One Special Identity between the complete elliptic integrals of the first and the third kind

One Sp ecial Iden tit y b et w een the complete elliptic in tegrals of the first and the third kind Y u Jia Institute of High Energy Ph ysics, Chinese Academ y of Science s Beijing 100049 , China Abstract I pro v e an ide n tit y bet w een the first kind and the third kind complete elliptic in tegrals with the follo wing form: Π  (1 + x )(1 − 3 x ) (1 − x )(1 + 3 x ) , (1 + x ) 3 (1 − 3 x ) (1 − x ) 3 (1 + 3 x )  − 1 + 3 x 6 x K  (1 + x ) 3 (1 − 3 x ) (1 − x ) 3 (1 + 3 x )  = ( 0 , (0 < x < 1) − π 12 ( x − 1) 3 / 2 √ 1+3 x x . ( x < 0 or x > 1) This relation can b e applied to e liminate the complete elliptic in tegral of the third kind from the analytic solutions of the imaginary part of tw o-lo op sunset diagrams in the equal mass case. The v alidit y of this relation in the complex domain is also briefly dis- cussed. Keyw ords: complete elliptic in tegrals, ordinary differen tial equation 1 Elliptic in tegrals, as one of the most imp ortan t classes of nonelemen tary functions, hav e found num erous applications in man y branch es of engineer- ing and ph ysics. Ha ving b een comprehensiv ely studied b y man y eminen t mathematicians o v er cen turies, a v ast amoun t of kno wledge on the elliptic in tegrals, as we ll as their close cousin, elliptic functions , has b een acc um u- lated [1, 2, 3, 4]. The simplest and particularly imp ortant class of elliptic in tegrals is the c omplete elliptic in tegrals, totally of three kinds. The complete elliptic in te- gral of the third kind, Π, b eing the most complicated one, can b e expressed in terms of the complete elliptic in tegral of t he first kind, K , plus elemen tary functions and Heuman’s Lam b da function or Jacobi’s Zeta function. The last t w o transcenden tal functions can in turn b e express ible from the inc om plete elliptic in tegrals of t he first and second kinds. The aim of this note is to establis h a simple relation b etw een the complete elliptic in tegrals of the first a nd the third kind, for some sp ecific texture of argumen ts of course. That is, in this special cas e, the c omplete elliptic in tegral of the third kind can b e transformed to the complete elliptic in tegral of the first kind, plus elemen tary function a t most, without resort to an y other nonelemen tary function. There a re a v ariet y of con v en tions adopted in literature in definin g the elliptic in tegrals. In this w ork I find it con ve nien t to use the same definitions as ta k en b y [2, 4] for the first, second and third kind of complete e lliptic in tegrals: K ( m ) ≡ Z 1 0 dt 1 p (1 − t 2 )(1 − mt 2 ) = π 2 F  1 2 , 1 2 ; 1; m  , E ( m ) ≡ Z 1 0 dt r 1 − mt 2 1 − t 2 = π 2 F  − 1 2 , 1 2 ; 1; m  , (1) Π( n, m ) ≡ Z 1 0 dt 1 (1 − nt 2 ) p (1 − t 2 )(1 − mt 2 ) = π 2 F 1  1 2 ; 1 , 1 2 ; 1; n, m  , where F a nd F 1 denote Gaussian h yp ergeometric f unction, and App ell h y- p ergeometric function of t w o v ariables, resp ectiv ely . In most practical ap- plications, the parameter m and the characteris tic n are restricted to b e less than 1 . How ev er, it is worth emphasizing when these argumen ts exceed 1 or ev en are off the real axis, these elliptic inte grals are still w ell d efined mathematically , though b ecome complex v alued in general. 