Definite integrals involving combinations of powers and logarithmic functions of complicated arguments expressed in terms of the Hurwitz zeta function

TL;DR

This paper derives closed-form expressions for complex definite integrals involving powers and logarithms, expressing them in terms of the Hurwitz zeta function and fundamental constants, with a comprehensive reference table.

Contribution

It introduces new formulas for integrals with complex arguments, linking them to the Hurwitz zeta function and providing a practical reference table.

Findings

Closed-form formulas for complex integrals derived

Expressed integrals in terms of Hurwitz zeta and constants

Provided a reference table for these integrals

Abstract

In this manuscript, the authors derive closed formula for definite integrals of combinations of powers and logarithmic functions of complicated arguments and express these integrals in terms of the Hurwitz zeta. These derivations are then expressed in terms of fundamental constants, elementary and special functions. A summary of the results is produced in the form of a table of definite integrals for easy referencing by readers.