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Questions tagged [contour-integration]

Questions on the evaluation of integrals along a locus in the complex plane.

This is the procedure of calculating the contour integral around a given path/contour. It allows us to evaluate integrals on the real line $\mathbb{R}$ that are not able to be evaluated using real-variable methods.

Links:

Contour Integration at Wolfram MathWorld

3880 questions
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How does contour integral work?

It might be a vague question but I can't help asking what is so powerful in contour integral that makes it possible to compute certain improper real integrals which are seemingly very difficult to compute by real variable calculus method.
Paladin
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How to prove $\int_0^1 \frac1{1+x^2}\arctan\sqrt{\frac{1-x^2}2}d x=\pi^2/24$?

Since I'm stuck at this final step of the solution here. I wished to try contour integral, taking the contour a quadrant with centre ($0$) and two finite end points of arc at…
RE60K
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ln and rational function using contour integration

I was playing around with $\displaystyle \int_{0}^{1}\frac{\ln^{2}(x)}{x^{2}-x+1}dx=\frac{1}{2}\int_{0}^{\infty}\frac{\ln^{2}(x)}{x^{2}-x+1}dx$ and managed to solve it using real methods via digamma and what not. It evaluates to…
Cody
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Integral with contours

I want to evaluate the integral $\displaystyle \int_0^\infty \dfrac{\ln x}{e^x+1}\,{\rm d} x$ using contour integration. At first I though using a rectangular. Problem is that I cannot establish the vertices there. There is one poles included in…
Tolaso
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Convergence of a line integral along semi-circular arc

There is a line integral in a form, $$\int_\mathrm{arc} \frac{\exp(iz)}{z^2+1} \, dz$$ "arc" is a semi-circular line with radius $R$ on the upper half complex plane. and i know that the integral converges to zero as R goes to infinity. What about…
Onur
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Contour integration: $\int_{-\infty}^\infty\frac{\sin^2(x)}{x^2+1}dx=\frac{\pi}{2}\left(1-\frac{1}{e^2}\right)$ using Cauchy's integral formula

I need to show that $$\int_{-\infty}^\infty\frac{\sin^2(x)}{x^2+1}dx=\frac{\pi}{2}\left(1-\frac{1}{e^2}\right)$$ but I don't really know why I'm not getting the result using contour integration (I'm not supposed to use the residue theorem). Can't…
safd
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$\int_0^{2\pi} e^{\cos(\phi)}\cos(\phi - \sin(\phi)) d\phi$ via contour integration

Can anyone help me calculating this integral using contour integration? $\int_0^{2\pi} e^{\cos(\phi)}\cos(\phi - \sin(\phi)) d\phi$ I've used the subctraction formula of the cosine: $$\cos(\phi - \sin(\phi)) = \cos(\phi)\cos(\sin(\phi)) +…
Manuel
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what is the proper contour for $\int_{-\infty}^{\infty}\frac{e^z}{1+e^{nz}}dz$

what is the proper contour for $$\int_{-\infty}^{\infty}\frac{e^z}{1+e^{nz}}dz:2\leq n$$ I tried with rectangle contour but the problem which I faced how to make the contour contain all branches point because $1+e^{nz}=0$ for every…
mnsh
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Contour integration of $ \int_{-1}^{1} \frac{dz}{\sqrt{1-z^2}(a+bz)}$

Hi all so I am looking at evaluating the following integral using contour integration: $$ \int_{-1}^{1} \frac{dz}{\sqrt{1-z^2}(a+bz)} $$ where $a > b > 0$. Multiplying the top and bottom by $\sqrt{1-z^2}$ the equation becomes: $$ \int_{-1}^{1}…
Jack
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Real integrals using Complex integration

What is the criterion for choosing a contour in the complex plane for the evaluation of real integrals of the form $\int_{-\infty}^{\infty}f(x)dx$? For example $\int_{-\infty}^{\infty}\sin{x^2}dx$.
user90001
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Contour integration of $\int \frac{dx} {(1+x^2)^{n+1}}$

I want to compute $$ \int_{-\infty}^\infty \frac 1{ (1+x^2)^{n+1}} dx $$ for $n \in \mathbb N_{\geq 1}$. If I let $$ f(z) := \frac 1 {(z+i)^{n+1}(z-i)^{n+1}} $$ then I see that $f$ has poles of order $n+1$ in $\pm i$. Initially I thought that…
user42761
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Contour integration - Branch cut

I'm asked to show the following equality given $a\in (-1,1)\subset\Bbb R$ $$\int\limits_0^\infty\frac{x^a\ \log(x)}{(1+x)^2}dx=\frac{\pi\sin(\pi a)-a\pi^2\cos(\pi a)}{\sin^2(a\pi)}$$ So I'm trying to use the keyhole contour (as shown here) and so…
user1923
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Contour integral, residue

I'm trying to solve the integral $\int^{2\pi}_0\cos^2(\theta) \sin^2(\theta) d\theta $. So far I've used $\cos\theta=\frac{1}{2}(z+z^{-1})$, $\sin\theta=-\frac{1}{2}i(z-z^{-1})$ and $d\theta=-iz^{-1}dz$ to get the contour integral…
Nana
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Show that $\int_0^{\pi}\frac {\cos {n \theta}}{1-2r\cos \theta+r^2}\, \mathrm d \theta = \frac {\pi r^n}{1-r^2}$

I am trying to calculate $$I=\int_0^{\pi}\frac {\cos {n \theta}}{1-2r\cos \theta+r^2}\, \mathrm d \theta$$ where $r\in(0,1)$ I tried substituting $u = e^{2 i \theta}$ and using the Cauchy integral formula: \begin{align}I&=\Re \int_{|u|=1}\frac…
user85798
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Typical Contour Inegral Proof

I'm trying to prove that the following contour integral approaches 0 as R -> $\infty$. How exactly would we go about doing this? $$ \int{\log\left(z^{2} + 1\right) \over 1 + z^{2}}\,{\rm d}z\quad \mbox{over the curve}\quad …
Incognito
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