Published 2025-02-09
Keywords
- Integral Identities,
- Bessel Function,
- Struve functions,
- Elliptic Integral
Copyright (c) 2025
This work is licensed under a Creative Commons Attribution 4.0 International License.
How to Cite
Glasser, M. (2025). A four parameter integral identity and a few consequences. Scientia Series A: Mathematical Sciences, 36, 9-12. https://doi.org/10.71712/f1ya-ec51
Abstract
The identity
\[
\int_0^\infty \frac{e^{-ax} \, e^{-b\sqrt{x^2 + 2cx + a^2}}}{\sqrt{x^2 + 2cx + b^2}} \, dx
=
\int_0^\infty \frac{e^{-bx} \, e^{-a\sqrt{x^2 + 2cx + b^2}}}{\sqrt{x^2 + 2cx + a^2}} \, dx
\]
is derived, applied to the Struve function $H_0$, and used to deduce the reduction formula
\[
\int_0^\infty \frac{F(\sqrt{x^2 + 2cx + b^2} + x)}{\sqrt{x^2 + 2cx + b^2}} \, dx
=
\int_0^\infty f(t)e^{-t} E_1[(c+b)t] \, dt,
\]
where $F$ is arbitrary.
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