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Questions tagged [complex-integration]

This is for questions about integration methods that use results from complex analysis and their applications. The theory of complex integration is elegant, powerful, and a useful tool for physicists and engineers. It also connects widely with other branches of mathematics.

The concept of definite integral of real functions does not directly extend to the case of complex functions, since real functions are usually integrated over intervals but complex functions are integrated over curves.

Surprisingly complex integration are not so complex to evaluate, oftenly simpler than the evaluation of real integrations. Some real integrations which are otherwise difficult to evaluate can be evaluated easily by complex integration, and moreover, some basic properties of analytic functions are established by complex integration only.

References:

https://en.wikipedia.org/wiki/Contour_integration

"An Introduction to the Theory of Analytic. Functions of One Complex Variable" by Lars Ahlfors

"Complex Variables and Applications " by James Ward Brown and Ruel V. Churchill

3109 questions
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Complex integral of a function where the radius is approaching 0

Question: If $f(z)$ is continuous in a neighbourhood around the origin, then show that $\lim_{r\to 0}\int_{0}^{2\pi}f(re^{i\theta}) d\theta = 2\pi f(0)$. Since the function $f$ is continuous, could I simply move the limit into the integral:…
Wulfgang
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Explaining the concept of complex integration

I need some help to explain a colleague that complex integration has different properties than real integration. As an example, I gave him the following…
Yair
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Find $\int_{C} \sec^2{z} \cdot dz$ Where C be any path from $\frac{\pi i}{4}$ to $\frac{\pi}{4}$ in the unit disk.

We have to find $$ \int_{C} \sec^2{z} \cdot dz$$ Where C be any path from $\frac{\pi i}{4}$ to $\frac{\pi}{4}$ in the unit disk. We know that, $$Z = x + iy$$ so, $$dz = dx + idy$$ so the integral changes to, $$ \int_{C} \sec^2{(x + iy)} \cdot (dx +…
Abhilekh Gautam
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Integrating complex exponentials

I want to compute the following integral involving complex exponentials but my approach is leading me to problems with infinities. $$ I = {a \over 2} \int_{-\infty}^\infty (e^{-bt^2+i \omega t} + e^{-bt^2-i \omega t})\; dt $$ $$ = {a \over 2}…
Veak
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Why is WolframAlpha saying that two equal expressions are not equal?

I am trying to solve the following integral: \begin{equation}\label{1} \int \sin(\alpha t)\exp(-i\alpha t)\, \text{dt}. \end{equation} Using the exponential form of $\sin(\alpha t)$, the integral becomes \begin{equation}\label{pet} \int…
lost_in_python
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Complex integral of $\int_{\partial D(0,1)}\frac{\mathrm{d}z}{z^2 -3z +1} $

I have to calculate $\int_{\partial D(0,1)}\frac{\mathrm{d}z}{z^2 -3z +1} = \int_0^1 \frac{2 i \pi e^{2 i \pi t}\mathrm{d}t}{(e^{2 i \pi t})^2 -3e^{2 i \pi t} +1}=\int_0^{2\pi} \frac{ i e^{i \theta}\mathrm{d}\theta}{(e^{i \theta})^2 -3e^{i…
Albert Durand
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Complex Integration of Improper Integrals

I need to calculate the integration using Cauchy's Residue Theorem $$\int_{-\infty}^{\infty} \frac{x^2}{(x^2+1)(x^2+9)}dx$$ I am stuck here how can I approach this. $$\int_{Cr}^{} f(z) dz = 0$$. I need to show that the function is zero using…
Ratul Das
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What is the residue for the pole of $\frac{1}{az^4+bz^2+c}$

I have to solve $\int_0^{\infty} \frac{1}{ax^4+bx^2+c} dx$ we can assume $(b^2 - 4ac \neq 0)$ I confirmed integral on arc with radius R converges to $0$ (when R -> $\infty$) But I can't find the residue to use Residue theorem. The calculation…
shrimp555
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Evaluating $\int_c \cot^2z $ dz

if C is the positively oriented circle $ \lvert z \rvert = 4 $ $cot^2z = \frac{cos^2z}{sin^2z}$ singularites will occur when $sin^2z = 0$, so $\pi$ and $ -\pi $ within $\lvert z \rvert = 4$ I have tried using my p over q rule $p(z_0) = \cos^2z \neq…
guyman
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Complex integrals and poles

I have to show, if $ f(z)$ has a pole of order 1 at $ z=z_0 $, then $ w(z) = [f(z)]^2 $ must have a pole of second order at $ z_0 $. so far all I have is that $f(z) $ will have a pole of order 1 if $ m=1 $ $ f(z)=\sum_{n=-m}^\infty a_n(z-z_0)^n…
guyman
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Direct integration of a complex integral

In my homework there is a integral looks so strange, this is the first time that I see this kind of thing. So I have no idea what to do. $\int_{0}^{-1+i\infty}(z+1)e^{iz}dz$ What does it mean the upper part of integral $-1+i \infty$? And how can i…
Aegean
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Complex Integration Using Residue Theorem with branch cut

How to calculate $$\int_{0}^{\infty }\frac{ln(1+x^{2})}{(1+x^{2})} dx $$ ? How to choose the branch cut and proceed?
Snpr_Physics
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complex integral giving different results

If I have an integral $\int_C \frac{1}{z} dz$ where $C$ is the left half of the circle from $i$ to $-i$, then I can choose the branch cut to be the positive axis, and have the antiderivative to be log(z). Then using the antiderivative, I can…
CuriousAlpaca
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Complex integration path notation

I'm struggling to interpret the notation for this integral and can't find a direct definition for it. $\int_{(1/\mathcal{L})}f(s)ds$. Here, $\mathcal{L}:=log(x)$ for a large $x\in\mathbb{N}$. What does this path mean? Is it the parametrization…
jshpmm
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How to integrate this function? $\int_{-1}^1 \frac{1}{\sqrt{1-x^2}(1+x^2)}\,dx$

How to find definite integral $$\int_{-1}^1 \frac{1}{\sqrt{1-x^2}(1+x^2)}\,dx$$ using complex intergral? And if $$ f(z) = \frac{1}{\sqrt{1-z^2}(1+z^2)}\,$$ There are simple poles at $$ z = i , z = -i $$ so I calculated residue at those poles, $$…
thisisyoung__
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