# Tag Archives: hyperbolic

## Contour integration for a rational function of cos and cosh

$$\int^{\infty}_{-\infty} \frac{\cos(ax)}{\cosh(x)} \,dx = \pi \, \mathrm{sech} \left( \frac{\pi a}{2}\right)$$ $$\textit{proof}$$ Consider $$f(z) = \frac{e^{iaz}}{\sinh(z)}$$ If we integrate around a contour of height $\pi$ and stretch it to infinity we get By taking \( T \to \infty … Continue reading