Mathematics Stack Exchange
Mathematics Stack Exchange

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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Why doesn't this residue method work for calculating $\sum_{k=1}^{k=\infty} \frac{\cos(k x)}{k^2}$

I want to calculate $S = \sum_{k=1}^{k=\infty} \frac{\cos(k x)}{k^2}$ using the method of residues. By using the function $f(z) = z \sum_{k=1}^{k=\infty} \frac{1}{k(z-k)}$, which has simple poles at $k = 1, 2, 3, ...$ with residue 1, and $h(z) =…
Kappie001
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What is the Residue of this function?

What is $$\operatorname*{Res}_{z=i \pi}{\frac{e^{(1-a)z}}{\left (1+e^{z} \right )^n}}$$ where $0
math.lv
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finding residue for complex analysis

I am having a tough time finding the residue for a function, suppose my test function is $$\frac{z^2}{{(z^2+a^2)}^2}$$ while I could determine the poles to be $+-ai$ and I know the formula to find the residue to be $$a_{-1}= \frac{1}{2\pi i} \int…
Xfrglkv
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Cauchy's residue application problem

I would like to know what did I do wrong. There's my problem : I= $\frac{1}{2\pi i}$ $\int_a \frac{1}{z^4+1}~dz$ Where a $x^2 + y^2 = 2x$ I already know: there're 4 poles, but only 2 fits for me (right ones): $e^\frac{i\pi}{4}$ and …
Edgar
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Pole and residue of the following function at infinity

I am confused about one particular problem regarding complex infinities. Suppose i have EXP[-z^2] with z being the complex number. Clearly it has poles at z=+i(infinity) and -i(infinity). . How to show that. If I replace w=1/Z then surely it doesn't…
kau
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Finding the residue of $\frac{1}{z(z^2+4)(z+2i)}$ at $z=-2i$

I've literally tried every technique I know of and they all lead to explosions of the kind $1/0$. Generally speaking the residue at $c$ for a function can be calculated as: $\frac{1}{(n-1)!}\frac{d^{n-1}}{dz^{n-1}}\Big((z-c)^n f(z) \Big)$ where $n$…
DLV
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Question about how to calculate a certain residue

I want to calculate the residue at $z=i$ for $f(z)=\frac{1}{(z^2+1)(z^2+9)(z^4+4)}$. I've calculated the residue correctly by using the formula $Res(f;i)=\lim_{z\to i}{(z-i)f(z)}$. I also know that it is possible to use the…
Jarvi79
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Evaluate the integral using Residue integration method

$$\int_0^{\pi} \frac{2 d\theta}{k - cos\theta}$$ What is k here? I'm not sure if I understand the question. I am told from my book the answer is $\frac{2 \pi}{\sqrt{k^2 - 1}}$ but not sure how we got here.
Jwan622
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How can I solve this integral using residue theorom?

$$ \int_0^{2 \pi} \frac{\sin ⁡x}{(1+\frac{1}{2}\cos x)} dx $$ I want to solve this. I have solved this here in the image but I have not reached a real number
Nima_Ebr
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Need help of finding residue

I need to find residues of $$7^n+11^n\pmod{19}$$ I have no idea where to start... n is a natural number.
Karagum
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