Three Methods of Integrating a Class of Rational Functions

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Abstract

We give three derivations of $I_n = \int_{-\infty}^{\infty} \frac1{(x^2 + a^2)^{n+1}} \mathrm{d} x$ and show that it evaluates to $\frac{\pi (2n-1)!!}{n! , 2^n a^{2n+1}}$ for $n \in \mathbb{N}_0$ and $\frac{1}{a^{2n+1}} B{\left(\frac12, n + \frac12\right)}$ for $n \in \mathbb{R}^+_0$. The first method exploits a recurrence relation involving the indefinite counterpart of $I_n$, while the second method uses contour integration. The third employs a substitution to express $I_n$ in terms of the Beta function. We then reconcile the difference between the discreteness and continuity of the two closed forms found for $I_n$. We proceed to prove a connection between $I_n$ and $\frac{1}{\sqrt{1 - t}}$. Lastly, we use our results to derive identities involving $\pi$, the Gaussian integral and the Beta function, such as $\pi = \sqrt{1 - t} \sum_{n = 0}^{\infty} B{\left(\frac12, n + \frac12\right)} t^n$ and $\sum_{n = 0}^{\infty} \frac{1}{n} \frac{B(n+1, k+1)}{B(n, k)} = 1$.

Citation

Chong, 2024. http://asdia.dev/papers/integral_rational .

@article{TMICRF,
author = {Eytan Chong},
year = {2024},
title = {Three Methods of Integrating a Class of Rational Functions},
url = {http://asdia.dev/papers/integral_rational}}