Use residues to evaluate $\int_0^\infty \frac{\cosh(ax)}{\cosh(x)}\,\mathrm{d}x$, where $|a|<1$

Question

Use residues to evaluate $$ \int_0^\infty \frac{\cosh(ax)}{\cosh(x)}\,\mathrm{d}x $$ where $|a|<1$.

Try considering the integral of the form $$ \int_C \frac{\exp(az)}{\cosh(z)}\,\mathrm dz, $$ where $C$ is the contour given by $y=0,\, y=\pi,\, x=-R,\, x=R$.

Answer 1

As the original query that asked to use residues has not been answered completely I will contribute some ideas.

Suppose $a$ is a rational number $p/q$ where $p<q$ and $p-q$ is odd. Use a rectangular contour that consists of four segments: $\Gamma_0$ along the real axis from $-R$ to $R$, $\Gamma_1$ parallel to the imaginary axis to $R + \pi i q$, $\Gamma_2$ parallel to the real axis but in the opposite direction to $-R + \pi i q$ and finally, $\Gamma_3$ parallel to the imaginary axis to $-R$ on the real axis.

Now set $$f(z) = \frac{e^{az}+e^{-az}}{e^z+e^{-z}}$$ so that we are looking for $$\frac{1}{2} \int_{-\infty}^\infty f(z) dz$$ and integrate $f(z)$ along $\Gamma_0 - \Gamma_1 - \Gamma_2 - \Gamma_3$. Examine each segment in turn as $R$ goes to infinity. Clearly the integral along $\Gamma_0$ is simply the integral we are looking for. The contributions of $\Gamma_1$ and $\Gamma_3$ vanish in the limit. Along $\Gamma_2$ we have $x= t + \pi i q$, getting $$ \int_\infty^{-\infty} \frac{e^{\frac{p}{q}t + \pi i p}+e^{-\frac{p}{q}t - \pi i p}}{e^{t+ \pi i q}+e^{-t- \pi i q}} dt = - (-1)^{p-q} \int_{-\infty}^\infty \frac{e^{\frac{p}{q}t}+e^{-\frac{p}{q}t}}{e^{t}+e^{-t}} dt =\int_{-\infty}^\infty \frac{e^{\frac{p}{q}t}+e^{-\frac{p}{q}t}}{e^{t}+e^{-t}} dt. $$ The last equality is because $p-q$ is odd.

To conclude we need to compute the poles and residues inside our contour. The poles are at $$\rho_k = \frac{1}{2}\pi i + \pi i k$$ and the residues are $$\lim_{z\to \rho_k} \frac{(z-\rho_k) (e^{az} + e^{-az})}{e^z + e^{-z}} = \lim_{z\to \rho_k} \frac{(z-\rho_k) (a e^{az} -a e^{-az}) + e^{az} + e^{-az}}{e^z - e^{-z}}.$$ But $$ \lim_{z\to \rho_k} \frac{1}{e^z - e^{-z}} = \frac{1}{i e^{\pi i k} - (-i) e^{-\pi i k}} = \frac{1}{i e^{\pi i k} + i e^{-\pi i k}} = \frac{e^{\pi i k}}{i (1+1)} = \frac{(-1)^k}{2i}$$ so that finally $$\operatorname{Res}_{z=\rho_k} f(z) = \frac{(-1)^k}{2i} \left( e^{a\rho_k} + e^{-a\rho_k}\right).$$ With $J$ being the integral we are looking for and $I$ the integral along $\Gamma_0$ we have $$ J = \frac{1}{2} I = \frac{1}{4} 2 I = \frac{1}{4} 2\pi i \sum_{k=0}^{q-1} \operatorname{Res}_{z=\rho_k} f(z)$$ The conclusion is that $$ J = \frac{1}{2} \pi i \sum_{k=0}^{q-1} \operatorname{Res}_{z=\rho_k} f(z) = \frac{\pi}{4} \sum_{k=0}^{q-1} (-1)^k \left( e^{a\rho_k} + e^{-a\rho_k}\right) = \frac{\pi}{2} \sum_{k=0}^{q-1} (-1)^k \cosh(a\rho_k).$$ where we have used the fact that $1/2 + k < q$ implies that $k$ runs up to $q-1.$

