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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Solve real integrals $\int_0^{2\pi} \frac{d\theta}{1+a\cos(\theta)} = \frac{2\pi}{\sqrt{1-a^2}}$ using complex variables.

I am trying to verify that $\displaystyle\int_0^{2\pi} \frac{d\theta}{1+a\cos(\theta)} = \frac{2\pi}{\sqrt{1-a^2}}$, for $-1\lt a \lt 1$. So far I replaced $\cos(\theta)$ with $\dfrac{z+\frac{1}{z}}{2}$ and $d\theta$ with $\dfrac{dz}{iz}$, and…
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When doing Cauchy's residue theorem what does $\gamma$ mean?

I'm going through a complex analysis question that says : Let $R$ be a positive real number greater than $2$, let $γ_1\colon[−R,R]\longrightarrow\mathbb{C}$ be defined by $\gamma_1= t$, let $\gamma_2= S(0,R)$ and let $\gamma=\gamma_1\oplus\gamma_2$.…
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How to solve this using residue calculus?

$$ \int_{-\infty}^{\infty} \, \frac{\sin(k)}{k}e^{ikx} dk$$ I'm not too sure how to go about this. I think there is a pole at $z=0$ but then the residue is also$ =0 $ so i'm a bit lost :/ Any help would be appreciated!
jdhokia
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Find the residue of $\frac{e^{1/z}}{(z-1)^2}$ at $z=0$.

Find the residue of $\frac{e^{1/z}}{(z-1)^2}$ at $z=0$. How to obtain Laurent Series?
shansh0201
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How to calculate $\int_C {\sin(z^2)\over(z-3)^2(z+1)(z+4)}$?

Finding $\int_C {\sin(z^2)\over(z-3)^2(z+1)(z+4)}$ which is the counterclockwise circle with center at origin and radius $=5$ I tried finding the Laurent series and then using the Residue Theorem. For the Laurent series I got $${1\over…
Heavenly96
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Difficult Integral with Branch Cut

The integral I'm working on is $$I(x)=\frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}ds\frac{1}{s}e^{x(s-s^{1/2})}\ .$$ I'm told the path of integration is a vertical line to the right of the origin. First I need to find the branch point,…
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Why does this method work?

I have been reading my professor's solutions to a problem where I was instructed to find the residue at a singularity of a particular function. The function is: $$f(z) = \frac{1}{1+\mathrm{cosh}(z)}$$ The singularities for this function are at…
Sriotchilism O'Zaic
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Application of residue theorem for improper integrals

While i am reading one example in the book, i came across the book teaching me how to evaluate $\int_{-\infty}^{\infty}\dfrac{\sin x}{x}dx$ by using residue theorem. However, while they say we still construct a semi circle with radius $R$ and…
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Showing there exists a function $h$ such that $h'(z) = f(z)$, when $f$ two poles with residues that sum to 0

I'm reviewing some complex analysis for finals season, and I ran into this problem in a chapter about the residue theorem. Let $f$ be analytic on $\mathbb{C}$ except for poles at 1 and -1. Assume further that $Res(f; 1) = -Res(f;-1)$. Then show…
Kit
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Complex integral, Cauchy's theorem

I am currently doing some complex analysis and stumbled upon the following problem: I want to calculate the following integral: $$\frac{1}{2\pi i}\int_{-\infty}^{\infty}e^{ix}(ix)^{c-1}dx$$ where $ 0< c < 1$ I tried using the pole in $x=0$ and using…
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Residue and direction contour

Does the direction of the contour clockwise/anticlockwise effect the value of the residue?
gbd
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How to calculate $\oint\frac{dz}{z^3(z+4)}$ for $|z-2|<3$?

Which is right $$\oint\frac{dz}{z^3(z+4)}=2\pi i(\text{Res}(f,0)+\text{Res}(f,-4))$$ or $$\oint\frac{dz}{z^3(z+4)}=2\pi i\,\text{Res}(f,0)$$? I am unsure because $z=-4$ is outside $|z-2|<3$ so does its Residue included in the value of the integral?
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Evaluate $\int \frac{\cos\pi z}{z^2-1}\, dz$ inside rectangle with vertices $2+i,2-i,-2+i,-2-i$

My attempt: The poles are $z=1,-1$, both lie inside the rectangle. Residue at the poles are $-\frac12$ each, since residue at $$f(a)=\left[\frac{\cos \pi z}{\frac{d}{dz}(z^2+1)}\right]_{z=a}=\frac{\cos\pi a}{2a}.$$ So, by residue theorem, $$\int…
Diya
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Residues and integration along singular analytic hypersurface

Suppose $X$ is a complex-analytic variety which is Stein and $f:X\to \mathbb{C}$ is a non-constant analytic function. Denote $X_t=f^{-1}(t)$, and suppose that $X_0$ is singular while $X_t$ for all $t$ such that $|t|\in(0,\epsilon)$ is smooth. Let…
KReiser
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How to evaluate the function $D(x-y)$ in chap.2 of Peskin's QFT by residue method?

The original integral is:$$D(x-y)=\int\frac{d^3p}{(2\pi)^3}\frac{1}{2E_{\vec{p}}}\exp(-i\vec{p}\cdot\vec{r})=\frac{1}{4\pi^{2}r}\int_{0}^{+\infty}dp\frac{p}{\sqrt{p^2+m^2}}\sin(pr).$$ By using change of variable,I can find that there is a Bessel…
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