ) + π . This am biguity in the solution of the inv erse mono dromy problem ca n b e rectified by c a lculating one additional par ameter, namely , one of the Stokes m ultipliers 14 ; how ever, w e choos e a different approach. There is a common doma in of v a lidit y of the results obtained in this pap er a nd 14 This ambiguit y is distinct fr om the one for the dir ect m ono dromy problem mentione d in Remark 1.3. Degenerate Thir d Painlev ´ e Equation: II 37 Part I [1]. In App endix B, this fa c t is used in o rder to justify the c hoice fo r ˆ r 0 ( τ ) that is given in Equation (4.1). F urthermore , as a by-product o f Equations (4.14), (4.8), and (4 .1), one shows that p 32 + ( ˆ r 0 ( τ ) − 4 ) 2 = τ → + ∞ − 4 √ 6 sin e ϑ ( τ ) 1 − i √ 2 co s e ϑ ( τ ) = τ → + ∞ 2 sin( e ϑ ( τ )) sin 2 ϑ 0 cos( 1 2 ( e ϑ ( τ ) + ϑ 0 )) cos( 1 2 ( e ϑ ( τ ) − ϑ 0 )) . ( 4.15 ) Via Equations (3.77), (4.1), (4.8) , and (4.1 5), and conditions (3.12), one shows that ˆ u 0 ( τ ) = τ → + ∞ − 3 2 (i √ 2 (cos( ϑ 0 ) + cos e ϑ ( τ )) − √ 6 (sin( ϑ 0 ) + sin e ϑ ( τ ))) (cos( ϑ 0 ) + cos e ϑ ( τ )) 2 + O ( τ − 1 3 + δ − δ 1 ) . (4.16) Now, from Equations (4.8), (4.10), and (4.1 6), and the trigono metr ic identities sin( ϑ 0 ) + sin e ϑ ( τ ) = 2 sin( 1 2 ( ϑ 0 + e ϑ ( τ ))) cos( 1 2 ( ϑ 0 − e ϑ ( τ ))) and co s( ϑ 0 ) + cos e ϑ ( τ ) = 2 co s( 1 2 ( ϑ 0 + e ϑ ( τ ))) cos( 1 2 ( ϑ 0 − e ϑ ( τ ))), one arrives at E quation (4 .2). T o obtain h 0 ( τ ) (cf. Eq ua tion (4.3)), one uses Equations (3.68) , (3.91), (4.8), (4.10), and (4.15), and conditions (3.12). Finally , using Equa tions (3.65), (4.2), and (4.3), one arrives at Equation (4 .4) for κ 2 0 ( τ ). It follows from the (1 2)-element of G and E quation (4.10) that p b ( τ ) = τ → + ∞ exp 5 π i 4 + ln(2 3 2 3 1 4 ) − 2 π i( e ν + 1) + ln √ 2 π g 12 Γ( e ν + 1 ) ! + z 0 ( τ ) + z ∞ ( τ ) + O ( τ − δ G ) ! , ( 4.17 ) where z 0 ( τ ) is given in E q uations (4.12) and (4 .13), and (cf. Equation (3.1 47)) z ∞ ( τ ) = t ∞ + c ∞ − I ♯ ∞ ( τ ), with t ∞ , c ∞ , and I ♯ ∞ ( τ ) defined, resp e c tively , by E quations (3 .131), (3.132), and (3.59). Hence, using Equations (4.14) and (4.17), one s hows that p b ( τ ) = τ → + ∞ ˆ u 0 ( τ ) ˆ r 0 ( τ ) exp(Φ( τ )) , (4.18) where Φ( τ ) is defined by E quation (4.9), whence, v ia Equations (4.1 ) and (4.2), and conditions (3.12), one arr ives at E quation (4.5). Remark 4.1. It is imp orta n t to note that, as a consequence of Equa tion (4.18) , τ 1 6 Im( a ) ˆ u 0 ( τ ) p b ( τ ) ˆ r 0 ( τ ) = τ → + ∞ O (1 ); (4.19) therefore, conditions (3.14 3) are satisfie d; mo reov er, via conditions (3.1 36) and the es tima te (4.1 9) , it follows that co nditions (3.128) and (3.129) ar e a ls o s atisfied. Making the following choice for the parameters ε , δ , and δ 1 , 0 < 20 δ < 10 ε < δ 1 < 1 / 3 , one shows that conditions (3.9 3) and (3.136) are v alid uniformly for 0 < Re( e ν + 1) < 1 . F or the functions ˆ r 0 ( τ ) , ˆ u 0 ( τ ) , and h 0 ( τ ) defined, respectively , by E quations (4.1) , (4.2) , and (4.3) , conditio ns (3.12) are satisfied. Prop ositio n 4. 2. Under the c onditions of Pr op osition 4 .1 , t he functions a ( τ ) , b ( τ ) , c ( τ ) , and d ( τ ) , defining, via Equatio ns (3.5 ) , the solution of t he system of isomono dr omy deformations (1.5) , have the fol lowing asymptotic r epr esentations: p − a ( τ ) b ( τ ) = τ → + ∞ 2 α 4 cos( κ ( τ ) + ϑ 0 ) cos( κ ( τ ) − ϑ 0 ) cos 2 ( κ ( τ )) + O ( τ − 1 3 + δ − δ 1 ) , (4.2 0) a ( τ ) d ( τ ) = τ → + ∞ i4 α 6 cos( ϑ 0 ) cos( κ ( τ ) + ϑ 0 ) cos( κ ( τ ) − ϑ 0 ) cos 2 ( κ ( τ )) cos ϑ 0 + sin 2 ϑ 0 cos( κ ( τ )) cos( κ ( τ ) − ϑ 0 ) + O ( τ − 2 3 +2 δ ) + O ( τ − 1 3 + δ − δ 1 ) , (4.21) b ( τ ) c ( τ ) = τ → + ∞ − i4 α 6 cos( κ ( τ ) + ϑ 0 ) cos( κ ( τ ) − ϑ 0 ) cos 2 ( κ ( τ )) cos( ϑ 0 ) sin 2 ϑ 0 cos( κ ( τ )) cos( κ ( τ ) − ϑ 0 ) Degenerate Thir d Painlev ´ e Equation: II 38 + sin 2 ϑ 0 cos( κ ( τ ) + ϑ 0 ) cos( κ ( τ ) − ϑ 0 ) − cos 2 ϑ 0 + O ( τ − 2 3 +2 δ ) + O ( τ − 1 3 + δ − δ 1 ) , (4.22) c ( τ ) d ( τ ) = τ → + ∞ − 4 α 4 cos 2 ϑ 0 + cos ϑ 0 cos( κ ( τ )) cos( κ ( τ ) − ϑ 0 ) cos( ϑ 0 ) sin 2 ϑ 0 cos( κ ( τ )) cos( κ ( τ ) − ϑ 0 ) + sin 2 ϑ 0 cos( κ ( τ ) + ϑ 0 ) cos( κ ( τ ) − ϑ 0 ) − cos 2 ϑ 0 + O ( τ − 1 3 + δ − δ 1 ) , (4.23) wher e b ( τ ) is given in Equation (4.5) ( se e, also, Equation (4.9 )) , κ ( τ ) := ( e ϑ ( τ ) + ϑ 0 ) / 2 , and e ϑ ( τ ) , ϑ 0 , and δ G > 0 ar e define d, r esp e ctively, by Equations (4.7) , (4.8) , and (3.144) . Pr o of . If g ij , i, j = 1 , 2, a re τ -dep endent, then an y functions whose asymptotics are given by Equations (4 .1) –(4.5) satisfy conditions (3.1 2), (3.9 2), (3.93), (3.128), (3.129), (3.136), and (3 .143); therefore, one can now use the justification scheme suggested in [10] (see, also, [1 4]). Equa tions (4.20)– (4.23) are obta ined from E quations (3.11), (3.14), (3.15), and (3.17) v ia the dire c t substitution o f Equations (4.1) a nd (4.2). F rom Pro p o s ition 1.2, Equations (3.5), and Equatio n (3.11), one shows that u ( τ ), the solution of the degener ate third Painlev´ e equation (1.1), is g iven b y u ( τ ) = ǫτ 1 / 3 p − a ( τ ) b ( τ ) = ǫ ( ǫb ) 2 / 3 2 τ 1 / 3 (1 + ˆ u 0 ( τ )) , ǫ = ± 1 . (4.24) It was s hown in [1] that, in terms of the function h 0 ( τ ), the Hamiltonian (cf. Equation (1.2)), H ( τ ), corres p o nding to u ( τ ), is g iven by H ( τ ) = 3( ǫb ) 2 / 3 τ 1 / 3 − 4 h 0 ( τ ) τ − 1 / 3 + 1 2 τ a − i 2 2 . (4.25) Finally , the function f ( τ ) (cf. Equation (1.3)) can b e presented in terms of ˆ r 0 ( τ ) with the help of Equations (3.10), (3.8), (3.5), a nd (1.7): f ( τ ) = − i( ǫb ) 1 / 3 2 τ 2 / 3 1 − ˆ r 0 ( τ ) 2 − 1 2 i a + 1 2 . (4.26) Thu s, Prop osition 4 .1 implies the corr esp onding a symptotic res ults for the functions ˆ r 0 ( τ ), H ( τ ), a nd f ( τ ) stated in Theorem 2.1. It is imp or tant to note that asy mptotics of ˆ r 0 ( τ ) can also b e refo r mulated as as ymptotics o f the logarithmic deriv ative of u ( τ ). Prop ositio n 4 . 3. The le ading term of asymptotics of (ln u ( τ )) ′ c oincides with the lo garithmic deriva- tive of the le ading t erm of asymptotics of t he function u ( τ ) given in The or em 2.1 , Equation (2.4): d ln u ( τ ) d τ = τ →∞ e i πε 1 3 √ 3 ( − 1) ε 2 ( ǫb ) 1 / 3 τ − 1 / 3 cos ϑ 2 sin ϑ 2 sin ϑ 2 − ϑ 0 sin ϑ 2 + ϑ 0 + O ( τ − 2 3 + δ − δ 1 ) , (4.27) wher e ϑ = ϑ ( ε 1 , ε 2 , τ ) is defin e d by Equations (2.2) , (2.5) , and (2.6) , and ϑ 0 is given in Equation (2.7) . Pr o of . As in the main bo dy of the pa per , the c ase ε 1 = ε 2 = 0 is co nsidered in de ta il; in pa rticular, ϑ = ϑ (0 , 0 , τ ) = e ϑ ( τ ) + ϑ 0 − π : the r emaining ca ses a re proved a nalogous ly up on applying the transforma tion F ε 1 ,ε 2 int ro duced at the beg inning of Sec tion 2. F rom Eq uations (4.26) and (1.3 ), one s hows that u ′ ( τ ) u ( τ ) = i b u ( τ ) − 2i( ǫb ) 1 / 3 τ 1 / 3 1 − ˆ r 0 ( τ ) 2 . (4.28) Substituting into E quation (4.28) the a s ymptotics for u ( τ ) and ˆ r 0 ( τ ) given in Equations (2.4) and (4.1), resp ectively , and using the iden tity 2 sin ϑ 2 sin ϑ 2 − ϑ 0 = cos ϑ 0 − cos( ϑ − ϑ 0 ) , Degenerate Thir d Painlev ´ e Equation: II 39 one arr ives at u ′ ( τ ) u ( τ ) = τ → + ∞ i( ǫb ) 1 / 3 2 sin 3 ϑ 2 + 5 co s( ϑ 0 ) sin ϑ 2 + ϑ 0 + sin ϑ 2 + ϑ 0 cos( ϑ − ϑ 0 ) τ 1 / 3 sin ϑ 2 sin ϑ 2 − ϑ 0 sin ϑ 2 + ϑ 0 + O ( τ − 2 3 + δ − δ 1 ) . Consider, now, the num era tor of the ab ove fra ction. One uses the identities sin ϑ 2 + ϑ 0 = sin ϑ 2 cos ϑ 0 + cos ϑ 2 sin ϑ 0 , sin ϑ 2 + ϑ 0 cos( ϑ − ϑ 0 ) = 1 2 sin 3 ϑ 2 − 1 2 sin ϑ 2 − 2 ϑ 0 = 1 2 sin 3 ϑ 2 − 1 2 (2 cos 2 ϑ 0 − 1) sin ϑ 2 + sin( ϑ 0 ) cos( ϑ 0 ) cos ϑ 2 , to tr ansform the sec ond a nd third ter ms , re s pec tiv ely , o f the s umma nd: reg rouping, now, the ter ms in the n umerator , one transforms it into the for m 1 2 4 sin 3 ϑ 2 + sin 3 ϑ 2 + (8 co s 2 ϑ 0 + 1) sin ϑ 2 + 6 sin( ϑ 0 ) cos( ϑ 0 ) cos ϑ 2 = − i3 √ 3 cos ϑ 2 , where, in or der to g et the last equality , one us es the v a lues of sin ϑ 0 and cos ϑ 0 given in Equatio n (4 .8). Thu s, a s ymptotics (4.27) is prov ed. T o finish t he pro of, one ha s to co nfir m that the logar ithmic deriv ative of the leading term o f asymptotics (2.4) c o incides, mo dulo O ( τ − 2 3 + δ − δ 1 ), with the leading term o f a symptotics (4 .27). T aking the logarithmic deriv ative of asymptotics (2.4), neglecting terms that are O ( τ − 1 ), and assuming that the deriv atives o f the e r rors can also be neglected, one shows that √ 3 ( ǫb ) 1 / 3 τ 1 / 3 cos ϑ 2 − ϑ 0 sin ϑ 2 − ϑ 0 + cos ϑ 2 + ϑ 0 sin ϑ 2 + ϑ 0 − 2 co s ϑ 2 sin ϑ 2 ! = √ 3 ( ǫb ) 1 / 3 cos ϑ 2 τ 1 / 3 2 sin ϑ 2 sin ϑ 2 − ϑ 0 sin ϑ 2 + ϑ 0 − 2 sin ϑ 2 ! = √ 3 ( ǫb ) 1 / 3 cos ϑ 2 2 sin 2 ϑ 2 + cos( ϑ ) − cos(2 ϑ 0 ) τ 1 / 3 sin ϑ 2 sin ϑ 2 − ϑ 0 sin ϑ 2 + ϑ 0 . One completes the pro of up on noting that 2 sin 2 ( ϑ/ 2) + cos( ϑ ) − cos(2 ϑ 0 ) = 2 sin 2 ϑ 0 = 3. Consider, now, more ca r efully , the cheese-like domain e D u (cf. Equation (3.4) ). As follows fro m the leading term of asymptotics (2.4) for the c a se Re( e ν + 1) 6 = 1 / 2, whe r e e ν := e ν (0 , 0), the countable sets P δ u and Z δ u are empty , that is, e D u coincides with the strip D (cf. Equation (2.10)). In this ca se, δ is v acuous: strictly sp eaking, one can improv e some of the error estimates in Section 3 and this section by a ssuming, in lieu of conditions (3.12), the fo llowing conditions for the co e fficie nts: ˆ r 0 ( τ ) ∼ τ → + ∞ O ( τ − 1 3 + δ ) , ˆ u 0 ( τ ) ∼ τ → + ∞ O ( τ − 1 3 + δ ) , ˆ h 0 ( τ ) ∼ τ → + ∞ O (1 ) . T o eschew this, we ca n formally substitute δ = 0 in our calculatio ns ; surely , thoug h, we will no t a chieve the b est p os sible erro r estimates, but they are sufficient for our pur po ses. Consider the definition o f δ G given in Equa tion (3.14 4), where δ G 12 = δ G 21 = δ 0 (cf. E quations (3.143) and (4 .1 9)). Put δ = 0 in the fo r mulae for ∞ δ 12 and ∞ δ 21 , and δ 0 defined, respectively , in Lemma 3.4 .1, and Theo rem 3.4.1 and Lemma 3.4.2. Analyzing the resulting system for δ G , one has to choose ε ∈ (0 , 1 / 9) so that δ G attains its maximum; the result re ads: (i) for Re( e ν + 1) ∈ (0 , 1 / 2 ) and ε = δ 1 / (7 + 6Re( e ν + 1)), 0 < δ G < δ 1 1 + 2Re ( e ν + 1) 7 + 6Re ( e ν + 1) ; (ii) for Re( e ν + 1) ∈ (1 / 2 , 1 ) and ε = δ 1 / (9 + 2Re( e ν + 1)), 0 < δ G < δ 1 3 − 2Re ( e ν + 1) 9 + 2Re ( e ν + 1) . Degenerate Thir d Painlev ´ e Equation: II 40 T ogether with the justification scheme o f [10], this completes the pro of of Theo r em 2.1. The structure of the cheese-like domain e D u for Re( e ν + 1 ) = 1 / 2 is more complicated: to study this case, it is conv enient to in tro duce the following definition. Definition 4.1. Define ˆ ϑ as the le ading term of asymptotics of ϑ ( cf. Equation (2.5)) , t hat is, ϑ = ˆ ϑ + O ( τ − δ G ln τ ) . Prop ositio n 4.4. Supp ose t hat g 11 ( ε 1 , ε 2 ) g 12 ( ε 1 , ε 2 ) g 21 ( ε 1 , ε 2 ) g 22 ( ε 1 , ε 2 ) 6 = 0 , Re i 2 π ln( g 11 ( ε 1 , ε 2 ) g 22 ( ε 1 , ε 2 )) = 1 2 , and the br anch of t he function ln( · · · ) is chosen so t hat Im(ln( − g 11 ( ε 1 , ε 2 ) g 22 ( ε 1 , ε 2 ))) = 0 . F or m ∈ N , c onsider t he fol lowing t r ansc endental e quations: ˆ ϑ 2 = π m, ˆ ϑ 2 + ϑ 0 = π m, ˆ ϑ 2 − ϑ 0 = π m. (4.29) Ther e exist unique solutions of these e quations, τ ∞ as,m , τ + as,m , and τ − as,m , r esp e ctively 15 , the signs of t he r e al p arts of which c orr esp ond to ( − 1) ε 1 ; furthermor e, τ ∞ as,m = m →∞ e i π ε 1 2 π ( − 1) ε 2 m 3 √ 3 ( ǫb ) 1 / 3 3 / 2 1 − 3 1 ( ε 1 , ε 2 ) 4 π ln m m − 3 2 ( ε 1 , ε 2 ) 4 π 1 m + O ln 2 m m 2 , (4.30 ) τ ± as,m = m →∞ e i π ε 1 2 π ( − 1) ε 2 m 3 √ 3 ( ǫb ) 1 / 3 3 / 2 1 − 3 1 ( ε 1 , ε 2 ) 4 π ln m m − 3 4 π ( 2 ( ε 1 , ε 2 ) ± 2 ϑ 0 ) 1 m + O ln 2 m m 2 , (4.31) wher e 1 ( ε 1 , ε 2 ) := 1 2 π ln( − g 11 ( ε 1 , ε 2 ) g 22 ( ε 1 , ε 2 )) ( ∈ R ) , (4.32) 2 ( ε 1 , ε 2 ) := 1 ( ε 1 , ε 2 ) ln(24 π ) + ( − 1 ) ε 2 Re( a ) ln(2 + √ 3) + π 2 − 1 2 arg g 11 ( ε 1 , ε 2 ) g 12 ( ε 1 , ε 2 ) g 21 ( ε 1 , ε 2 ) g 22 ( ε 1 , ε 2 ) − arg Γ 1 2 + i 1 ( ε 1 , ε 2 ) + i ( − 1) ε 2 Im( a ) ln(2 + √ 3) + 1 2 ln g 11 ( ε 1 , ε 2 ) g 12 ( ε 1 , ε 2 ) g 21 ( ε 1 , ε 2 ) g 22 ( ε 1 , ε 2 ) , (4.33) and ϑ 0 is given in Equation (2 .7) . Pr o of . F o llows fr o m a s uc c essive a pproximations a rgument applied to Equatio ns (4.29). Prop ositio n 4 . 5. L et the c onditions on the mono dr omy data b e the same as those in Pr op osition 4.4 . Supp ose that 0 < 13 δ < δ 1 < 1 / 3 . Then 0 < δ < δ G < δ 1 − 8 δ 5 . Pr o of . Cons ider the definition of δ G given in Equa tion (3.1 4 4), where δ 0 is defined in Theor em 3.4 .1 and Le mma 3.4.2, ∞ δ 12 and ∞ δ 21 are defined in Lemma 3 .4.1, and δ G 12 = δ G 21 = δ 0 (cf. Equations (3.14 3) and (4.19)). Under the assumptions s tated in the Prop o s ition, one ca n choos e ε which app ear s in these equations to s atisfy the inequa lities 3 δ 2 < ε < δ 1 + 2 δ 10 ; then δ G = 2( ε − δ ). This completes the pro of o f Theorem 2.3 modulo the distribution of the cheese-holes in e D u . 15 τ ∞ as,m , τ + as,m , and τ − as,m are the p oles and zero es of th e leading term of asymptotics of u ( τ ) (cf. Equation (2.4)). Degenerate Thir d Painlev ´ e Equation: II 41 Theorem 4.1. F or any given solution u ( τ ) with the mono dr omy da ta satisfying the c onditions (2.11) , c onsider the che ese-like domain D u ( cf. R emark 2.4) . Ther e exists T > 0 su ch that, ∀ τ ± as,m , τ ∞ as,m with | τ ± as,m | > T and | τ ∞ as,m | > T in e ach che ese-hole c enter e d at τ ± as,m ( or τ ∞ as,m ) , ther e ex ist s one, and only one, zer o τ ± m ( or p ole τ ∞ m ) of u ( τ ) lo c ate d in this che ese-hole: | τ ± as,m − τ ± m | < C | τ ± as,m | 1 3 − δ , | τ ∞ as,m − τ ∞ m | < C | τ ∞ as,m | 1 3 − δ , wher e C > 0 . Pr o of . As in the main bo dy of the pa per , the c ase ε 1 = ε 2 = 0 is co nsidered in de ta il; in pa rticular, ϑ = ϑ (0 , 0 , τ ) = e ϑ ( τ ) + ϑ 0 − π : the re ma ining cases are prov ed ana logously up on applying the transfor- mation F ε 1 ,ε 2 int ro duced at the b eginning of Section 2. The stra tegy here is to integrate b oth sides of Equatio ns (4.27) and (2.1 8), and the equation resulting up o n multiplying Eq uation (2.19) by the factor 2 τ − 1 , around the b o unda ry of a cheese-hole centered at τ ∗ = τ ± as,m (or τ ∞ as,m ). Hereafter , the symbol H means that the ab ov e-mentioned integration is taken in the count er-c lo ckwise se nse. Denote by n + , n − , and n p , res p ectively , the num b e r s o f zero es of types τ + , τ − , and p oles τ p of a solution u ( τ ) inside the cheese-hole ar ound whos e b o undary the in tegr a tion is p erformed. The integrals on the left-hand sides of Eq uations (4 .27) a nd (2 .18), a nd the equa tion res ulting upon m ultiplying Equation (2.19) b y the factor 2 τ − 1 , can b e ca lculated via the Cauch y Residue Theor em; the r esults, resp ectively , read: 2 π i( n + + n − − 2 n p ) , 2 π i( n − + n p ) , 2 π i( n − − n p ) . (4.3 4) T o integrate the right-hand side s , note that the er ror estimates in the ab ove-men tioned equations are all of the form O ( τ − 1 3 − υ ), for s o me υ > 0; therefore, I O 1 τ 1 3 + υ d τ ∼ τ →∞ O C τ υ + δ ∗ → τ ∗ →∞ 0 . Thu s, for so me T > 0 , the in tegra l is less than 2 π / 3. The leading terms of the right-hand s ide s o f Equations (4 .2 7) a nd (2.18), and the equation resulting upo n multiplying Equation (2.1 9) by the factor 2 τ − 1 , are meromorphic functions in the strip D , where D is defined b y Equa tion (2.10); ther efore, o ne can also calculate H by using the Ca uch y Residue Theo rem 16 . T o do so, notice that the non-trivial pa rt of the leading terms (non-v a nishing under int egr ation) has the for m τ − 1 / 3 F ( ϑ ), where the function F is expresse d in terms o f trigonometric functions: I 2 √ 3 ( ǫb ) 1 / 3 τ − 1 / 3 F ( ϑ ) d τ = I ˆ ϑ ′ F ( ϑ ) d τ + O I F ( ϑ ) τ d τ = I ˆ ϑ ′ F ( ˆ ϑ ) d τ + O I ˆ ϑ ′ max( | F ′ ( ˆ ϑ ) | ) τ − δ G ln τ d τ + O I F ( ϑ ) τ d τ = I F ( ˆ ϑ ) d ˆ ϑ + O ( τ δ − δ G ∗ ln τ ∗ ) + O ( τ − 2 / 3 ∗ ) , (4.35) where us e has b een made of the fact that δ < δ G , max( | F ′ ( ˆ ϑ ) | ) = O ( τ 2 δ ∗ ), a nd F ( ϑ ) = O ( τ δ ∗ ) in the annular reg ion of width O ( τ − δ G ln τ ) around the circle of integration. Again, for τ ∗ > T , each estimate in Equation (4.3 5) is less than 2 π / 3; th us, for large enough τ ∗ , the mo dulus of the con tribution of the error estimates to H is less than 2 π . (Their contribution is actually equal to zer o, since the r esults of the integration of the right-hand sides should coincide with the r e sults of the integration of the corres p o nding left-hand sides, and, acco rding to Equations (4.34), should b e equal b e 2 π i m ultipled by an integer.) Now, calcula ting the integrals H F ( ˆ ϑ ) d ˆ ϑ in Equations (4.27) and (2.18), and the equation resulting up on mult iplying Equation (2.19) by the factor 2 τ − 1 , one arr ives at the following systems: (i) for τ ∗ = τ ∞ as,m , 2 π i( n + + n − − 2 n p ) = − 4 π i , 2 π i( n − + n p ) = 2 π i , 2 π i( n − − n p ) = − 2 π i; (ii) for τ ∗ = τ + as,m , 2 π i( n + + n − − 2 n p ) = 2 π i , 2 π i( n − + n p ) = 0 , 2 π i( n − − n p ) = 0; 16 It follows from the deriv ation in Section 3 and this section that the error estimate O ( τ − δ G ln τ ) in the “phase”, ϑ , is a meromorphic function in D ; this fact, how eve r, is not used i n the pro of. Degenerate Thir d Painlev ´ e Equation: II 42 and (iii) for τ ∗ = τ − as,m , 2 π i( n + + n − − 2 n p ) = 2 π i , 2 π i( n − + n p ) = 2 π i , 2 π i( n − − n p ) = 2 π i . One completes the pro of up on solving these sy s tems. Corollary 4.1. Define ˆ τ ∞ as,m and ˆ τ ± as,m , r esp e ctively, as the le ading terms of asymptotics of τ ∞ as,m and τ ± as,m ( cf. Pr op osition 4.4) , t hat is, τ ∞ as,m =: ˆ τ ∞ as,m + O ln 2 m √ m , τ ± as,m =: ˆ τ ± as,m + O ln 2 m √ m . Under the c onditions of The or em 4.1 , τ ∞ m = m →∞ ˆ τ ∞ as,m + O m 1 2 − 3 δ 2 , τ ± m = m →∞ ˆ τ ± as,m + O m 1 2 − 3 δ 2 , wher e 0 < δ < 1 / 3 9 . Pr o of . As follows from P r op osition 4.5, 0 < δ < 1 / 39; th us, ln 2 m/ √ m ≪ m 1 2 − 3 δ 2 as m → ∞ . This completes the pro o f of Theorem 2.2, which is equiv a lent to the sp ecification of the cheese-holes in e D u . Ac kno wledgmen ts Ac kno wledgmen ts Ac kno wledgmen ts The author s were suppo rted, in part, by a College of Charlesto n (Co fC) Mathema tics Department Summer Research Aw ar d. Degenerate Thir d Painlev ´ e Equation: II 43 App endix A: P oles and Zero es In this Appendix so me bas ic information concerning the poles a nd zer o es of the solution u ( τ ) of Equation (1.1) and the asso c iated functions H ( τ ) and f ( τ ) (cf. Equations (1.2) a nd (1.3), res p ectively) are presented. Let τ ∞ be a p ole of u ( τ ); then, its La urent expansio n reads u ( τ ) = − τ ∞ 4 ǫ ( τ − τ ∞ ) 2 + a 0 + ∞ X k =1 a k ( τ − τ ∞ ) k , ǫ = ± 1 , (A.1) where a 0 is a parameter , and the rema ining co efficients, a k , ar e recursively and uniquely defined; for example, a 1 = − a 0 τ ∞ , a 2 = ab 5 τ ∞ − 12 ǫa 2 0 5 τ ∞ + 9 a 0 10 τ 2 ∞ , a 3 = − 8 ab 45 τ 2 ∞ + 24 ǫa 2 0 5 τ 2 ∞ − 4 a 0 5 τ 3 ∞ , a 4 = − ǫb 2 7 τ ∞ + 32 a 3 0 7 τ 2 ∞ − 4 ǫa 0 ab 7 τ 2 ∞ + 10 ab 63 τ 3 ∞ − 47 ǫa 2 0 7 τ 3 ∞ + 5 a 0 7 τ 4 ∞ , a 5 = ǫb 2 35 τ 2 ∞ − 96 a 3 0 7 τ 3 ∞ + 124 ǫa 0 ab 105 τ 3 ∞ − ab 7 τ 4 ∞ + 57 ǫa 2 0 7 τ 4 ∞ − 9 a 0 14 τ 5 ∞ . The Laurent expansion of the asso ciated Hamiltonia n, H ( τ ), reads H ( τ ) − 1 2 τ a − i 2 2 = 1 τ − τ ∞ + H 0 + ∞ X k =1 H k ( τ − τ ∞ ) k , (A.2) where the c o efficien ts, H k , can b e uniquely deter mined via Equa tio n (37) of [1]; for example, H 0 = 12 ǫa 0 , H 1 = − 8 ǫa 0 τ ∞ , H 2 = 2i ǫb τ ∞ + 6 ǫa 0 τ 2 ∞ , H 3 = − 16 aǫb 15 τ 2 ∞ − 16 a 2 0 5 τ 2 ∞ − 24 ǫa 0 5 τ 3 ∞ , H 4 = 8i ba 0 τ 2 ∞ + 8 aǫb 9 τ 3 ∞ + 8 a 2 0 τ 3 ∞ + 4 ǫa 0 τ 4 ∞ , H 5 = 52 b 2 35 τ 2 ∞ − 128 aba 0 35 τ 