how to calculate the following integral$\int_{-\infty}^{\infty}\frac{1}{\left(t^2+\pi^2\right)^2 \cosh(t)}dt$

Question

calculate the following integral $\int_{-\infty}^{\infty}\frac{1}{\left(t^2+\pi^2\right)^2 \cosh(t)}dt$ I need to very hollowing steps.thank you in advance

Answer 1

Contour integration may save you, but here I want to present some real-analytic method. Let $I$ denote the integral. With the change of variable $t = \pi x$, we have

$$ I = \frac{1}{\pi^{3}} \int_{-\infty}^{\infty} \frac{dx}{(x^{2} + 1)^{2} \cosh(\pi x)}. $$

1. Preliminary

Now for an $L^{2}$ function $f$, we denote its Fourier transform by

$$ \hat{f}(\omega) = \int_{-\infty}^{\infty} f(x)e^{-2\pi i \omega x} \, dx. $$

Then we have the following lemmas:

Lemma 1. The function $f(x) = \mathrm{sech}(\pi x)$ is invariant under this Fourier transform. That is, $\hat{f}(\omega) = f(\omega)$.

Proof. Direct calculation is possible.

Lemma 2. For two $L^{2}$ functions $f$, $g$, we have $$ \int_{\Bbb{R}} f \hat{g} = \int_{\Bbb{R}} \hat{f} g. $$

Proof. Prove this first for nice functions (Schwartz functions) with Fubini's Theorem and then use approximation argument.

Lemma 3. We have $$ \bigg( \frac{1}{(1+x^{2})^{2}} \bigg)^{\wedge}(\omega) = \frac{\pi}{2} (1+2\pi|\omega|) e^{-2\pi|\omega|}. $$

Proof. This follows from the following famous formula: $$ \int_{-\infty}^{\infty} \frac{\cos \alpha x}{1+x^{2}} \, dx = \pi e^{-|\alpha|}. $$

2. Calculation

Let $f(x) = (1+x^{2})^{2}$ and $g(x) = \mathrm{sech}(\pi x)$. Then

\begin{align*} I &= \frac{1}{\pi^{3}} \int_{\Bbb{R}} f(x)\hat{g}(x) \, dx = \frac{1}{\pi^{3}} \int_{\Bbb{R}} \hat{f}(x)g(x) \, dx \\ &= \frac{1}{\pi^{2}} \int_{0}^{\infty} (1+2\pi x)e^{-2\pi x}\mathrm{sech}(\pi x) \, dx \\ &= \frac{2}{\pi^{2}} \int_{0}^{\infty} (1+2\pi x)\frac{e^{-\pi x}}{1+e^{-2\pi x}} \, dx \\ &= \frac{2}{\pi^{2}} \sum_{n=0}^{\infty} (-1)^{n} \int_{0}^{\infty} (1+2\pi x) e^{-(2n+3)\pi x} \, dx \\ &= \frac{2}{\pi^{3}} \sum_{n=0}^{\infty} (-1)^{n} \left( \frac{1}{2n+3} + \frac{2}{(2n+3)^{2}} \right)\\ &= \frac{2}{\pi^{3}} \left( 3 - \frac{\pi}{4} - 2G \right), \end{align*}

where $G$ is the Catalan constant