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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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Evaluate the integral using contour integration

Evaluate ∫$\frac{\cos2x}{(1−2a\cos x+a^2)}\,\mathrm dx$ from $0$ to $2π$ I need to evaluate the following using contour integral given that |a|<1 I wrote $\cos2x$ as $\frac{1}{2}\bigl(\frac{z^4+1}{z^2}\bigr)$ , And changed $\mathrm dx$ into $\mathrm…
Shariq Shuaib
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evaluate $\int_0^{\infty}\frac{x^a}{x^2+1}dx$ for $a\in(0,1)$

I am stuck on this complex analysis homework problem :( here is my attack so far: If I integrate around a keyhole contour, I can show that $$\int_0^{\infty}\frac{x^a}{x^2+1}dx=\frac{2\pi i}{1-e^{2a\pi i}}\sum_{w\in\{i,-i\}}RES(\frac{e^{a\log…
T. Alexander
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Evaluation of contour integration help involving exponential and cosh$z$

Let the contour $\gamma$ be a positively oriented cifrcle of radius 2 centered at zero, traversed once. Evaluate I = $\int_{\gamma}$ $\frac{dz}{(1-e^{iz})cosh(z)}$. This is what Ive done so far let $z = 2e^{i\theta}$ for 0 < $\theta$ < 2$\pi$ $dz$…
sam
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Elementary contour integral

I have an integral $$ \int_{-\infty}^{\infty}\frac{1}{(\omega^{2}-4)(\omega-2-i)(\omega+2-i)}d\omega $$ And I wish to evaluate this using Cauchy's Integral Theorem. I understand that with a simple pole on the real axis like…
kowalski
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Contour integral by parts: convergent becomes divergent

Please kindly take some time to read this long question. Suppose I have a meromorphic function $f$. Let $\gamma$ be a rectifiable closed curve. Suppose $f$ and $\ln(z-s)$ are continuous on $\gamma$. ($s$ is a constant.) $\gamma$ contains some…
Szeto
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Showing that $\lim_{R \rightarrow \infty} \bigg(\int_{0}^{R}e^{ix^{2}}dx-e^{i \pi/4}\int_{0}^{R}e^{-r^{2}}dr \bigg)=0$?

In the text "Complex Variables Introduction and Applications Second Edition", I'm having trouble proving the proposition in $(1)$, could this be done through Cauchy's Theorem ? We wish to evaluate the integral $I=\int_{0}^{\infty}e^{ix^{2}}$.…
Zophikel
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Contour integration of a trigonometric function

Following is true.(My book says..) $$2 - \frac12 h = \frac{h^2}{2\pi}\int^{\pi / h}_{-\pi / h}\sin ^2\xi \cot^2\frac12 h \xi \, d\xi $$ Using contour integration, how to show this is true? Can you show the process?
plhn
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Contour integration for inverse cosine

I wanna calculate the integral $$\int_0^\infty \mathrm d q \frac{1}{\cos ka -\cos qa}.$$ I think this problem can be solved by using the contour integration, but I don't really have a good idea on how to manipulate the integrand. Any comment would…
rbwang
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question about extension of Cauchy's residue theorem

Prove that $\int\limits_{0}^{\infty} \frac{\sin x}{x(1+x^2)^2}dx = \frac{\pi(2e-3)}{4e}$. There goes an extension of Cauchy's residue theorem that states if a finite number of poles lie directly on the contour $C$, then $\oint_C f(z)dz = 2\pi i \sum…
am_11235...
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Determine the integral of the function $f(z)=\tan z$

Determine the integral of the function $f(z)=\tan z$ around a contour that is a rectangle extending from $0$ to $2 \pi$ in the real direction and from $-0.5i$ to $0.5i$ in the imaginary direction. $$ $$ How to find the rectagle ? Here $-0.5i \leq y…
MAS
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How do I solve $\int_{-1}^1 \frac{1}{(x-2)(x+2)\sqrt{1-x^2}} dx$?

I'm hopelessly stuck with the integral $$\int_{-1}^1 \frac{1}{(x-2)(x+2)\sqrt{1-x^2}} dx$$ I'm guessing the way to go is complex contour integration, but I have no idea.
con-f-use
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Calculating complex integral.(Finding the principal Value)

The question is $$\int_0^\infty \frac{1-\cos x}{x^2} \, dx$$ My approach was as follows. I converted the integral into a a complex integral.$\int_0^\infty \frac{1-\cos z}{z^2} \, dx$ $$\int_0^\infty \frac{\sin^2(z/2)}{z^2} \, dz$$ $$\implies…
ANUPAM BISHT
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Integral of $\int_{-1}^{1}x^{2n}e^{i k x}$

Im trying to integrate \begin{equation} \int_{-1}^{1}x^{2n}\,\mathrm{e}^{\mathrm{i}kx}\,\mathrm{d}x = 2\left(\, -1\,\right)^{n}\, \frac{\mathrm{d}^{2n}}{\mathrm{d}k^{2n}}\left[\,{\frac{\sin\left(\, k\,\right)}{k}}\,\right] \end{equation} I have the…
anonymous
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How can I solve this integration?

How can I solve this integration? Which method can I use to solve it? $$\int_0^{2\pi}\frac{d\theta}{\{(K^2 +{K_1}^2 -2KK_1\cos\theta) +c\}^2 +d^2}$$
Mariya
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Integral estimation

Is it true that $$\int_{-\pi}^{\pi}\left |\frac{iR^{x}e^{i \theta x}}{1-Re^{i\theta}}\right |\,\mathrm{d}\theta \leq 2\pi R\frac{R^{x}}{1-R}\xrightarrow{R\to\infty}0$$ $$\int_{-\pi}^{\pi}\left |\frac{ir^{x}e^{i \theta x}}{re^{i\theta}-1}\right…
Hopeless
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