A simple proof of the characteristic function of Student's $t$-distribution

TL;DR

This paper provides a straightforward proof of the characteristic function of Student's t-distribution using a differential equation approach, which can be applied to other distributions as well.

Contribution

It introduces a simple, differential equation-based proof technique for the characteristic function of Student's t-distribution, enhancing understanding and potential applicability.

Findings

Simplified proof of the characteristic function

Method applicable to other distributions

Potential for broader distribution analysis

Abstract

This note presents a simple proof of the characteristic function of Student's $t$-distribution. The method of proof, which involves finding a differential equation satisfied by the characteristic function, is applicable to many other distributions.