The integral $\int_0^a f(t) t^{-s} \mathrm{d}t$ diverges for $\text{Re}(s) \geq λ+ 1$, where $λ$ is the order of the first non-vanishing derivative of $f(t)$ at the origin. With the assumption that $f(t)$ is analytic at the origin, the finite-part of the divergent integral assumes the contour integral representation of the form $\bbint{0}{a} f(t) t^{-s} \mathrm{d}t = \int_C f(z) z^{-s} G(z) \mathrm{d}z$ where $G(z)$ depends on whether $z=0$ constitutes a pole or a branch point singularity of $z^{-s}$ [E. A. Galapon, \textit{Proc. R. Soc.}, \textbf{A 473} (2017), no. 2197, 20160567.]. In this paper, we extend these representations to accommodate logarithmic singularities of arbitrary order $n \in \mathbb{N}$, specifically for $\bbint{0}{a} f(t) t^{-s} \ln^n t \, \mathrm{d}t$. We then demonstrate the utility of the representations in the numerical evaluation of finite-part integrals and their use in determining the finite parts of non-Mellin-type divergent integrals -- those which exhibit singular behavior at the origin but lack a well-defined Mellin transform. Finally, these representations provide a closed-form evaluation of the Stieltjes transform $\int_0^a k(t) \ln^n t \left( t^ν(ω^2 + t^2) \right)^{-1} \mathrm{d}t$ in terms of finite-part integrals, from which the dominant asymptotic behavior is readily extracted for vanishingly small values of the parameter $ω$.
Finite-part integration is a method of solving convergent integrals by evaluating the finite-part of the divergent integral induced from the convergent integral itself [1,2,3]. The technique is applied for the exact evaluation of the Stieltjes transform [2,4,5,6],
Divergent integrals are introduced by expressing the kernel function as an infinite series
where the expansion is about ω = 0. The divergence arises from the formal interchange in the order of integration and summation without conforming to the uniformity conditions of the expansion in the given limit of integration. The righthand side of equation (1.1) becomes ∞ s=0 (-ω) s a 0 f (t) t s+1 dt. (1.3) The integral inside the summation is expected to diverge if the function f (t), or any of its derivatives, does not vanish at t = 0. In simple terms, equation (1.3) becomes a series in which each term is divergent. To use the equality in equation (1.1), the divergent integrals can be replaced by their finite-parts plus the missing terms that can be recovered by lifting the integration to the complex plane. It turns out that the missing terms are encoded in the singularity of the kernel function [1]. The finite-part of the divergent integral needed in this context can be defined by replacing the lower limit of integration with some positive number ϵ that is 0 < ϵ < a and expressing the solution of the integral as a ϵ f (t) t s+1 dt = C ϵ + D ϵ . (1.4) where C ϵ converges and D ϵ diverges as ϵ → 0. The finite-part of the divergent integral in equation (1.3) can now be denoted and defined as [2] \ \
Under certain conditions, the finite-part integral can be represented by the analytic continuation of the Mellin transform beyond its strip of analyticity. The values of the parameter λ in equation (1.5) determine the relationship between these two operations. When λ is a natural number, the evaluation of the finite-part integral requires the concept of the regularized limit. In the Laurent series expansion of a function about a singular point, the regularized limit corresponds to the coefficient of the zero-order term, i.e., the constant term.
To compute the regularized limit for functions with poles of order n, a matrix formulation of size (n + 1) × (n + 1) is employed. This matrix arises from the algebraic division of power series and is discussed in detail in [2], [7, p. 18, 0.313]. While this matrix-based approach is rigorous, it becomes increasingly tedious for functions with higher-order poles. One of the aims of this paper is to derive a closed-form expression for the regularized limit. Such a representation is expected to provide a more practical and efficient method for evaluating regularized limits involving poles of arbitrary order.
Contour integration has been shown to be an effective method complementary to finite-part integration in the evaluation of convergent integrals. Using the simple contour of integration illustrated in Figure 1, the finite-part integral appearing in equation (1.3) can be expressed as a contour integral [1]:
where G(z) is dertermined by whether z = 0 is a pole or a branch point of z -λ . For a pole singularity, i.e., when λ ∈ Z + 0 ,
(1.7) G(z) = 1 2πi (log z -πi). On the other hand, for a branch point, i.e. λ / ∈ Z, it is given by
.
This expression serves as the starting point for the development of contour integral representations analyzed in this paper. A notable limitation of equation (1.6) is that it does not account for the presence of a logarithmic singularity. Another objective of this work is to extend the contour integral representation of the finitepart integral to accommodate integrands that exhibit logarithmic singularities of arbitrary order. The contour integral representation extends the domain of the Mellin transform beyond its strip of analyticity by leveraging its connection with the finite-part integral. While the Mellin transform may not always yield a well-defined result, even under analytic continuation, the finite-part integral is expected to exist or, at the very least, to evaluate to zero. As demonstrated in [2], the finite-part integral can exist independently of the Mellin transform. The crucial insight lies in the use of the contour integral representation of the finite-part integral, which enables the evaluation of otherwise ill-defined expressions.
An additional advantage of the contour integral approach is its utility in the numerical computation of finite-part integrals. By employing contour deformation techniques, one can determine the desired solution to the given integral problem with greater flexibility and precision.
Finally, the last objective of this paper is to demonstrate the application of finite-part integration in evaluating the Stieltjes transform of the form (1.9)
This type of integral has a potential connection to the Euler-Heisenberg effective Lagrangian [8]. The Euler-Heisenberg Lagrangian represents a non-linear correction to the classical Maxwell Lagrangian, a
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