3 ∞ + 128 ǫa 3 0 35 τ 3 ∞ − 8i ba 0 τ 3 ∞ − 16 aǫb 21 τ 4 ∞ − 468 a 2 0 35 τ 4 ∞ − 24 ǫa 0 7 τ 5 ∞ . The function f ( τ ) also has a first-orde r p ole at τ = τ ∞ : f ( τ ) = − τ ∞ 2( τ − τ ∞ ) + f 0 + ∞ X k =1 f k ( τ − τ ∞ ) k , (A.3) where the c o efficien ts, f k , can b e uniquely determined via Equatio n (1.3); for example, f 0 = − i a 2 − 3 4 , f 1 = − 2 a 0 ǫ, f 2 = i ǫb + a 0 ǫ τ ∞ , f 3 = 8 a 2 0 5 τ ∞ − 4 aǫb 5 τ ∞ + i ǫb τ ∞ − 3 a 0 ǫ 5 τ 2 ∞ , f 4 = 4i a 0 b τ ∞ − 12 a 2 0 5 τ 2 ∞ + 4 aǫb 45 τ 2 ∞ + 2 a 0 ǫ 5 τ 3 ∞ , f 5 = 6 b 2 7 τ ∞ − 64 ǫa 3 0 35 τ 2 ∞ − 48 aba 0 35 τ 2 ∞ − 4 aǫb 63 τ 3 ∞ + 94 a 2 0 35 τ 3 ∞ − 2 ǫa 0 7 τ 4 ∞ . There are t wo types o f first-o rder zero es of u ( τ ), denoted τ s , s = ± 1, which differ by their T aylor expansions: u ( τ ) = ∞ X k =1 b s k ( τ − τ s ) k , (A.4) where, for ex ample, b s 1 = i sb, b s 2 = − b τ s a − i s 2 , b s 3 ∈ C , b s 4 = 1 2 τ s 4 ǫb 2 + i sab s 3 − b s 3 , Degenerate Thir d Painlev ´ e Equation: II 44 b s 5 = 1 20 bτ 2 s 7 bb s 3 − 8 b 3 ǫ − 6i s ( b s 3 ) 2 τ 2 s + 48i sab 3 ǫ − 10i sabb s 3 . These t wo types of first-order zero es are “analytically” distinguished b y the Hamiltonian function; indeed, at τ = τ + , the function H ( τ ) is holo morphic, H ( τ ) = H + 0 + ∞ X k =1 H + k ( τ − τ + ) k , where the fir st few co efficients are H + 0 = − 3i b + 3 τ + 2 b + 1 2 τ + a − i 2 2 + i τ + a − i 2 , H + 1 = 1 2 τ 2 + a − i 2 2 , H + 2 = − 3 b + 3 2 bτ + a − i 2 + 1 2 τ 3 + a − i 2 2 − i τ 3 + a − i 2 3 , while, at τ = τ − , the function H ( τ ) has a firs t-order p ole, H ( τ ) − 1 2 τ a − i 2 2 = 1 τ − τ − + H − 0 + ∞ X k =1 H − k ( τ − τ − ) k , (A.5) where, for ex ample, H − 0 = 3i b − 3 τ − 2 b , H − 1 = − i b − 3 b , H − 2 = − 2i ǫb τ − + 3i b − 3 4 bτ − . The zero es o f u ( τ ) ar e also distinguished by the function f ( τ ): f ( τ ) = ∞ X k =1 f + k ( τ − τ + ) k , (A.6) where the fir st tw o co efficients are f + 1 = 4 a 2 + 1 8 τ + − 3i b + 3 τ + 4 b , f + 2 = − 2i ǫb − 3 b + 3 4 b a − i 2 − i(4 a 2 + 1) 8 τ 2 + a − i 2 , and f ( τ ) = τ − 2( τ − τ − ) + f − 0 + ∞ X k =1 f − k ( τ − τ − ) k , (A.7) where, for ex ample, f − 0 = − i 2 a + i 2 , f − 1 = i b − 3 τ − 4 b , f − 2 = i ǫb − i b − 3 8 b . As follows fro m the definitions of the asso ciated functions H ( τ ) and f ( τ ), they have p oles only at the p oles a nd τ − -zero es of the function u ( τ ). App endix B: Comparison of Asymptotic Results The ranges of the v alidity fo r asymptotics of u ( τ ) o btained in Part I [1] o f our studies a nd in this pap er o verlap: this enables us to r esolve the ambiguit y in the choice of sign for the function ˆ r 0 ( τ ) discussed in the pro of of P rop osition 4.1. The asymptotics of u ( τ ) for | Re ( e ν ( ε 1 , ε 2 ) + 1) | < 1 / 6, ε 1 , ε 2 = 0 , ± 1, are obtained in Theo rem 3.1 of [1], whilst the a symptotics of u ( τ ) stated in Theor em 2.1 of this pap er ar e applicable for Re( e ν ( ε 1 , ε 2 ) + 1) ∈ (0 , 1) \ { 1 / 2 } . In fact, if o ne is concer ne d only with the leading exponent o f cosh( · · · ) in the asymptotics of u ( τ ) presen ted in Theor e m 3.1 of [1], then the rang e of its v alidity can b e extended to | Re( e ν ( ε 1 , ε 2 ) + 1 ) | < 1 / 2. Analo g ously , only the leading exp onent in the expansion of sin − 2 ( · · · ) in the asymptotics of u ( τ ) o btained in Theorem 2.1 of this pap er is “la rger” than the err or correc tio n term Degenerate Thir d Painlev ´ e Equation: II 45 in ca se Re( e ν ( ε 1 , ε 2 ) + 1) > 0. There fo re, the cor resp onding leading e xpo nents of the asymptotics of u ( τ ) in Theor em 3.1 of [1] and Theorem 2.1 of this pap er should co inc ide , provided 0 < Re( e ν ( ε 1 , ε 2 ) + 1) < 1 2 . (B.1) Recall the ma in asymptotic (as τ → ∞ ) result for u ( τ ) obta ined in Part I [1] 17 : Theorem B.1 (Theorem 3 .1 [1]) . L et ε 1 , ε 2 = 0 , ± 1 , ǫb = | ǫb | e i π ε 2 , and u ( τ ) b e a solution of the de gen- er ate thir d Painlev ´ e e quation (1.1) c orr esp onding to t he mono dr omy data ( a, s 0 0 , s ∞ 0 , s ∞ 1 , g 11 , g 12 , g 21 , g 22 ) . Supp ose that g 11 ( ε 1 , ε 2 ) g 12 ( ε 1 , ε 2 ) g 21 ( ε 1 , ε 2 ) g 22 ( ε 1 , ε 2 ) 6 = 0 , Re i 2 π ln( g 11 ( ε 1 , ε 2 ) g 22 ( ε 1 , ε 2 )) < 1 6 . Then, ∃ δ > 0 such that u ( τ ) has the asymptotic exp ansion u ( τ ) = τ →∞ e i πε 1 ( − 1) ε 1 ǫ p | ǫb | 3 1 / 4 r ϑ ( τ ) 12 + p e ν ( ε 1 , ε 2 ) + 1 e 3 π i 4 cosh i ϑ ( τ ) + ( e ν ( ε 1 , ε 2 ) + 1) ln ϑ ( τ ) + z ( ε 1 , ε 2 ) + o ( τ − δ ) ! , (B.2) wher e ϑ ( τ ) := 3 √ 3 | ǫb | 1 / 3 | τ | 2 / 3 , e ν ( ε 1 , ε 2 ) + 1 := i 2 π ln( g 11 ( ε 1 , ε 2 ) g 22 ( ε 1 , ε 2 )) , z ( ε 1 , ε 2 ) := 1 2 π ln 2 π − i π 2 − 3 π i 2 ( e ν ( ε 1 , ε 2 ) + 1) + ( − 1) ε 2 i a ln(2 + √ 3) + ( e ν ( ε 1 , ε 2 ) + 1) ln 12 − ln ω ( ε 1 , ε 2 ) p e ν ( ε 1 , ε 2 ) + 1 Γ( e ν ( ε 1 , ε 2 ) + 1) , with ω ( ε 1 , ε 2 ) := g 12 ( ε 1 , ε 2 ) g 22 ( ε 1 , ε 2 ) , (B.3) and Γ( · · · ) is the gamma fun ction. L et H ( τ ) b e the Hamiltonian fun ction