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seen Oct 17 '13 at 17:07

Oct
28
comment Show that $\int_{-\infty}^\infty\frac{\cos(x)}{e^x+e^{-x}}dx=\frac{\pi}{e^{\pi/2}+e^{-\pi/2}}$?
@sos440 - The top of the rectangle can be parameterized by $\gamma(z)=R+i{\pi}-2Rt$, $d{\gamma}=-i{\pi}t$ $0<t<1$. So we can write $\int_{\gamma}{f(z)dz}=\int_{t=0}^{1}\frac{e^{i(-R+i{\pi}-2Rt)}}{e^{R+i{\pi}-2Rt‌​}+e^{-{R+i{\pi}-2Rt}}}(-2Rdt)$. However, I'm struggling to figure out what happens as R approaches infinity. It appears to me as if the integral also approaches infinity, due to the $-2Rdt$ term. I suppose there is an $e^R$ term in the denominator, but I'm just not sure how to work it all out.
Oct
28
comment Show that $\int_{-\infty}^\infty\frac{\cos(x)}{e^x+e^{-x}}dx=\frac{\pi}{e^{\pi/2}+e^{-\pi/2}}$?
@sos440 - I think I understand the hint better now. We aim to compute the residues at each pole of f(z) in the upper half plane. But because the denominator of f consists of terms which are both exponential functions, we can think of them as sinusoids, so we only need to look at the points for which |Im(z)|<=pi (theta<=pi). However, we send R to infinity to analyze the entire upper half plane. I understand that the integral tends to 0 when parameterizing the left and right sides of the rectangle, but I'm struggling to understand for the top and bottom (see next comment for details).
Oct
27
comment Show that $\int_{-\infty}^\infty\frac{\cos(x)}{e^x+e^{-x}}dx=\frac{\pi}{e^{\pi/2}+e^{-\pi/2}}$?
The integral above appears to go to 0 as R goes to infinity. However, why would I want R to go to infinity? I suppose I don't fully understand the hint
Oct
27
comment How to relate $2\sin(3\pi/8)-2\sin(7\pi/8)$ and $\csc(3\pi/8)$?
Thanks for the answer! According the textbook solution, cosecant is correct, so that must be wrong.
Oct
15
comment Power series expansion of $\frac{z^2}{1-z}$?
Sheesh, wish I wouldn't have wasted so much time on this - either way, at least I don't have to waste any more! Thanks!