# 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

[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