define d in Equation (1.2 ) c orr esp onding t o the function u ( τ ) given ab ove. Then, H ( τ ) − 1 2 τ a − ( − 1) ε 2 i 2 2 = τ →∞ e i πε 1 3( ǫb ) 2 / 3 τ 1 / 3 + 2 | ǫb | 1 / 3 τ − 1 / 3 a − ( − 1) ε 2 i 2 − i2 √ 3 ( e ν ( ε 1 , ε 2 ) + 1) + o ( τ − δ ) . Now, taking in to acco unt co ndition (B.1), o ne shows that asymptotics (B.2) implies u ( τ ) = τ →∞ e i πε 1 ( − 1) ε 2 ǫ | ǫb | 2 / 3 2 | τ | 1 / 3 + ( − 1) ε 1 ǫ p | ǫb | 2 · 3 1 / 4 | τ | 2 3 Re( e ν ( ε 1 ,ε 2 )+1) e Θ( τ ) + O ( τ − 2 3 Re( e ν ( ε 1 ,ε 2 )+1) ) + o ( τ 2 3 Re( e ν ( ε 1 ,ε 2 )+1) − δ ) , (B.4) where Θ( τ ) := i3 √ 3 | ǫb | 1 / 3 | τ | 2 / 3 + iIm( e ν ( ε 1 , ε 2 ) + 1) ln | τ | 2 / 3 + ( e ν ( ε 1 , ε 2 ) + 1) ln(36 √ 3 | ǫb | 1 / 3 ) + ( − 1) ε 2 i a ln(2 + √ 3) + i π 4 − 3 π i 2 ( e ν ( ε 1 , ε 2 ) + 1) − ln ω ( ε 1 , ε 2 )Γ( e ν ( ε 1 , ε 2 ) + 1) √ 2 π . (B.5) Using the Euler formula for sin( · · · ) to ex pa nd as ymptotics (2.3) (assuming condition (B.1)) in Theo- rem 2.1 o f this work, one ar rives, aga in, at asymptotics (B.4), but with the err or corr e c tion O ( τ 2 3 Re( e ν ( ε 1 ,ε 2 )+1) − δ G ln τ ) + O ( τ 4 3 Re( e ν ( ε 1 ,ε 2 )+1) − 1 3 ); 17 In order to facil itate the comparison, the parameter ε whic h appears in Theorem 3.1 of [1] has b een c hanged to ǫ . Degenerate Thir d Painlev ´ e Equation: II 46 furthermore, there is a discr epancy in the definition o f the par ameter ω ( ε 1 , ε 2 ) (cf. Equation (B.5)): instead of (B.3), one finds that ω ( ε 1 , ε 2 ) := g 11 ( ε 1 , ε 2 ) g 12 ( ε 1 , ε 2 ) . (B.6) In fact, (B.6) is the corr e ct definitio n of ω ( ε 1 , ε 2 ) (see the discuss io n below); there fo re, (B.6) should be used in lieu of the definition o f ω ( ε 1 , ε 2 ) in T he o rem B.1 and in Theorem 3 .1 of [1]. The r o ot of this discrepancy is r elated with the fact that the right factor of the second term in Equation (79) of [1] should be changed, namely , ln 3( εb ) 1 / 6 √ 2 → ln 3( εb ) 1 / 6 e − i π √ 2 . (B.7) As a co nsequence of (B.7), corresp onding changes should b e made to the following formulae in [1]: (i) to the right-hand sides of E quations (96), (9 8 ), and (99 ) one must add the term i π ( ν + 1); (ii) to the arg ument of the exp onential function on the right-hand side of E quation (101) one must add the term − 2 π i( ν + 1); (iii) to the r ig ht -hand side of the formula for z n given in Cor ollary 4.3.1 one m ust add the term 2 π i( ν + 1 ); and (iv) the definition of the parameter b ω ( ε 1 , ε 2 ) a ppe a ring in Theor em A.1 should b e changed 18 to b ω ( ε 1 , ε 2 ) := b g 11 ( ε 1 , ε 2 ) b g 12 ( ε 1 , ε 2 ). Making analog o us expansions for the co t( · · · ) functions in Equation (2.8) of Theorem 2.1, one obtains an asymptotic for mu la for H ( τ ) that coincides with the one pr esented in Theorem B.1. F or the other c hoice of ˆ r 0 ( τ ) (cf. the pro of of Prop o s ition 4.1) one would obtain the same asymp- totics (B.4)–(B.5) but with the change Θ( τ ) → Θ( τ ) + i π . App endix C: Asymptotics for Imaginary τ Here, asymptotics as τ → ± i ∞ o f the functions u ( τ ), H ( τ ), and f ( τ ) a re presented. These results a re obtained by applying transforma tions 6.2 .2 (changing 19 a → − a ) and 6 .2.3 (c hanging 20 τ → i τ ) g iven in Section 6 of [1] to the asymptotic results (for ε 1 = ε 2 = 0) stated in Theorems 2.1 – 2.3 of this pap er. F or this purp ose, it is conv enient to in tro duce the auxiliary mapping 21 b F ε 1 ,ε 2 : M → M , ( a, s 0 0 , s ∞ 0 , s ∞ 1 , g 11 , g 12 , g 21 , g 22 ) 7→ (( − 1) 1+ ε 2 a, s 0 0 , b s ∞ 0 ( ε 1 , ε 2 ) , b s ∞ 1 ( ε 1 , ε 2 ) , b g 11 ( ε 1 , ε 2 ) , b g 12 ( ε 1 , ε 2 ) , b g 21 ( ε 1 , ε 2 ) , b g 22 ( ε 1 , ε 2 )) , ε 1 = ± 1 , ε 2 = 0 , ± 1 , which is equiv alent to the form ulae for the mono dromy data in transforma tio ns 6.2 .2 and 6.2.3 in Section 6 of [1] (with l = ε 1 and p = ε 2 ). Define 22 : (1) b F − 1 , 0 : b s ∞ 0 ( − 1 , 0 ) = s ∞ 1 e 3 πa 2 , b s ∞ 1 ( − 1 , 0 ) = s ∞ 0 e πa 2 , b g 11 ( − 1 , 0 ) = − g 22 e 3 πa 4 , b g 12 ( − 1 , 0 ) = − ( g 21 + s ∞ 0 g 22 )e − 3 πa 4 , b g 21 ( − 1 , 0 ) = ( s 0 0 g 22 − g 12 )e 3 πa 4 , and b g 22 ( − 1 , 0 ) = ( s 0 0 ( g 21 + s ∞ 0 g 22 ) − g 11 − s ∞ 0 g 12 )e − 3 πa 4 ; (2) b F − 1 , − 1 : b s ∞ 0 ( − 1 , − 1) = s ∞ 0 e − πa 2 , b s ∞ 1 ( − 1 , − 1) = s ∞ 1 e πa 2 , b g 11 ( − 1 , − 1) = − i g 21 e − πa 4 , b g 12 ( − 1 , − 1) = − i g 22 e πa 4 , b g 21 ( − 1 , − 1) = − i( g 11 − s 0 0 g 21 )e − πa 4 , and b g 22 ( − 1 , − 1) = − i( g 12 − s 0 0 g 22 )e πa 4 ; (3) b F − 1 , 1 : b s ∞ 0 ( − 1 , 1 ) = s ∞ 0 e − πa 2 , b s ∞ 1 ( − 1 , 1 ) = s ∞ 1 e πa 2 , b g 11 ( − 1 , 1 ) = g 11 e − πa 4 , b g 12 ( − 1 , 1 ) = g 12 e πa 4 , b g 21 ( − 1 , 1 ) = g 21 e − πa 4 , and b g 22 ( − 1 , 1 ) = g 22 e πa 4 ; (4) b F 1 , 0 : b s ∞ 0 (1 , 0) = s ∞ 1 e πa 2 , b s ∞ 1 (1 , 0) = s ∞ 0 e 3 πa 2 , b g 11 (1 , 0) = − i g 12 e πa 4 , b g 12 (1 , 0) = − i( g 11 + s ∞ 0 g 12 )e − πa 4 , b g 21 (1 , 0) = − i g 22 e πa 4 , and b g 22 (1 , 0) = − i( g 21 + s ∞ 0 g 22 )e − πa 4 ; 18 There are a f ew innocuous m isprints in [1], which do not, how ever, affect the final results: (i) the factor 3 √ 3 − 2 appearing on the ri gh t-hand sides of Equat ions (76) and (92) should r ead 3( √ 3 − 1); (ii) the f actor i( εb ) 1 / 3 τ 2 / 3 appearing on the right-hand side of Equation (98) and in the argument of the exp onen tial function on the right-hand side of Equation (101 ) should read 3i 2 ( εb ) 1 / 3 τ 2 / 3 ; and (ii i) the factor i(3 √ 3 − 1)( εb ) 1 / 3 τ 2 / 3 appearing on the r ight -hand side of Equation (99) should r ead i(3 √ 3 − 3 2 )( εb ) 1 / 3 τ 2 / 3 . 