2 The main r esult states as follow s. F or ∀ x ∈ R , there exists a n iden tity Π  (1 + x )(1 − 3 x ) (1 − x )(1 + 3 x ) , (1 + x ) 3 (1 − 3 x ) (1 − x ) 3 (1 + 3 x )  − 1 + 3 x 6 x K  (1 + x ) 3 (1 − 3 x ) (1 − x ) 3 (1 + 3 x  = ( 0 , (0 < x < 1) − π 12 ( x − 1) 3 / 2 √ 1+3 x x . ( x < 0 or x > 1) (2) The K a nd Π functions with this sp ecific arrangemen t of the argumen ts, o dd as it ma y lo ok, are not unfamiliar to particle phys icists. In fact these func- tions hav e b een encoun tered in the analytical expressions for the phase space of three equal-mass particles [5, 6, 7]. T o b e precise, it is w orth p oin ting out that the co efficien t of Π function acciden tally v anishes in this example. Note the 3-b o dy phase space can b e obtained from cutting the simplest scalar tw o- lo op sunset diagram. F or a general sunset diagram with a non trivial v ertex structure, or if the p o w er o f propagators exceeds than 1, the complete ellip- tic in tegral o f the third kind will inevitably arise in ev aluating its imaginary part. Equation (2) can then b e inv ok ed to trade the Π function for the K function, th us considerably simplifyi ng the answ er. In the aforemen tioned ph ysical application, simple kin ematics enforces 0 < x ≤ 1 3 , for whic h b oth the parameter and c haracteristic of K and Π are p ositiv e and less than 1. This restriction is not nece ssary fo r general purp ose, therefore it will b e discarded in the follo wing discuss ion. There exist some formulas whic h transform one Π function to another Π function plus a K function [1 , 2]. Ho w ev er, the relation de signated in (2), whic h links o ne Π f unction t o o ne K function only , is rather p eculiar. T o the b est of m y kno wledge, this relation cannot b e deriv ed b y any kno wn form ula, and it also has nev er b een explicitly stated in an y published w ork. F or this reason, I feel it ma y b e w orth while to rep ort it here. The strategy of the pro of is to construct a differen tial equation satisfied b y the le ft side of eq uation (2), called y ( x ) in shorthand. Emplo ying the w ell-know n differen tia l prop erties of complete elliptic in tegrals [1, 4]: dK ( m ) dm = E − (1 − m ) K 2 m (1 − m ) , ∂ Π( n, m ) ∂ m = − E + (1 − m )Π 2(1 − m )( n − m ) , (3) ∂ Π( n, m ) ∂ n = nE − ( n − m ) K + ( n 2 − m )Π 2 n (1 − n )( n − m ) , 3 together with the c hain rule, I find that y satisfie s the following first-o rder differen tial equation: y ′ = y 1 + 2 x + 3 x 2 x ( x − 1)(1 + 3 x ) . (4) The magic is tha t, after t he differen tiatio n, the complete elliptic in tegral of the second kind canc els, and the elliptic in tegrals of the first and the third kind conspire, in a rather p eculiar wa y , to cluster in to the original form. This is a basic t yp e of linear differen tial equation, and the corresponding solution is y ( x ) = C ( x − 1) 3 / 2 √ 1 + 3 x x , (5) where C is a constan t to b e determine d. Notice the ab ov e solution p ossess es a p ole at x = 0 and t wo branc h p oints lo cated at x = 1 and x = − 1 3 , and in general one should not exp ect C w ill assume a univ ersal v alue in the entire domain of x , R . I shall attempt to fix the v alue of C region b y region. First let us consider the c ase w hen x b elongs to the op en in terv al I 0 = (0 , 1). Inspecting Eq. (2), it is e asy to see y ( x 0 ) = 0 for x 0 = 1 3 ∈ I 0 . This initial v alue cannot b e satisfied unless if C = 0. Also note the right hand side of (4) is a conti n uous function of x in this in terv al. Therefore, b y the exis tence and uniqu eness theorem for line ar equation (fo r instance, see theorem 2.1 in Ref. [8]), the d ifferen tia l equation (4) admits the unique solution y = 0 in this in terv al. Next I turn to the solution for x ∈ I 1 = ( −∞ , − 1 3 ). Examining the left side of Eq .(2), one readily finds y ( x 1 ) = π 3 for x 1 = − 1 ∈ I 1 . Thi s initial v alue can b e satisfied only if C = − π 12 . Th us by the theorem of existence and uniquenes s, the function in (5) with this v alue of C constitutes the unique solution in the in terv al I 1 . There still remain t wo other inte rv als, I 2 = ( − 1 3 , 0) and I 3 = (1 , ∞ ) to b e inv estigated. When x resides in I 2 , both K and Π functions become complex-v alued, and the left side of (2) turns to be purely imaginary; whereas as x ∈ I 3 , though b o th K and Π b ecome als o c omplex, the left s ide of (2) nev ertheless is real. There are no simple initial v alues can be inferred in these regions. I p erformed a n umerical c hec k with the a id of the computing pac k age Ma thema t ica [4], and find the solution (5) with C = − π 12 in these t w o regions agree with the left side of Eq.