Non-Zero Order of an Extended Temme Integral

Abstract

A new three-dimensional integral containing f(x,y,z)Iv(xα) is derived where Iv(xα) is the Modified Bessel Function of the first kind and the integral is taken over the infinite cubic space 0x∞,0y∞,0z∞. The integral is not easily evaluated for complex ranges of the parameters. A representation in terms of the Hurwitz–Lerch zeta function, polylogarithm function and Riemann zeta functions are evaluated. This representation yields triple integral representations in terms of fundamental constants that can be derived. Almost all Lerch functions have an asymmetrical zero distribution.

Keywords

• modified bessel function
• triple integral
• Apéry’s constant