19 In transformation 6.2.2, τ → τ , that is, τ n = τ o . 20 In transformation 6.2.3, a → a , that is, a n = a o . 21 There is a mi sprint on page 1202 (Appendix) of [ 1]: for items (1) and (4) in the definition of the auxili ary mapping b F ε 1 ,ε 2 , the chang e a → − a should b e made everywhere. 22 s 0 0 ( ε 1 , ε 2 ) = s 0 0 . Degenerate Thir d Painlev ´ e Equation: II 47 (5) b F 1 , − 1 : b s ∞ 0 (1 , − 1 ) = s ∞ 0 e πa 2 , b s ∞ 1 (1 , − 1 ) = s ∞ 1 e − πa 2 , b g 11 (1 , − 1 ) = g 11 e πa 4 , b g 12 (1 , − 1 ) = g 12 e − πa 4 , b g 21 (1 , − 1 ) = g 21 e πa 4 , and b g 22 (1 , − 1 ) = g 22 e − πa 4 ; (6) b F 1 , 1 : b s ∞ 0 (1 , 1) = s ∞ 0 e πa 2 , b s ∞ 1 (1 , 1) = s ∞ 1 e − πa 2 , b g 11 (1 , 1) = i( g 21 + s 0 0 g 11 )e πa 4 , b g 12 (1 , 1) = i( g 22 + s 0 0 g 12 )e − πa 4 , b g 21 (1 , 1) = i g 11 e πa 4 , and b g 22 (1 , 1) = i g 12 e − πa 4 . Theorem C. 1. L et ε 1 = ± 1 , ε 2 = 0 , ± 1 , ǫb = | ǫb | e i π ε 2 , and u ( τ ) b e a solution of Equation (1.1) c orr esp onding to the mono dr omy data ( a, s 0 0 , s ∞ 0 , s ∞ 1 , g 11 , g 12 , g 21 , g 22 ) . Supp ose that b g 11 ( ε 1 , ε 2 ) b g 12 ( ε 1 , ε 2 ) b g 21 ( ε 1 , ε 2 ) b g 22 ( ε 1 , ε 2 ) 6 = 0 , Re( b ν ( ε 1 , ε 2 ) + 1) ∈ (0 , 1) \ 1 2 , (C.1) wher e b ν ( ε 1 , ε 2 ) + 1 := i 2 π ln( b g 11 ( ε 1 , ε 2 ) b g 22 ( ε 1 , ε 2 )) . (C.2) Then ther e ex ist δ G satisfying, for 0 < Re( b ν ( ε 1 , ε 2 ) + 1) < 1 2 , the ine qu ality 0 < δ G < 1 3 1 + 2Re( b ν ( ε 1 , ε 2 ) + 1) 7 + 6Re( b ν ( ε 1 , ε 2 ) + 1) , and, for 1 2 < Re( b ν ( ε 1 , ε 2 ) + 1) < 1 , the ine quality 0 < δ G < 1 3 3 − 2Re( b ν ( ε 1 , ε 2 ) + 1) 9 + 2Re( b ν ( ε 1 , ε 2 ) + 1) , such that u ( τ ) has the asymptotic ex p ansion u ( τ ) = τ →∞ e i πε 1 2 e − i πε 1 2 ǫ ( ǫb ) 2 / 3 2 (e − i πε 1 2 τ ) 1 / 3 1 − 3 2 sin 2 ( 1 2 β ( ε 1 , ε 2 , τ )) ! (C.3) = τ →∞ e i πε 1 2 e − i πε 1 2 ǫ ( ǫb ) 2 / 3 2 (e − i πε 1 2 τ ) 1 / 3 sin( 1 2 β ( ε 1 , ε 2 , τ ) − ϑ 0 ) sin( 1 2 β ( ε 1 , ε 2 , τ ) + ϑ 0 ) sin 2 ( 1 2 β ( ε 1 , ε 2 , τ )) , (C.4 ) wher e β ( ε 1 , ε 2 , τ ) := b φ ( τ ) − i ( b ν ( ε 1 , ε 2 ) + 1) − 1 2 ln b φ ( τ ) − i ( b ν ( ε 1 , ε 2 ) + 1) − 1 2 ln 12 + ( − 1 ) 1+ ε 2 a ln(2 + √ 3) + π 4 − 3 π 2 ( b ν ( ε 1 , ε 2 ) + 1) + i ln b g 11 ( ε 1 , ε 2 ) b g 12 ( ε 1 , ε 2 )Γ( b ν ( ε 1 , ε 2 ) + 1) √ 2 π + O ( τ − δ G ln τ ) , (C.5) with b φ ( τ ) = 3 √ 3 ( − 1 ) ε 2 ( ǫb ) 1 / 3 (e − i πε 1 2 τ ) 2 / 3 , (C.6) ϑ 0 given in Equation (2.7) , and Γ( · · · ) the Euler gamma fu nction [11] . L et H ( τ ) b e t he H amiltonian function define d by Equation (1.2) c orr esp onding to the function u ( τ ) given ab ove. Then H ( τ ) has t he asymptotic exp ans ion H ( τ ) = τ →∞ e i πε 1 2 e − i πε 1 2 3( ǫb ) 2 / 3 (e − i πε 1 2 τ ) 1 / 3 − i( − 1) ε 2 4 √ 3 ( ǫb ) 1 / 3 (e − i πε 1 2 τ ) − 1 / 3 ( b ν ( ε 1 , ε 2 ) + 1) − 1 2 + 1 2 √ 3 i( − 1) 1+ ε 2 a + 1 2 + i 4 cot 1 2 β ( ε 1 , ε 2 , τ ) + i 4 cot 1 2 β ( ε 1 , ε 2 , τ ) − ϑ 0 + O ( τ − δ G ) . (C.7) The function f ( τ ) define d by Equation (1.3) has the fol lowing asymptotics: f ( τ ) = τ →∞ e i πε 1 2 − ( − 1) ε 2 ( ǫb ) 1 / 3 2 (e − i πε 1 2 τ ) 2 / 3 i + 3 √ 2 sin( 1 2 β ( ε 1 , ε 2 , τ )) sin ( 1 2 β ( ε 1 , ε 2 , τ ) − ϑ 0 ) ! . (C.8) Degenerate Thir d Painlev ´ e Equation: II 48 Remark C.1. Define the s tr ip ( in the b φ -plane ) b D := n τ ∈ C : Re( b φ ( τ )) > c 1 , | Im( b φ ( τ )) | < c 2 o , (C.9) where b φ ( τ ) is given in Equation (C.6) , and c 1 , c 2 > 0 are par ameters. The asymptotics of u ( τ ) , H ( τ ) , and f ( τ ) pre s ent ed in Theorem C.1 a re actually v a lid in the strip doma in b D . Theorem C. 2. L et ε 1 = ± 1 , ε 2 = 0 , ± 1 , ǫb = | ǫb | e i π ε 2 , and u ( τ ) b e a solution of Equation (1.1) c orr esp onding to the mono dr omy data ( a, s 0 0 , s ∞ 0 , s ∞ 1 , g 11 , g 12 , g 21 , g 22 ) . Supp ose that b g 11 ( ε 1 , ε 2 ) b g 12 ( ε 1 , ε 2 ) b g 21 ( ε 1 , ε 2 ) b g 22 ( ε 1 , ε 2 ) 6 = 0 , Re i 2 π ln( b g 11 ( ε 1 , ε 2 ) b g 22 ( ε 1 , ε 2 )) = 1 2 . (C.10) L et the br anch of the fun ction ln( · · · ) b e chosen 23 such that Im(ln( − b g 11 ( ε 1 , ε 2 ) b g 22 ( ε 1 , ε 2 ))) = 0 . Define b 1 ( ε 1 , ε 2 ) := 1 2 π ln( − b g 11 ( ε 1 , ε 2 ) b g 22 ( ε 1 , ε 2 )) ( ∈ R ) . (C.11) Then ∃ δ ∈ (0 , 1 / 39) such that the function u ( τ ) has, for al l lar ge en ough m ∈ N , se c ond-or der p oles, b τ ∞ m , ac cumu lating at t