Show that $$p.v.\int_{-\infty}^{\infty}\frac{e^{2x}}{\cosh(\pi
x)}dx=\text{sec}1$$ by integrating $\frac{e^{2z}}{\cosh(\pi z)}$ around
rectangles with vertices at $z=\pm p,p+i,-p+i.$
I asked this similar question, and was recommended this link stackexchange, but the answer provided is confusing.
I am not sure how they got those vertices, but with those vertices I got the parameterization:
$\begin{cases} \Gamma_1 = t, & -p\leq t\leq p \\\\ \Gamma_2= p+it, &0\leq t \leq 1 \\\\ \Gamma_3 = i-t, & -p\leq t\leq p \\\\ \Gamma_4=i(1-t)-p, & 0\leq t\leq 1. \end{cases}$
Thus I have $\Gamma_p =\Gamma_1 +\Gamma_2+\Gamma_3+\Gamma_4 .$
$f(z)=\frac{e^{2z}}{\cosh(\pi z)} = \frac{2e^{z(2+i\pi)}}{e^{2i\pi z}+1}$, where the denominator has a $\text{pole}=(2n+1)\frac{1}{2}$.
And this is as far as I can get.
Best Answer
Consider the integral
$$\oint_C dz \frac{e^{2 z}}{\cosh{\pi z}}$$
where $C$ is the above-described rectangle. On the one hand, this integral is equal to the integral about the individual legs of the contour, viz.:
$$\int_{-p}^p dx \frac{e^{2 x}}{\cosh{\pi x}} + i \int_0^1 dy \frac{e^{2 (p+i y)}}{\cos{\pi (p+i y)}} + \int_{p}^{-p} dx \frac{e^{2 (x+i)}}{\cosh{\pi (x+i)}} + i \int_1^0 dy \frac{e^{2 (-p+i y)}}{\cos{\pi (-p+i y)}}$$
Take the limit as $p \rightarrow \infty$. It should be clear that the 2nd and 4th integrals - those over the vertical legs of $C$ - will vanish in this limit. That leaves the 1st and 3rd integrals over the horizontal sections; these may be combined to produce
$$\left(1+e^{i 2}\right) \int_{-\infty}^{\infty} dx \frac{e^{2 x}}{\cosh{\pi x}}$$
This equals, by the residue theorem, the residue at the only pole within $C$, namely, $z=i/2$:
$$i 2 \pi \lim_{z->i/2} \frac{e^{2 z}}{\cosh{\pi z}} = i 2 \pi\frac{e^{i}}{\pi \sinh{(i \pi/2)}} = 2 e^{i}$$
Therefore
$$\int_{\infty}^{\infty} dx \frac{e^{2 x}}{\cosh{\pi x}} = \frac{2 e^{i}}{1+e^{i 2}}= \sec{1}$$
You should note that no $PV$ was needed here.