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.

2742 questions
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Evaluation of real integral by residues

I have to calculate the integral $$I=\int_{0}^{2\pi}\frac{\cos^2(3\theta)}{5-4\cos(2\theta)}d\theta$$ using residues, but I'm having trouble calculating the residue in $z=0$. If you rewrite the integral by stating $z=exp(i\theta)$, and letting…
Lessa121
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Residue of $\frac{z}{Log{z}}$ at $z=1$?

What is the residue of $\frac{z}{Log{z}}$ at $z=1$? I tried expanding log into it's series, factoring out a $z-1$ and then applying the geometric series expansion, but I'm just getting a series that starts at $k=1$....
user82004
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Calculating residue of $(z^3+1)^{-3}$ at $z=\exp(i\pi/3)$

I'm struggling to calculate the residue of $(z^3+1)^{-3}$ at $z=\exp(i\cdot\pi/3)$, which is the pole for the given contour. Using the formula for residue has resulted in a prohibitive double derivative (3rd order pole) and limit evaluation. I'm not…
user149760
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Residue Calculation Problem

I need to find the residue of the following equation at $z = 0$. $$ \frac{\cot(z)\cot(hz)}{z^3}$$ My attempt is as follows: The residue will be the coefficient of $1/z$ in the Laurent Series expansion. Expansion of $$\cot(z) = 1/z - z/3 -…
bhavesh
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How to calculate $I(x)=\int_{-1}^1 \frac{dt}{\sqrt{1-t^2}(t-x)}$ by Residue theorem [$|x|>1$]

How to calculate $\int_{-1}^1 \frac{dt}{\sqrt{1-t^2}(t-x)}$ for $|x|>1$ by Residue theorem? I could do is just as: $$I(x)=\int_{-\pi/2}^{\pi/2}\frac{d\theta}{\sin\theta-x}\\ =\frac{1}{2}\int_{\pi}^\pi \frac{d\tau}{\sqrt{\frac{1-\cos…
XLDD
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Verity of Residue theorem of [0,2pi]

After I turn $$ cos\theta=\frac12(z+\frac1{z})$$and $$ d\theta=\frac1{iz}dz$$ the denominator become a mess $$ \frac{dz}{(a^2+\frac{b^2}4(z^2+2+\frac1{z^2})+\frac{ab}2(z+\frac1z))(iz)}$$ How can a find out the pole?
user111605
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Residue Theorem for Denominator with $e^z$

$$ f(z)=\frac{z^3}{e^z-1} $$ Is this a simple pole at $z=0$ or some other types of pole? If it is a simple pole, what is its residue? Is it using this formula or other else? $$ \lim_{z\to 0}=zf(z) $$
user111605
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Integrate $\csc(1/z)$ on a contour through $0$

$\gamma$ is the (triangle) contour $i\longrightarrow-i\longrightarrow1\longrightarrow i$. $\def\rmd{\mathop{}\!\mathrm{d}}$ Using Mathematica to evaluate the directional limit at $0$ on $\gamma$ Limit[1/Sin[1/z], z -> 0, Direction -> -I] I get $0$.…
hbghlyj
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Finding contour integral result using residue theorem

I was wondering what the best way to solve this would be. I was using the cauchy integral residue formula which revolved around taking 2 derivatives since the poles are at +/- i and are of order 3. Quickly realized that taking double derivatives are…
qwerty2277
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Computing residues for the equation: $(\tan(z)+2)/(4z^2+\pi z)$

I'm looking at the wolframalpha residue calculator for the following equation: $$\frac{\tan(z)+2}{4z^2+\pi z}$$ which gives me the residue: $2/\pi $ for Root 1 which I found, but it also says residue: $-1/\pi$ for Root 2. Root 1: $z=0$ Root 2:…
Claire
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how to integrate when the singularity is right on the integration path

$$ \int_{0}^{2\pi}\frac{1}{1+\cos x}dx $$ I don't know how to do the integration when the singularity is right on the integration path
Ironman1965
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Residue, last step

I'm doing some fouriertransform, and can't get the last step correct when Im comparing to an old exam with the question. Its the fourier transform of $f(x)=\frac{1}{1+7x^2}$. Where my solution is wrong is when I compute the actual residue. My…
uoiu
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Residue of $\frac1{(z^3+1)^2}$

I am practicing with the residue calculation and I am having a bit of misunderstanding. Let $f(z) = \frac{1}{(z^3+1)^2}$. I want to calculate the Res of $f$ in all its singularities which are the poles $-1 , e^{\frac{i \pi}{3}}$ and $ e^{\frac{-i…
MWSC1996
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Residue of $\frac{i}{(z-2)(2z-1)}$ at $z=1/2$

Trying to do this but can't seem to figure out where I'm going wrong. \begin{align*} \textrm{Res}(f,1/2) &= \frac{i}{\frac{1}{2}-2}\\ &= -\frac{2}{3}i \end{align*} The correct answer however is $-i/3$.
confusedaf
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Residues of $\sin\left(\frac{z^2+1}{z}\right)$

Looking for assistance with the following function.. i need to calculate the Residues of $$f(z)= \sin\left(\frac{z^2+1}{z}\right)$$ regrading my way: $$\sin\left(\frac{z^2+1}{z}\right)=\sin(z)\cos(\frac{1}{z})+\cos(z)\sin(\frac{1}{z}) $$ $$=…
user862242
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