he p oint at infinity, b τ ∞ m = m →∞ e i πε 1 2 2 π ( − 1) ε 2 m 3 √ 3 ( ǫb ) 1 / 3 3 / 2 1 − 3 b 1 ( ε 1 , ε 2 ) 4 π ln m m − 3 b 2 ( ε 1 , ε 2 ) 4 π 1 m + O m 1 2 − 3 δ 2 , (C.12 ) wher e b 2 ( ε 1 , ε 2 ) := b 1 ( ε 1 , ε 2 ) ln(24 π ) + ( − 1 ) 1+ ε 2 Re( a ) ln(2 + √ 3) + π 2 − 1 2 arg b g 11 ( ε 1 , ε 2 ) b g 12 ( ε 1 , ε 2 ) b g 21 ( ε 1 , ε 2 ) b g 22 ( ε 1 , ε 2 ) − arg Γ 1 2 + i b 1 ( ε 1 , ε 2 ) + i ( − 1) 1+ ε 2 Im( a ) ln(2 + √ 3) + 1 2 ln b g 11 ( ε 1 , ε 2 ) b g 12 ( ε 1 , ε 2 ) b g 21 ( ε 1 , ε 2 ) b g 22 ( ε 1 , ε 2 ) ; (C.13) furthermor e, t he function u ( τ ) has, for al l lar ge enou gh m ∈ N , a p air of first -or der zer o es, b τ ± m , ac cumulating at t he p oint at infi n ity, b τ ± m = m →∞ e i πε 1 2 2 π ( − 1) ε 2 m 3 √ 3 ( ǫb ) 1 / 3 3 / 2 1 − 3 b 1 ( ε 1 , ε 2 ) 4 π ln m m − 3 4 π ( b 2 ( ε 1 , ε 2 ) ± 2 ϑ 0 ) 1 m + O m 1 2 − 3 δ 2 , (C.14) wher e ϑ 0 given in Equation (2.7 ) . Remark C.2 . T o present a symptotics o f u ( τ ) , H ( τ ) , and f ( τ ) outside of neighbo rho o ds of poles and zero es, introduce the cheese-lik e domain, b D u , for a solution u ( τ ) : b D u := n τ ∈ b D : | b φ ( τ ) − b φ ( b τ κ m ) | > C | b τ κ m | − δ o , where the strip do main b D is defined by Equation (C.9) , b φ ( τ ) is giv en in Equation (C.6) , C > 0 is a parameter, κ = ∞ , ± ( b τ κ m are the po les and z e r o es int ro duced in Theorem C.2 ) , and 0 < δ < 1 / 3 9 . Theorem C. 3. L et ε 1 = ± 1 , ε 2 = 0 , ± 1 , ǫb = | ǫb | e i π ε 2 , and u ( τ ) b e a solution of Equation (1.1) c orr esp onding to the mono dr omy data ( a, s 0 0 , s ∞ 0 , s ∞ 1 , g 11 , g 12 , g 21 , g 22 ) . Su pp ose that c onditions (C.1 0) ar e valid, the br anch of ln( · · · ) is cho sen as in The or em C.2 , and b 1 ( ε 1 , ε 2 ) is define d by Equation (C.11) . Then ther e ex ist δ, δ G ∈ R + satisfying the ine qualities 0 < δ < 1 39 , 0 < δ < δ G < 1 15 − 8 δ 5 , such that u ( τ ) has the asymptotic ex p ansion u ( τ ) = τ →∞ e i πε 1 2 τ ∈ b D u e − i πε 1 2 ǫ ( ǫb ) 2 / 3 2 (e − i πε 1 2 τ ) 1 / 3 sin( 1 2 b β ( ε 1 , ε 2 , τ ) − ϑ 0 ) sin( 1 2 b β ( ε 1 , ε 2 , τ ) + ϑ 0 ) sin 2 ( 1 2 b β ( ε 1 , ε 2 , τ )) , (C.15) 23 The second condition of Equations (C.10 ) suggests that this branch of ln( · · · ) exists. Degenerate Thir d Painlev ´ e Equation: II 49 wher e b β ( ε 1 , ε 2 , τ ) := b φ ( τ ) + b 1 ( ε 1 , ε 2 ) ln b φ ( τ ) + b 1 ( ε 1 , ε 2 ) ln 1 2 + ( − 1) 1+ ε 2 Re( a ) ln(2 + √ 3) + π 2 − 1 2 arg b g 11 ( ε 1 , ε 2 ) b g 12 ( ε 1 , ε 2 ) b g 21 ( ε 1 , ε 2 ) b g 22 ( ε 1 , ε 2 ) − arg Γ 1 2 + i b 1 ( ε 1 , ε 2 ) + i ( − 1) 1+ ε 2 Im( a ) ln(2 + √ 3) + 1 2 ln b g 11 ( ε 1 , ε 2 ) b g 12 ( ε 1 , ε 2 ) b g 21 ( ε 1 , ε 2 ) b g 22 ( ε 1 , ε 2 ) + O ( τ − δ G ln τ ) , (C.16) with b φ ( τ ) and ϑ 0 given, r esp e ctively, in Equations (C.6 ) and (2.7) . L et H ( τ ) b e t he H amiltonian function define d by Equation (1.2) c orr esp onding to the function u ( τ ) given ab ove. Then H ( τ ) has t he asymptotic exp ans ion H ( τ ) = τ →∞ e i πε 1 2 τ ∈ b D u e − i πε 1 2 3( ǫb ) 2 / 3 (e − i πε 1 2 τ ) 1 / 3 + ( − 1) ε 2 4 √ 3 ( ǫb ) 1 / 3 (e − i πε 1 2 τ ) − 1 / 3 b 1 ( ε 1 , ε 2 ) + 1 2 √ 3 ( − 1) 1+ ε 2 a − i 2 + 1 4 cot( 1 2 b β ( ε 1 , ε 2 , τ )) + 1 4 cot( 1 2 b β ( ε 1 , ε 2 , τ ) − ϑ 0 ) + O ( τ − δ G ) . (C.17) The function f ( τ ) define d by Equation (1.3) has the fol lowing asymptotics: f ( τ ) = τ →∞ e i πε 1 2 τ ∈ b D u − ( − 1) ε 2 ( ǫb ) 1 / 3 2 (e − i πε 1 2 τ ) 2 / 3 i + 3 √ 2 sin( 1 2 b β ( ε 1 , ε 2 , τ )) sin ( 1 2 b β ( ε 1 , ε 2 , τ ) − ϑ 0 ) ! . (C.18) Remark C.3. F or real, non-zero v alues o f b , singular imagina ry solutions u ( τ ) ( for imagina r y τ ) of Eq ua tion (1.1) ar e specified b y the follo wing “s ingular imag inary reduction” for the mono dro my data 24 : s 0 0 = − s 0 0 , b s ∞ 0 ( ε 1 , ε 2 ) = − b s ∞ 1 ( ε 1 , ε 2 ) e 2 π a , b g 11 ( ε 1 , ε 2 ) = − b g 22 ( ε 1 , ε 2 ) , b g 12 ( ε 1 , ε 2 ) = − b g 21 ( ε 1 , ε 2 ) , Im( a ) = 0 . (C.19) In this c ase, as y mptotics of b τ ∞ m , b τ ± m , u ( τ ) , H ( τ ) , and f ( τ ) are as g iven in Equations (C.12) , (C.14) , (C.15) , (C.17) , and (C.18) , r esp ectively , but with the changes b 1 ( ε 1 , ε 2 ) → b 0 ( ε 1 , ε 2 ) , b 2 ( ε 1 , ε 2 ) → b ♯ 0 ( ε 1 , ε 2 ) , and b β ( ε 1 , ε 2 , τ ) → b β 0 ( ε 1 , ε 2 , τ ) , where b 0 ( ε 1 , ε 2 ) := 1 π ln | b g 11 ( ε 1 , ε 2 ) | , (C.20) b ♯ 0 ( ε 1 , ε 2 ) := b 0 ( ε 1 , ε 2 ) ln(2 4 π ) + ( − 1) 1+ ε 2 Re( a ) ln(2 + √ 3) − π 2 − arg b g 11 ( ε 1 , ε 2 ) b g 12 ( ε 1 , ε 2 )Γ 1 2 + i b 0 ( ε 1 , ε 2 ) , (C.21) b β 0 ( ε 1 , ε 2 , τ ) := b φ ( τ ) + b 0 ( ε 1 , ε 2 ) ln b φ ( τ ) + b 0 ( ε 1 , ε 2 ) ln 1 2 + ( − 1) 1+ ε 2 Re( a ) ln(2 + √ 3) − π 2 − arg b g 11 ( ε 1 , ε 2 ) b g 12 ( ε 1 , ε 2 )Γ 1 2 + i b 0 ( ε 1 , ε 2 ) + O ( τ − δ G ln τ ) . (C.22) 24 There exist regular i m aginary solutions (cf. Part I [1], Appendix) whic h are specified b y another imaginary reduct ion. 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