(2) to the 14 decimal place. 4 Equation (2) can ha v e some in teresting consequen ces. Here I illustrate one example: Γ( 1 4 ) 2 4 √ π = K  1 2  = 3 − p 6 √ 3 − 9 2 Π 1 − p 2 √ 3 − 3 2 , 1 2 ! (6) = 3 + p 6 √ 3 − 9 2 Π 1 + p 2 √ 3 − 3 2 , 1 2 ! − π v u u t 2 + √ 3 + s 7 + 38 √ 3 9 . The analytic expres sion for K ( 1 2 ) in term of Γ function is kno wn [3]. Ho wev er, it is amus ing that these strange lo oking Π functions can also b e put in closed form. Lastly , one natural question may b e raised– ho w about the v alidit y of equation (2) when the domain o f x is extended from real to complex? It is a curious question o wning to the ric h a nalytic structure of elliptic in tegrals. Through a n umerical study using Ma thema tica , I find this relation still holds in most regions of complex plane. It is most lucid t o demonstrate this examination in plots. The left side of equation (2) still v anishes in an ov al region in ternally ta ngen t to a rectangle (0 < Re z < 1 , | Im z | < 0 . 33), whic h is characteriz ed b y the basin and plateau in Figure 1. This can b e view ed as a generalization to the firs t p ortion of equation (2 ). The second part of this relation is v alid almost eve rywhere else except in a p ear shap ed region em b edded inside a rectangle (1 < Re z < 6 . 74 and | Im z | < 0 . 97), whic h can b e clearly visualized in Figure 2 as the tip of an iceb erg surrounded b y the flat, b oundless sea. A t this s tage I am unable to prov ide a rigoro us justification for this observ ation. A thorough understandi ng of these f eatures will b e definitely desirable. Ac kno wled g men t I thank Kexin Cao for encouraging me to write do wn this note. This researc h is supp o rted in part b y National Natural Science F oundation of Chi na under Gran t No. 106050 31. 5 References [1] P . F. Byrd and M. D . F riedman, Handb o ok of El liptic Inte gr als for En- gine ers and Sc i e ntists , Springer V erlag, Berlin (1 9 71). [2] M. Abramow itz and I. A. Stegun, Handb o ok of Mathematic al F unctions , Do v er Publications, New Y ork (1972 ) . [3] I. S. Gradsh teyn a nd I. M. Ryzhik , T able of Inte gr al s , Series, and Pr o d- ucts , Academic Press, San Diego (2000). [4] W olfram Researc h, Inc., Mathematic a Edition: V ersion 5.0 , W olfram Researc h, Inc., Champaign, IL (2003). [5] B. Almgren, Arkiv f¨ or Ph ysik 38 , 16 1 (1968). [6] S. Bauberger, F. A. Berends, M. Bohm and M. Buza, Nucl. Ph ys. B 434 , 383 (19 95) [arXiv:hep-ph/9409388]. [7] A. I. Dav ydyc hev and R. Delb ourgo, J. Ph ys. A 37 , 4 8 71 (2004 ) [arXiv:hep-th/0311075]. [8] W. E. Boy ce and R. C. D iPrima, Elemen tary Differ ential Equations and Boundary V alue Pr obl e ms (2 nd Edition), John Wiley & Sons , New Y ork (1969). 6 -0.5 0 0.5 1 Re z -0.5 0 0.5 Im z 0 0.5 1 1.5 2 Re @ Pi - 1 + 3 z      6 z K D -0.5 0 0.5 1 Re z -0.5 0 0.5 1 Re z -0.5 0 0.5 Im z -0.5 0 0.5 Im @ Pi - 1 + 3 z      6 z K D -0.5 0 0.5 1 Re z Figure 1: Profile of the left side of Equation (2) as a complex -v alued function, with the real v ariable x promoted to a complex v ariable z . 7 0 5 10 Re z -2 -1 0 1 2 Im z -10 0 10 Re @ Pi - 1 + 3 z      6 z K - f H z LD 0 5 10 Re z 0 5 10 Re z -2 -1 0 1 2 Im z -2 0 2 Im @ Pi - 1 + 3 z      6 z K - f H z LD 0 5 10 Re z Figure 2: Examination of the v alidit y o f second portio n of Equation (2) with a complex v ariable z , where f ( z ) ≡ − π 12 ( z − 1) 3 / 2 √ 1+3 z z . 8