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Answer by xpaul for Showing that $\int_{-\infty}^{\infty} \frac{\sinh...
I have a simple way to calculate your old question. Note$$ \frac{\sinh(ax)}{\sinh(\pi x)} \cos(bx) = \frac{\sinh(ax)}{\sinh(\pi x)}\cosh(ibx)=\frac{\sinh((a+bi)x)+\sinh((a-bi)x)}{\sinh(\pi x)} $$and $$...
View ArticleAnswer by Random Variable for Showing that $\int_{-\infty}^{\infty}...
I indeed integrated the wrong function.Integrating $ \displaystyle f(z) = \frac{e^{(a+ib)z}}{\sinh \pi z}$ around the same contour, we get$$ \begin{align} \text{PV} \int_{-\infty}^{\infty} f(x) \, dx +...
View ArticleShowing that $\int_{-\infty}^{\infty} \frac{\sinh ax}{\sinh \pi x} \, \cos bx...
I want to show that $$ \int_{-\infty}^{\infty} \frac{\sinh ax}{\sinh \pi x} \, \cos bx \, dx = \frac{\sin a}{\cos a + \cosh b} \, , \quad -\pi < a < \pi . $$I tried integrating $f \displaystyle...
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