Abstract
A class of definite integrals involving a quotient function with a reducible polynomial, logarithm and nested logarithm functions are derived with a possible connection to contact problems for a wedge. The derivations are expressed in terms of the Lerch function. Special cases are also derived in terms fundamental constants. The majority of the results in this work are new.
Highlights
In this paper, we derive the definite integral given by Received: 8 August 2021 Z ∞Accepted: 17 September 2021Published: 21 September 2021Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. x m−1 logk dx + x2 )(b2 − x2 )( a2 where the parameters k, c, a, b, m are general complex numbers
Mellin transform was first established by Mellin [1]
Lerch function with a possible connection to contact problems for a wedge [5]. We will use this new transform and evaluate it to yield special cases in terms of Catalan’s constant C, π, Euler constant γ, the zeta function of Riemann ζ (s), the Hurwitz zeta function ζ (s, v), and the log-gamma function log(Γ( x )). These special cases are new with the aim of providing a new set of integral for use by researchers where applicable
Summary
Mellin transform was first established by Mellin [1]. The theory of this transform is well documented in [2,3]. Lerch function with a possible connection to contact problems for a wedge [5] We will use this new transform and evaluate it to yield special cases in terms of Catalan’s constant C, π, Euler constant γ, the zeta function of Riemann ζ (s), the Hurwitz zeta function ζ (s, v), and the log-gamma function log(Γ( x )). These special cases are new with the aim of providing a new set of integral for use by researchers where applicable
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