Mathematics Stack Exchange
Mathematics Stack Exchange

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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Using residue theorem to solve double integral, involving singular roots?

I am a physics grad student (high energy), I've come across a problem while doing a certain loop integral that I don't understand. I've removed as much of the physics as I can so that this is just a math problem, hopefully someone can spot my…
Ray Goerke
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I need help finishing a contour integral

The original question is $$\int\limits_{-\infty}^{+\infty}\frac {\sin x}{x^3-2x^2+4x-8}\,\mathrm dx$$ I have gotten the problem to the part where it is $$\oint\limits_Cf(z)\,\mathrm dz=2\pi i\frac {\mathrm d}{\mathrm dz}\left[\frac…
ViscoseCar
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When does the value of a complex contour integral depend on the choice of the contour of integration?

I have seen examples of complex contour integrals whose value depends on the choice of the contour of integration and some integrals where the value does not depend on the choice of the contour. Is there any indication by which we can know whether…
Solidification
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Why are values of complex integration different along different contours?

I integrated $f(z) = z$ conjugate on two different line segments a) from $0$ to 1 then to $1 + i$ b) from $0$ to $1 + i$ directly And I correctly obtained answers of $1 + i$ for a) and $1$ for b) but why are the answers different
Elli
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Contour integration $\frac{e^{iz}}{2\sqrt{z}}$

When $z=u+iv$, I would like to compute the integral of $\frac{e^{iz}}{2\sqrt{z}}$ along above curve. The imaginary axis $$\frac{1}{2}\int_{R}^{0} \frac{e^{-v}}{\sqrt{iv}}d(iv)$$ $R$ goes to $\infty$. Here because of $\sqrt{iv}$. I confuse to use…
primavera
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Contour integral redefining variables

I have the integral ${\operatorname{Im}} \left (\int^\infty_0 e^{ix} x^{s-1} \, \mathrm{d} x \right)$ and I wish to redefine $x \to iy$ but I am unsure of how to justify this using contour integration arguments. I would be very grateful for any…
Max Clifford
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question about keyhole contour for $\int_0^\infty \frac{x^\lambda}{(1+x)^2}dx$

Evaluate using contour integration : $$\int_0^\infty\frac{x^\lambda}{(1+x)^2}dx$$ where $-1<\lambda\neq 0<1$. For $\boxed{-1<\lambda<0}$ case, we choose a suitable keyhole contour taking $z=0$ as branch point and positive $x$-axis as branch cut.…
am_11235...
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contour integral for $\int_0^\infty\frac{\ln x}{x^2+a^2}dx$

Prove using contour integration that $$\displaystyle\int_0^\infty \frac{\ln x}{x^2+a^2}dx=\frac{\pi}{2a}\ln a$$ I used the substitution $x=e^t$ to transform the integral to $$\int_{-\infty}^\infty \frac{te^t}{e^{2t}+a^2}dt$$ Now it is easier for…
am_11235...
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integrating $\frac{(2z-1)}{(z^2 - 1)}$ around the circle of radius $1$ centred at $1$, anticlockwise.

integrating $\frac{(2z-1)}{(z^2 - 1)}$ around the circle of radius $1$ centred at $1$, anticlockwise. so I used the residue theorem, and I got the answer a $i(\pi)$, so I was just wondering whether that was right.
lard
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simple pole of a complex function

Given $$f(z)=\dfrac{\tanh (2z)}{z\sin(\pi z)}$$ Are $z=\pm 1$ simple poles of $f(z)$? It appears so as $\sin(n\pi)=0$ for integer n. But the integral along the circle $|z|=\pi/2$ of this function is given as $4i$. Did I go wrong?
Purushothaman
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Evaluation of definite integral using contour integration

It seems simple to evaluate $$\int_0^\pi \dfrac{d\theta}{5+4\cos \theta}$$ I tried with $$\int_0^\pi \dfrac{d\theta}{5+4\cos \theta}=\dfrac{1}{2}\int_0^{2\pi} \dfrac{d\theta}{5+4\cos \theta}=\dfrac{1}{2i}\oint_c \dfrac{dz}{(z+2)(2z+1)} $$ (c is a…
Purushothaman
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Improper integral involving exponential function

How can one compute $\displaystyle \int_0^\infty \dfrac{x^3\;dx}{e^x-1}$. I tried contour integration replacing $x$ with $z$ but confused about the proper contour for integration.
Purushothaman
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Evaluate $\int_{\gamma_R}f(z) dz$ when $R>2$

Let $$f(z)=\frac{e^{\pi iz}}{z^2-2z+2}$$ and $\gamma _R$ is the closed contour made up by the semi-circular contour $\sigma_1$ given by, $\sigma_1(t)=Re^{it}$, and $0\leq t\leq\pi$ the straight line $\sigma_2$ from $-R$ to $R$ (a semi circle). When…
Levi
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A Contour integration

$$\frac{1}{2\pi i}\int_{I(\lambda,\infty)}^{} \left(\frac{1}{t}+\frac{1}{2}+\frac{1}{12}t\right)\frac{e^{-zt}\log t}{t^2}\,dt$$ where $I(\lambda, \infty)$ is the integral path consisting of $(\infty, \lambda)$ , counterclockwise circle of radius…
O-V-E-R-H-E-A-T
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Complex contour integral evaluation 2

$$\int_0^1 \frac{\ln x}{x-1}dx$$ I thought of solving the problem by using $|z|=1$ and indenting the singularity around $z=1$. Or can we use log trick (as used in $-\infty$ to $+\infty$) problems in order to solve this? in yes then $\log x$ can…
Shreyas Hegde
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