# Tagged Questions

Questions on the evaluation of residues, on the evaluation of integrals using the method of residues or in the method's theory.

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### Residue of $\cot z$ at $z= 0$

I need to calculate the Residue of $\cot z$ at the pole $0$ . Now, I think that the order of the pole is $1$. But I do not know how to show it. So, assuming the order of pole to be $1$, I calculated ...
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### Integrating $\int_{-\infty}^{\infty}\frac{e^{ax}}{1+e^x}dx$ for $0<a<1$.

I'm supposed to find the value of $\int_{-\infty}^{\infty}\frac{e^{ax}}{1+e^x}dx$ for $0<a<1$. I wanted to integrate over the upper semicircle of radius $R$, and take the limit as $R\to\infty$. ...
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Let $R=\frac{P}{Q}$ where $P$ and $Q$ are polynomials such that $Q$ has not zeros in $\mathbb{R}$ and $\mbox{deg}(Q)=\mbox{deg}(P)+1$. Show that the Cauchy principal value of $\int_{-\infty}^{\... 1answer 33 views ### Compute$\mathrm{Res}(\frac{e^{iz}}{z(z^2+1)^2},i)$I have to compute$\mathrm{Res}(\frac{e^{iz}}{z(z^2+1)^2},i)$. Do I have to use the result from$Res[f/g,z_0]=\frac{f(z_0)}{g'(z_0)}$-Proof (I think I have a pole of order$2$)? Otherwise, how ... 1answer 27 views ### Prove that$\mathrm{Res}(\frac{f(z)}{g(z)}, z_0) = \frac{f(z_0)}{g'(z_0)}$[duplicate] Show that if$f$and$g$are analytic on a neighborhood of$z_0$with$f(z_0)\not= 0$and$z_0$is a simple zero of$g$, then we have$\mathrm{Res}(\frac{f(z)}{g(z)}, z_0) = \frac{f(z_0)}{g'(z_0)}$... 1answer 20 views ### Compute$\mathrm{Res}(\frac{1+2z+3z^2}{1+z+z^2-3z^3},1)$I have to compute$\mathrm{Res}(\frac{1+2z+3z^2}{1+z+z^2-3z^3},1)$. I know that$\mathrm{Res}(f,z_0)+a_{-1}= \int_{C_p} \frac{f(z)dz}{z-z_0}$, where$C_p$is simply the circle at$z_0$with radius ... 0answers 28 views ### How to solve the laplace transform of$f_m(t_m)$=$f_1(t)\int_{0}^{\alpha} f_2(\tau) d\tau$+$f_2(t)\int_{0}^{\alpha} f_1(\tau) d\tau$. Could you please help me to solve the following : if$t_m$= min($t_1$,$t_2$) The probability density function$t_1$is$f_1(t_1)$and$t_2$is$f_2(t_2)$then$f_m(t_m)$is the probability density ... 2answers 38 views ### Residue Theorem for function quotients. let$G$be an open disc centered around$z_0$of radius$r$. Let$f(z),g(z)$be holomorphic functions on$G$. such that$f(z)$has a simple zero at$z_0$. Find an expression for the residue of$\frac{...
$$\int_{-\infty}^{\infty} \frac{\cos(mx) dx}{e^{-x}+e^x} = \frac{\pi}{e^{m\pi /2}+e^{-m\pi /2}}$$ I need to evaluate the above integral specifically using a rectangluar contour and at some point ...