# Questions tagged [contour-integration]

Questions on the evaluation of integrals along a locus in the complex plane.

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### how can I use contour integration in complex plane to solve this problem?

enter image description here $$\int_0^{\tfrac\pi2} \frac{\ln(\cos(x))}{x^2+\ln^2(\cos(x))} \, dx$$
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### Generalizing contour integration to indefinite integrals [closed]

Is it possible to evaluate indefinite integrals using contour integration. For example take the integral of $\left(\dfrac{\ln x}x\right)^{2011}$. Is it possible to find its primitive solely using ...
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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: $$\int_{\Gamma}(z^7+z^4) dz$$ ...
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### hard integral: $\sin(x)/\sqrt{1+2\sin(2x)}$ between $0$ and $\pi/2$

I found this integral in one of my math textbooks. $$\int_0^{\pi/2}\frac{\sin(x)}{\sqrt{1+2\sin(2x)}}\mathrm{d}x$$ I tried to solve it using the substitution $u=\tan(\frac{x}{2})$ but it got me ...
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### an inequality on the area represented by a contour integral

This question was inspired by Carnot's great theorem on the efficiency of reversible thermodynamic cycles. We have two continuous positive functions $T(u)$ and $S(u)$ of the parameter $u\in [0,1]$. ...
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### Integrating the exponential of a fractional order complex polynomial

This post potentially has two questions related to the following integral: $$\int_{-\infty}^{\infty}e^{i(ax-bx^{s})}dx$$ where $1<s\leq2,$ $a\in\mathbb{R}$ and $b>0$. The first question: is ...
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### $\int^{c+i\infty}_{c-i\infty}\frac1za^z$: how to use this contour?

"Show that $\int^{c+i\infty}_{c-i\infty}\frac1za^z$ is 0 for $a<1$, $\frac12$ for $a=1$, 1 for $a>1$. For $a>1$, use the rectangle from $c-Bi$ to $-X+Ai$. For $a<1$, use $X$ instead ...
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### Evaluating the sum $\sum_{k=1}^\infty (-1)^k / k^2$ via a contour integral

I'm evaluating the sum: \begin{align*} \sum_{k=1}^\infty (-1)^k \frac{1}{k^2} \end{align*} I expressed the sum via a complex contour integration. But I'm not getting out the right answer. My method: ...
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### How is a Hankel contour different from a keyhole contour?

From what I'm guessing a keyhole contour is one that looks like this and because it can be shown that the contribution from $C_R$ and $C_\epsilon$ vanishes as $R\to \infty$, a Hankel contour looks ...
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### Saddle Point analysis of integral

I want to make saddle point approximation of the integral $$I = \frac{1}{2 \pi i } \int_{\gamma - i \infty} ^{\gamma + i \infty} \frac{1}{z^6} e^{zt} dz$$ but as I see the function in the exponent i....
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### Contour integration with $\int_{-\infty}^{\infty}{\frac{1-\cos(2z)}{z^2(z^2+1)}}\, dz$

$$\int_{-\infty}^{\infty}{\frac{1-\cos(2z)}{z^2(z^2+1)}}\, dz$$ The contour is shaped like this (image from this answer) With $\epsilon$ being the radius of the smaller semi-circle. The integral over ...
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### Integrals of $\int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} f(k)\text{d}k$

Consider a type of integrals $$\int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} f(k)\text{d}k$$ where $K=K(k),K^\prime=K(\sqrt{1-k^2})$ are complete elliptic integrals, and $k$ is an elliptic ...
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### any suggestions for solving this integral involving a complex exponential of trigonometric functions?

I'm looking to find a closed-form (or series) solution for the definite integral $$\int^{\pi/2}_0 \sin(x) \cos(x) \sin(a*\cos(x)) \exp(i*b*\sin(x)) dx$$ where $a$ and $b$ are real constants and $i$ is ...
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### how to choose the correct contour for the given integrals

can anybody tell me how to consider a correct contour for these integrals $$\int_{0}^{\infty} \frac{1}{x^\alpha} e^{-ikx} dx$$ $$\int_{-\infty}^{0} \frac{1}{x^\alpha} e^{-ikx} dx$$
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### Evaluating $\int_{-\infty}^{\infty}\frac{\ln\left(\frac{1}{2}+x+x^{2}\right)}{1+x^{2}}dx$

(Motivation) This is an integral I made up for fun. WolframAlpha doesn't seem to come up with a closed form for it and I'm surprised there doesn't seem to be a duplicate after using Approach0, but I ...
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### Prove complex integral equality

Suppose $\triangle$ is the open unit disk and $\overline{\triangle}$ be it’s closure (closed unit disk). Let $f$ be holomorphic in an open set containing the set $D = \mathbb{C} - \overline{\triangle}$...
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### Inverse fourier transform of $1/(\sqrt x+ia)$
I am trying to analytically calculate $$\int_{-\infty}^\infty d\omega \frac{e^{-i\omega t}}{\sqrt{\omega}+ia},$$ where $a>0$. The integrand has a branch cut with the branch point at $\omega=0$. ...