# Questions tagged [residue-calculus]

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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### Undefined residue at infinity

I am trying to find the residue of $\frac{(cos(sin(z)))}{z^2}$ at $\infty$. What I did was $Res(f(x),\infty) = Res (f(\frac{1}{z}), z=0)$. What I got was $Res(z=0)=-z^2cos(sin(\frac{1}{z}))$ which is ...
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### Why is the residue $\mathop{\mathrm{Res}}_{z=\pi/2} \frac{z}{\cos z}$ not $0$?

I'm supposed to get the residue of the function $\dfrac{z}{\cos z}$ at $z = \pi/2$. Here is my solution: \begin{align*} \mathop{\mathrm{Res}}_{z=\pi/2} \frac{z}{\cos z} &= \frac{1}{(m-1)!} \lim_{z ...
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### Find residue of $f(z) = \frac{\sin z}{(z^2+1)^2}$ at $z = \infty$

Find residue of $f(z) = \dfrac{\sin z}{(z^2+1)^2}$ at $z = \infty$. Then this is the same as finding the residue at $z=0$ for $\dfrac{-1}{z^2}f(1/z)= \dfrac{-z^2 \sin 1/z}{(z^2+1)^2}$ $z = 0$ is a ...
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### A quick way to calculate residues of logarithmic derivatives

Assume that $U\subset \mathbb{C}$ is open an let $f(s)$ be a meropmorphic function on $U$. Consider the logarithmic derivative of $f(s)$ where I mean the meromorphic function $\frac{f'(s)}{f(s)}$. I ...
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### Сalculate the integral PV $\int_{0}^{\infty} \frac{dx}{x^\alpha(x-a)}dx.$

Сalculate the integral $$\mathrm{PV}\hspace{-0.5ex}\int_{0}^{\infty} \frac{dx}{x^\alpha(x-a)},$$ where $0<\alpha <1$ and $a>0$. So we have simple poles $z = 0$ and $z = a$; We can build ...
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### Confusion regarding pole of a complex function.

I am a graduate student.I am studying complex analysis.I encountered the following problem in a lecture: Find the residue of $f(z)=\frac{z-\sinh(z)}{z^2\sinh(z)}$ at $z=\pi i$. Now,this problem is ...
Set up Suppose $\gamma$ a simple closed curve, oriented in a counterclockwise direction. $f(z)$ is a complex polynomial $$f(z)=a_n(z-z_n)^n+a_{n-1}(z-z_{n-1})^{n-1}+\cdots+a_0.$$ We already know ...