Tag Archives: rectangle

Integrating a fraction of exponential and trignometric using rectangular contour

[Ex 41 ] Watson’s complex integration $$\int^\infty_0 \frac{\sin(ax)}{e^{2\pi x}-1}\,dx = \frac{1}{4}\coth\left(\frac{a}{2} \right)-\frac{1}{2a}$$ $$\textit{solution}$$ By integrating the following function $$f(z) = \frac{e^{iaz}}{e^{2\pi z}-1}$$ The function is analytic in and on the contour, indented at the poles of the function Hence by … Continue reading

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Integration related to gamma function using rectangle contour

[Ex 9] Watson’s complex integration $$\int^{\infty}_{-\infty}e^{-x^2}\,\cos(2ax)dx=e^{-a^2}\sqrt{\pi}$$ $$proof$$ Integrate the following function $$f(z) = e^{-z^2}$$ Use the following contour Note that the function is entire, hence $$\int^{L}_{-L} e^{-t^2}\,dt+\int^{L+ai}_{L} e^{-t^2}\,dt+\int^{-L}_{-L+ai} e^{-t^2}\,dt+\int^{-L+ai}_{L+ai}e^{-t^2}\,dt=0$$ For the forth integral use the substitution \(x= t-ai\) $$\int^{-L+ai}_{L+ai}e^{-t^2}\,dt=\int^{-L}_{L}e^{-(x+ai)^2}\,dx=-e^{a^2}\int^{L}_{-L}e^{-x^2}\,e^{-2iax}dx$$ Take … Continue reading

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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

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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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