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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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where is the mistake with my fake proof?

I tried to show that $$\int_{-\infty}^{+\infty} \frac{dx}{(x^2+1)^{n+1}}=\frac {(2n)!\pi}{2^{2n}(n!)^2}$$ using contour integration so $$\int_{C} \frac{dz}{(z^2+1)^{n+1}}=\int_{-\infty}^{+\infty} \frac{dx}{(x^2+1)^{n+1}} +\int_{\gamma}..=2\pi…
mnsh
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Contour integrals on unit circle.

When dealing with a contour along a unit circle, we can set $|z| = 1$ and $z(t) = e^{it}$ so that $\frac{d z}{dt} = ie^{it}$ with $t\in [0,2\pi]$. Find the integral of: $$\int_{\Gamma}(z^7+z^4) dz$$ My own attempt. $$\int_{\Gamma} (z^7+z^5)dz =…
Chris Cross AppleSause
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Can contour integration be used to integrate $~u^u(1-u)^{1-u}~$?

I am told contour integration can be used to show $$\int_0^1 e^{i \pi z} z^z(1-z)^{1-z}=\frac{i\pi e}{24}$$ but am skeptical as the integrand contains log functions which do not admit simple closed contours for example to use the Residue Theorem. …
Dominic
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Evaluate the following integral using contour Integration

$$I = \int_0^\infty\frac{1}{x^4+x^2+1}dx$$ I have evaluated the integral directly and know the result is $\frac{\pi}{2\sqrt3}$ but I must be messing up my residues or perhaps my path of integration when attempting the contour evaluation. When…
Kirin171
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Contour integral for cos(sin) resulting in infinite sum

I'm trying to show that $$\int_{0}^{2\pi}\cos(n\theta-2\sin\theta)d\theta = 2\pi\sum_{r=0}^{\infty}\frac{(-1)^n}{r!(n+r)!}$$ The question hints that I should consider $$e^{z-z^{-1}}$$ but I don't see how to get to the answer. Considering the nature…
Number 34
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Evaluating a complex contour

I need to show the following result: $$ \int_{-\infty}^\infty \frac{1}{(1+x^2)^{n+1}}dx\, = \frac{1\cdot 3\cdot\ldots\cdot(2n-1)}{2\cdot 4\cdot\ldots\cdot(2n)}\pi $$ With n=1,2,3,... This function has a pole at i and -i. I've tried a semicircle in…
boudewijn
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Effective Branch cut

I have problems understanding how to get to the "effective Brach cut" in the top answer of this post: Dog Bone Contour Integral ? The answer says that one has a Branch cut for $\frac{1}{\sqrt{z-1}}$ at $(-\infty,1]$ and for $\frac{1}{\sqrt{z+1}}$…
benbln
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Finding a definite integral using complex analysis.

Now, I want to integrate $\int_{0}^{\infty} \dfrac{\cos (2x) -1}{x^2} \mathrm{d}x$, now I attempted to set $f(z)=\dfrac{e^{i2z}-1}{z^2}$ and then integrate around a similar contour to the classical one used for $\dfrac{\sin x}{x}$ i.e. a semicircle…
user2958701
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Residues to solve an improper integral

I'm asked to solve the following improper integral: $$\int_0^\infty \frac{\rm {Log}^2(t)}{1+t^2}dt. $$ Do I consider the function $f(z) = \frac{\rm{Log}^2(z)}{1+z^2}$ or some variant? Is the correct contour the so called "keyhole" that skips the…
Bouvet Island
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Is my solution correct? $\oint_C \frac{5\sec 2z}{e^{-3z}-1}\,\mathrm{d}z$ where $C:|z|=10^{-4}$

Problem.src) Compute $$\oint_C \frac{5\sec 2z}{e^{-3z}-1}\,\mathrm{d}z,$$ where $C:|z|=10^{-4}$. I solved the problem like this, but I don't know if this is the right way $$\oint_C \frac{5\sec 2z}{e^{-3z}-1}\,\mathrm{d}z =…
pleaseiwanttograduate
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Contour Integrals of Non-holomorphic functions

Suppose I wish to compute the integral $$ I = \oint_C dz f(z,{\bar z}) $$ where the contour $C$ is given. Obviously, the answer will depend on the contour chosen. I know the standard way to do this (standard from where I learnt it anyway). One…
Prahar
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Why does the principle of path independence require that the function being integrated is analytic in a simply-connected domain

I'm confused why the principle of path independence requires that the function being integrated is analytic in a simply-connected domain containing both contours used as the endpoints. Intuitively, I'm thinking that since the domain contains the…
Elli
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Can you use the Cauchy-Goursat theorem to evaluate a limit or is the ML theorem more suitable?

I am wondering how to evaluate the limit in the picture below. I've read online that the limit at infinity is zero when the denominator $\geq$ (numerator $+2$) but why is this? I am skeptical that I can use the Cauchy-Goursat theorem. Is there a way…
Elli
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Contour Integration with double real roots

I have the following function which I must integrate: F($\omega$) = $\int_{-\infty}^{\infty}\frac{1}{V\left[(\frac{\omega-\omega_{0}}{V})^{4}EI-\omega^{2}m\right]}d\omega$ With $V$ being velocity, $\omega_{0}$ the excitation frequency,…
rens van leijden
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Simple question on full differentials:

This one should be a quickie: if we are calculating $\oint xdy+ydx$ around the curve $x^4 + y^4 = 1$, I can spot quickly that $\oint xdy+ydx = \oint df$ where $f = xy$. How would I then evaluate $\oint df$? Is it simple, or am I better off with a…
Poo2uhaha
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