Integral formula for the Bessel function of the first kind

TL;DR

This paper introduces a new integral representation for the Bessel function of the first kind, extending classical formulas to complex parameters, which could facilitate advanced analytical and computational methods.

Contribution

The paper presents a generalized integral formula for $J_ u(z)$ applicable to all complex $ u$ and $z$, broadening the scope of classical representations.

Findings

Derived a new integral representation for $J_ u(z)$

Extended classical formulas to complex parameters

Potential applications in analysis and computation

Abstract

In this paper, we prove a new integral representation for the Bessel function of the first kind $J_\mu(z)$. This formula generalizes to any $\mu,z\in\mathbb{C}$ the classical representations of Bessel and Poisson.