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Integral of rational function with cosine hyperbolic function using rectangle contour

$$ \int^{\infty}_{-\infty} \frac{1}{(5 \pi^2 + 8 \pi y + 16y^2) }\frac{\cosh\left(y+\frac{\pi}{4} \right)}{\cosh^3(y)}dy=\frac{2}{\pi^3}\left(\pi \cosh\left(\frac{\pi}{4} \right)-4\sinh\left( \frac{\pi}{4}\right) \right)$$   $$\textit{proof}$$ Consider $$f(z) = \frac{\sinh(z)}{z \sinh^3(z-\pi/4)}$$ If we integrate around a contour of height \( \pi \) and stretch it to infinity we … Continue reading

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