A Double Logarithmic Transform Involving the Exponential and Polynomial Functions Expressed in Terms of the Hurwitz–Lerch Zeta Function

Abstract

The object of this paper is to derive a double integral in terms of the Hurwitz–Lerch zeta function. Almost all Hurwitz–Lerch zeta functions have an asymmetrical zero-distribution. Special cases are evaluated in terms of fundamental constants. All the results in this work are new.

j-Dimensional Integral Involving the Logarithmic and Exponential Functions: Derivation and Evaluation

“…In this section, we use Equation ( 2) in [1] to derive the contour integral representations for the Hurwitz-Lerch zeta function. The significance of this section is to derive a special function equivalent to the definite integral of the contour integral derived in Section 2 in terms of the same contour integral.…”

“…Using Equation (2) in [1] and replacing y with log(a) + iπ2 j (2y + 1) n , and then multiplying both sides by…”

“…The idea of multiple or repeated integrals has been around for decades. Work involving multiple definite integrals has been used in the studies of Reynolds et al [1], Jain et al [2], Agarwal et al [3], Sergey et al [4], Wiener [5], Kiyoshi [6] and Olver [7], to name a few.…”

A Quadruple Integral Involving Chebyshev Polynomials Tn(x): Derivation and Evaluation

“…An alternate derivation of Equation ( 2) in [9] when Re(m + w) ≤ 0 can be achieved by recalling a variant of Hankel's formula involving the Gamma function:…”

“…The derivations follow the method used in [7,9]. In the present case the cut approaches the origin from the interior of the first quadrant and the cut lies on opposite sides of the cut going round the origin with zero radius.…”