Edit. Use the following bound to see that the integral along $\Gamma_1$ vanishes (set $z= R + it$ with $0\le t \le \pi q$): $$ \left| \int_{\Gamma_1} f(z) dz \right| = \left| \int_0^{\pi q} \frac{e^{aR + ait} + e^{-aR -ait}}{e^{R+it} + e^{-R-it}} i dt \right| \le \int_0^{\pi q} \frac{e^{aR} + e^{-aR}}{e^{R} - e^{-R}} dt = \pi q e^{-(1-a) R} \frac{1-e^{-2aR}}{1-e^{-2R}}$$ Now certainly we have $$\lim_{R\to\infty}\frac{1-e^{-2aR}}{1-e^{-2R}} = 1$$ so that the integral is $\theta(e^{-(1-a) R})$ which goes to zero as $R$ goes to infinity. The integral along $\Gamma_3$ is done the same way.

Answer 2

There is some additional simplification that can be done which I'll do in a new answer because my browser is not coping well with those large formulas where speed is concerned.

We have $$ \frac{\pi}{2} \sum_{k=0}^{q-1} (-1)^k \cosh(a\rho_k) = \frac{\pi}{4} \sum_{k=0}^{q-1}(-1)^k e^{a\rho_k} + \frac{\pi}{4} \sum_{k=0}^{q-1}(-1)^k e^{-a\rho_k}$$

The first sum is $$ \sum_{k=0}^{q-1} (-1)^k e^{a\rho_k} = \sum_{k=0}^{q-1} (-1)^k e^{a i \pi /2} e^{a \pi i k} = e^{a i \pi/2} \frac{1-(-e^{a \pi i})^q}{1 + e^{a \pi i}} $$ which is $$ e^{a i \pi/2} \frac{1-(-1)^q e^{p \pi i}}{1 + e^{a \pi i}} = e^{a i \pi/2} \frac{1-(-1)^{p+q}}{1 + e^{a \pi i}} = e^{a i \pi/2} \frac{2}{1 + e^{a \pi i}} = \frac{1}{\cos (a\pi/2)}$$

The second sum is $$ \sum_{k=0}^{q-1} (-1)^k e^{-a\rho_k} = \sum_{k=0}^{q-1} (-1)^k e^{-a i \pi /2} e^{-a \pi i k} = e^{-a i \pi/2} \frac{1-(-e^{-a \pi i})^q}{1 + e^{-a \pi i}} $$ which is $$ e^{-a i \pi/2} \frac{1-(-1)^q e^{-p \pi i}}{1 + e^{-a \pi i}} = e^{-a i \pi/2} \frac{1-(-1)^{p+q}}{1 + e^{-a \pi i}} = e^{-a i \pi/2} \frac{2}{1 + e^{-a \pi i}} = \frac{1}{\cos (a\pi/2)}$$

It follows that the original sum and the integral is $$J = \frac{\pi}{2} \frac{1}{\cos(a \pi/2)}.$$

Answer 3

This doesn't use residues until we use $$ \sum_{k\in\mathbb{Z}}\frac{(-1)^k}{z+k}=\pi\csc(\pi z) $$ which can be proven using residues.

We just expand things in powers of $e^x$: $$ \begin{align} &\int_0^\infty\frac{\cosh(ax)}{\cosh(x)}\,\mathrm{d}x\\ &=\int_0^\infty e^{(a-1)x}\frac{1+e^{-2ax}}{1+e^{-2x}}\,\mathrm{d}x\\ &=\int_0^\infty\left(e^{(a-1)x}-e^{(a-3)x}+e^{(a-5)x}-\dots\right)\,\mathrm{d}x\\ &+\int_0^\infty\left(e^{(-a-1)x}-e^{(-a-3)x}+e^{(-a-5)x}-\dots\right)\,\mathrm{d}x\\ &=\frac1{1-a}-\frac1{3-a}+\frac1{5-a}-\dots\\ &+\frac1{1+a}-\frac1{3+a}+\frac1{5+a}-\dots\\ &=\frac1{a+1}-\frac1{a+3}+\frac1{a+5}-\dots\\ &-\frac1{a-1}+\frac1{a-3}-\frac1{a-5}-\dots\\ &=\frac12\left(\dots+\frac1{\frac{a+1}2-2}-\frac1{\frac{a+1}2-1}+\frac1{\frac{a+1}2}-\frac1{\frac{a+1}2+1}+\frac1{\frac{a+1}2+2}-\dots\right)\\ &=\frac12\pi\csc\left(\pi\frac{a+1}2\right)\\ &=\frac\pi2\sec\left(\frac\pi2a\right) \end{align} $$