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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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Evaluating Contour Integral

How do I go about evaluating the following by contour integration? $$ \int^1_0 \frac{dx}{(x^{2} - x^{3})^{1/3}} $$ The question does not fit in the standard form of : $\int^{2\pi}_0$ or $\int^\infty_{-\infty}$ etc. I tried substituting $ x =…
bhavesh
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Contour Integration Limitations?

So, I'm trying to evaluate the following integral by complex contour integration ONLY: $$\int_0^\infty{\frac{x^\alpha}{x(x+1)}} dx$$ where alpha is real and not an integer. Obviously, we need to use a key-hole contour to avoid the resulting branch…
Incognito
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Contour integration of $\frac{\sinh(az)}{\sinh(\pi z)} $

So I have the following integral I need to evaluate in the complex plane: $$\int_{-\infty}^\infty \frac{\sinh(az)}{\sinh(\pi z)}~\mathrm{d}z.$$ This equation will have poles when $z = ni$, so any integer along the imaginary axis. Any tips on how to…
Jack
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Can one evaluate Serret's integral using contour integration?

$$\int_0^1 \frac{\ln(x+1)}{x^2+1} dx$$ This is the integral, and if possible could someone tell me whether we could solve any such type of problem via contour integration.
The Grim Reaper
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Contour Integration $\frac1{\sin z \cos z}$

$\newcommand{\ds}[1]{\displaystyle{#1}}$ I have the contour integral $$ \oint_{\left\vert\,z\,\right\vert\ =\ 2}\frac{1}{\cos(z)\sin{z}}dz $$ To make it easier to work with, I use $\ds{\sin(z)=\frac{e^{iz}-e^{-iz}}{2i}}$ and…
user89582
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Contour Integration of Inverse Hyperbolic Cosine

I have the following integral $$\int_1^\infty\frac{\text{arccosh}(x)}{x^2+1}\,dx$$ I am convinced that it can be done through contour integration, since we are integrating along the principle branch of the inverse hyperbolic cosine. But I am unsure…
Jessie Christian
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Contour integration: $\int_0^{\infty} x^p /(x^4+1) dx$ where $-1 < p < 3$

I want to calculate $\int_0^{\infty} x^p /(x^4+1) dx$ where $-1 < p < 3$. My first guess is to let $f(z) := \frac {z^p}{z^4+1}$ and integrate this over $\gamma_R$ where $$ \gamma_r = [-R,R] \cup \{Re^{i\theta} : \theta \in [0,\pi] \} $$ The…
user42761
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Sokhotski–Plemelj theorem (With Sqrt Term)

I am reading up on the Sokhotski-Plemelj theorem, and so far I've seen it being applied on equation with the general form (ref: http://scipp.ucsc.edu/~haber/ph214/Plemelj.pdf): $$\lim_{\epsilon\to0}\frac{1}{x-x_0\pm i\epsilon} = P\frac{1}{x-x_0} \mp…
Wayne Y.
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Evaluating $\oint_C\frac{z}{z-1}dz$, where $C$ is the origin-centered circle of radius $4$. Why is my answer of $0$ wrong?

A textbook I've been reading contains the question: What is $$\oint_C\frac{z}{z-1}dz$$ where $C$ is a circle with radius $4$ centered at the origin? Apparently, the correct answer is $2\pi i$, but I think the answer is $0$. Here is my working: $C$…
YWu
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Integrating the complex function $(\tan(z) + 1)/(z^2-z)$ along the contour $|z| = 1/2$

I would like some help integrating the complex function $$\frac{\tan(z) + 1}{z^2-z}$$ along the contour $|z| = 1/2$. I factored the denominator and then thought of setting $(\tan(z) + 1)/z$ as $f(z)$ but then it is not analytic inside the contour…
Elli
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Any quicker way to find the residues here?

I have to integrate $\dfrac{1}{(z-4)(z^7-1)}$ w.r.t $z$ along $C : |z|=2$. 7 poles of the function lie inside $C$ and $z=4$ lies outside. I am trying to find the answer by applying residue theorem. i.e. $2\pi i \sum (\text{residues inside }C)$, but…
ali mansoor
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$\exp(i z^2)$ on $R e^{i \theta}$ with $\theta \in [0, \pi /4]$.

Let $\gamma_R : [0, \pi/ 4] \ni \theta \mapsto R e^{i \theta}$. I want to show that $$ \lim_{R \rightarrow \infty} \int_{\gamma_R} e^{i z^2} dz = 0 $$ for $R > 1$. In order to use Jordan's lemma I want to show that $$ \lim_{R \rightarrow…
user42761
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How to evaluate $\int_{c-i\infty}^{c+i\infty}\frac{\log(z)}{z}e^{zt}\,dz$

I am trying to compute $$\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{\log s}{s}e^{st}\,ds$$ in relation to $$\mathcal{L}^{-1}\left\{\frac{-\gamma-\log s}{s}\right\}$$ which certainly evaluates to $\log t$. But the integrand "has no poles"…
user816822
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Evaluation of a real integral using complex contour

Evaluate $\displaystyle \int_0^\infty \frac {\tan^{-1}(ax)}{x(1+x^2)}\, dx$ for $\displaystyle a>0, a \ne 1, $. Need to do this using contour integral. I know how to do it using Leibnitz rule. I cannot figure out poles of the function and how to…
user1454312
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A contour integral problem from trinity

A function $\phi(z)$ is zero when $z=0$, and is real when $x$ is real, and is analytic when $|z| \leq 1$; if $f(x,y)$ is the coefficient of $i$ in $\phi(x+iy)$, prove that if $-1
meye
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