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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.

Questions on the evaluation of residues, on the evaluation of integrals using the method of residues or in the method's theory. The method consists of using a closed contour on the complex plane to evaluate complex or real integrals.

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Evaluating the Residue of $\csc^n(z)z^{n-k}$ at $z=0$

I'm hoping to give $\underset{z=0}{\text{Res}}\left[\csc^n(z)z^{n-j}\right]$ a nice form for nonnegative integers $n$ and $j$ but the only way I can think of is going through and evaluating $$\frac1{(j-1)!}\lim_{z→0}\left(\frac…
Spador Yedi
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Evaluate the following improper integral using Residue Theorem

$$\int_0^{\pi} \frac{2 d \theta}{k - cos \theta}$$ So I know this theorem: $$\int_{-\infty}^{\infty} f(x) dx = 2 \pi i \sum Res f(z)$$ which makes me believe while using this theorem, I eventually need to divide the resulting value by half. So first…
Jwan622
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Find the singularities and residues that correspond. Show details

Find all the singularities in the finite plane and the corresponding residues. Show the details. $$f(z) = \frac{1}{1-e^z}$$ So I'm trying to find the singularities. So the singularities are located when the denominator = 0 and that occurs when…
Jwan622
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Residue associated with a double pole

This is a follow-up to a previous question that I deleted because I realised that it was badly posed. Here's a (hopefully) better phrased version. I need to compute the residue $\text{Res}[F(z),z_0]$ of a function $F(z)$ at a point $z=z_0$, where…
Sharat V Chandrasekhar
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Question related to Weierstrass approximation Theorem

While studying Weierstrass approximation Theorem, I realized continuity on a compact set is necessary to solve the Theorem. Then, here is my question. Does there exist a sequence $(p_n)_{n\ge 1}$ of polynomial such that it converges uniformly to…
fivestar
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Residue of a function with (essential) singularities at both $0$ and $\infty$

I need to take the residue of the following function at infinity: $$ f(z) = \oint_\infty \left(\frac{e^{-\alpha/z}e^{-\alpha z}}{z}\right)dz $$ Which, up to a sign, is invariant under inversions $z\rightarrow 1/w$, since $dz \rightarrow…
Akoben
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A question on using Residues Theorem

I am having a doubt on an example on Conway's "Funtions of one Complex Variables". At example 2.9, page 117. At some point, it is stated that, since $$ a+\cos(\theta)=\dfrac{z^2+2az+1}{2z} $$ it must imply…
Marra
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Evaluating an integral of a real function using the residue theorem

Using the residue theorem, I have shown that $\oint_C f(z) dz = \frac{\pi i}{e^2}$ where $f(z) = \frac{ze^{iz}}{(z^2+4)^2}$ and $C$ is the closed curve consisting of the horizontal line $y = 0$ from $x = -R$ to $x = R$ and the semicircle $y =…
1123581321
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Evaluate $\int_{-1}^1 \frac{1}{x-z}\frac{dx}{\sqrt{1-x^2}}$

I will be happy if someone checks my solution to $$I=\int_{-1}^1 \frac{1}{x-z}\frac{dx}{\sqrt{1-x^2}}=\frac{- \pi}{\sqrt{z^2-1}}$$ where $z \in \mathbb{C} \setminus[-1,1]$. Here is my computation: Substituting $x=\cos(t)$, we get…
Aleph-null
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coulomb potential

How to compute the following integral : \begin{equation} I = \int \frac{e^{-q|\mathbf r|}}{|\mathbf r|}e^{i\mathbf k\cdot\mathbf r}. d\mathbf r \end{equation} where this 3D volume integral is considered carried out on an unbound domain. This looks…
Ronan Tarik Drevon
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Simple Residue Question

I'm trying to calculate the residue of $g(z)=1/\sin(z)$ at the simple pole of $z=0$. For some reason, I can't figure out how to do it since when I compute the taylor expansion of $\frac {z}{\sin(z)}$ I do not get a $z^{-1}$ term. Thanks for your…
juper
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Problem with residue theorem

I'm having trouble with the following problem: $$ \int \frac{z}{(z^2+1)^2}\ dz $$ I need to integrate around the circle centered at the origin with radius 3/2. When I try to solve using the residue theorem, I find that there are two poles of…
Evan
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Residue of a function

I am calculating the residue of $$ \frac{\sin z}{z^{2}(\pi-z)} $$ Since there is only $1$ pole (simple) at $0$, I went about calculating the residue according to the rule: $$Res\left [ \frac{f(x)}{g(x)}, z_{0} \right ] = \lim_{z\rightarrow z_{0}} …
Rohin Garg
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reformulation of square residues for odd numbers?

If you look at the table of square residues: http://en.wikipedia.org/wiki/Quadratic_residue#Table_of_quadratic_residues you will find that for x^2 mod N where N is odd, we have N different residues but with simetry in half: Example: N=25 mod…
Bojan Vasiljević
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Complex Numbers - Equation manipulation

Please see link, I'm wondering how you end with RHS of equation on second line of image. Please can explain the step and algebra manipulation used Thanks
Arsenal123
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