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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Analytic continuation of Euler's reflection formula with the Gamma function

Let $\widetilde\Gamma$ be an analytic continuation of $\Gamma$ on $\mathbb C\setminus(-\mathbb N_0)$. Show that the function $$\widetilde\Gamma(z)\widetilde\Gamma(1-z)-\frac{\pi}{\sin(\pi z)}$$ ...
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Sum over all residues is zero

Let $f$ be a rational function mapping to $\mathbb{C}$, $$f(z)=\frac{P(z)}{Q(z)}$$ with $\deg P\leq \deg Q -2$. I want to show that the sum over all residues is zero. What am I asked to show? I think:...
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Relationship between the residues $Res(g\circ\varphi,z_0)$ and $Res(g,w_0)$

Let $\varphi:U\rightarrow\mathbb{C}$ be holomorphic with $\varphi'(z_0)\neq 0$ for some $z_0\in U$. Let $g$ be another function having a pole of order $1$ in $w_0=\varphi(z_0)$. What is the ...
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3d-Integral with residue method

How to deal with angular dependency $\theta_{kr}$ in the attempt to calculate this integral: $$\int \frac{\mathrm {d^3} k}{(2 \pi)^3} \frac{e^{i \vec{k}\cdot\vec{r}}}{a-k^2+ib}$$ by residue method?
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complex analysis

Find the residue of the function $f(z) = z/ (8-z^3)$ at $z = 2.$ I tried with the formula $\lim_{z\rightarrow a} \frac{1}{(m-1)!} \frac{ d^{m-1}}{ dz^{m-1}} (z-a)^m f(z)$. but it becomes very tedious. ...
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Determine this real integral with the Residue-theorem.

$$\int_{-\infty}^{\infty}{\frac{\sin x}{x^4-6x^2+10}\,\mathrm dx}$$ I get that when I evaluate the $\frac{\sin x}{x}$ one, I work with $\frac{e^{ix} - e^{-ix}}{2ix}$, I create a huge semicircle and a ...
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How does one integrate $x^2 \frac{e^x}{(e^x+1)^2}$?

How can I show this? $$\int_{-\infty}^{\infty} x^2 \frac{e^x}{(e^x+1)^2} dx = \pi^2/3$$ I tried applying residuals, but the pole is of infinite(?) order.
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The function I have is $f(z)=\zeta(z)\Gamma(z − 1)\sin(\pi z)$ and I need to find all singularities and their residues so I can evaluate a clockwise contour integral for the contour $\left\lvert z+\... 5answers 400 views Contour Integration$\int_0^1\frac1{\sqrt[n]{1-x^n}}dx$I want to compute: $$\int^{1}_{0}\frac{1}{\sqrt[n]{1-x^n}}dx$$ for natural$n>1$using Residue Calculus. I am thinking of using some kind of a keyhole or bone contour that could go around the$n$... 0answers 26 views Improper integral using residue calculus and contours [duplicate] Can someone please show me how to solve this improper integral "using residue calculus and appropriate contours"? $$\int_0^{\infty} \frac{1}{x^4+1}dx$$ 1answer 19 views Complex contour integral Problem Show that $$\oint_{|z|=1} \dfrac {(z+w)(z^{n-1})} {z-w}dz=0$$ using Residue calculus, where$n<0$and$|w|<1$. 1answer 50 views Evaluate$I(x)=\int_{-\infty}^{\infty}\frac{e^{xv}}{1+e^v}dv$This is the last part of a multistage evaluation of$I(x)=\Gamma(x)\Gamma(1-x)$. Through various substitutions we get the integral $$I(x)=\int_{-\infty}^{\infty}\frac{e^{xv}}{1+e^v}dv$$ We're also ... 2answers 52 views Residue of$\frac {e^{sz}} {e^z+1}$I'm usually okay with calculating residues but in this case the denominator isn't a polynomial and I'm not sure what to do. Can someone tell me how to calculate the residue of$\dfrac {e^{sz}} {e^z+1}...
Locate the poles and the find the residues of the following function: $\dfrac {e^{5z}}{\left( z+i\pi \right) ^{3}}$ From my understand there should be a pole at $-i\pi$ so y the residue theorem we ...
$$\int_0^{\infty} \frac{x^2+1}{x^4+1}dx$$ What i've found are the singularities at: $e^{\pi/4+\pi/2n}$ for $n=0,1,2,3$. But i'm unsure how to calculate the integral since I don't want to include the ...