Laplace transforms based some novel integrals via hypergeometric technique

TL;DR

This paper derives new analytical Laplace transform formulas for integrals involving hypergeometric functions and powers of sine and cosine, using hypergeometric techniques and algebraic properties.

Contribution

It introduces novel Laplace transform solutions for integrals with hypergeometric functions and powers of trigonometric functions, expanding existing analytical methods.

Findings

Derived Laplace transforms of hypergeometric integrals.

Obtained transforms of powers of sine and cosine functions.

Presented special cases and applications of the main results.

Abstract

In this paper, we obtain the analytical solutions of Laplace transforms based some novel integrals with suitable convergence conditions, by using hypergeometric approach (some algebraic properties of Pochhammer symbol and classical summation theorems of hypergeometric series ${}_{2}F_{1}(1)$, ${}_{2}F_{1}(-1)$ , ${}_{4}F_{3}(-1)$) . Also, we obtain the Laplace transforms of arbitrary powers of some finite series containing hyperbolic sine and cosine functions having different arguments, in terms of hypergeometric and Beta functions. Moreover, Laplace transforms of even and odd positive integral powers of sine and cosine functions with different arguments, and their combinations of the product (taking two, three, four functions at a time), are obtained. In addition, some special cases are yield from the main results.