# Questions tagged [contour-integration]

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

838 questions with no upvoted or accepted answers
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• 415
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### Validating solution to PDE using integral transforms

I'm trying to obtain the analytical solution of a Fokker-Planck PDE, which the solution is a probability density function, and then use this to find the mean of some quantity in the paper. The paper ...
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### Cauchy's integral formula and essential singularities

Let $f$ be holomorphic at $z_0\in\mathbb C$. I would like to compute the integral $$\oint_{\gamma_{z_0}} f(z)\, e^{\frac{1}{z-z_0}}dz,$$ where $\gamma_{z_0}$ is a small circle around $z_0$. By ...
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### Using the Residue theorem to solve contour integral $\oint_{|z|=R}\frac{z^2}{1-e^{2\pi i z^3}}dz$ with $n<R^3<n+1$ for positive integer $n$.

I'm a second-year undergraduate in a non-major complex analysis course. We just started looking at using residues to compute the values of contour integrals, and I've run into a homework problem that'...
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• 1,868
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### Compute exact integrals with quaternions

It's common knowledge that complex analysis is helpful in computing a bunch of exact real integrals. Is there any occurence of quaternions/quaternion formalism helping in the same way? If not, what ...
• 176
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### Proving that $\int_0^{\infty} \frac{(3-\sqrt{8}\cos(\log(2)t))Z(t)^2}{t^2+\frac{1}{4}}~dt=\pi \log(2)$ where $Z(t)$ is the Riemann-Siegel Z-function.

While looking at articles relevant to the Riemann Hypothesis, I've seen that the Riemann-Siegel Z-function $Z(t)$ is frequently used to study the Riemann zeta function along the critical line. ...
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### On $\sum a^n \tan(n\theta)$

It is well known that $$\sum_{n=0}^{\infty} a^n \cos(n\theta) = \frac{1-a\cos(\theta)}{1-2a\cos(\theta)+a^2}$$ $$\sum_{n=0}^{\infty} a^n \sin(n\theta) = \frac{a\sin(\theta)}{1-2a\cos(\theta)+a^2}$$ ...
• 476
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### Proof that $\oint_r d(x,N + n) < 0$?

Let $f(x)$ be a real-entire function such that for all $x>0$ we have $f(x) > 0$, $f'(x) > 0$ , $f '' (x) > 0$. And also $0 < D^M f(0) < D^{M-1} f(0)$. Let $0<T<1$ and $n$ a ...
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### Integration Around Part of a Branch Cut

I am studying the integral, given by a Laplace transform, $$\int_0^\infty\!e^{-\alpha x}\sinh^{-2/3}x\left(1+\frac 12\sinh^2x\right)^{-1/6}\left(1-\beta\sinh^{4/3}x\right)^{1/2}\,\mathrm dx$$ From ...
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### Evaluate: $I(\alpha)=\int_{0}^{\infty} \frac{\arctan\left ( \alpha (x-\operatorname{arsinh} x) \right ) }{x\sqrt{1+x^2} }\text{d}x$

I am interested this type integrals. Let $$I(\alpha):=\int_{0}^{\infty} \frac{\arctan\left ( \alpha (x-\operatorname{arsinh} x) \right ) }{x\sqrt{1+x^2} }\text{d}x.$$ For example, some simple ...
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### Evaluate $\int_{-\infty}^{\infty}\frac{\sqrt{a+ix}}{a^2+x^2}\,dx$ using residue calculus

I'm asked to evaluate $$\int_{-\infty}^{\infty}\frac{\sqrt{a+ix}}{a^2+x^2}\,dx$$ $\mathbb{R}\ni a>0$, using residue calculus (where $\sqrt{\cdot}$ is the PV $\sqrt{}$). My approach is as follows: ...
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• 161
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### Closed form solution of a contour integral involving Bessel function integral representation

I have the following integral for which I want a closed-form solution: $$I_n(kr,\cos\theta) = \int_{C} \frac{e^{ikrt}P_n(t)}{t-\cos\theta} dt,$$ where $P_n$ is a Legendre polynomial and $C$ is a ...
• 141
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### Solving for a function inside a convolution

I have this relationship: \begin{align} \frac{1}{|x|}=f(x)*f(x)\ , \end{align} where $*$ denotes the convolution. I want to solve for $f(x)$. My first instinct was to apply the convolution theorem: \...
• 711
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### An analytic function $f$ on $\{z \in\mathbb{C} : |z|>1 \}$

Let $\Gamma$ denote the positively oriented circle of radius $2$ with center at the origin. Let $f$ be an analytic function on $\{z \in\mathbb{C} : |z|>1 \}$, and let $$\lim_{z\to \infty}f(z)=0.$$ ...
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### Incomplete contour in second contour integration

I'm trying to evaluate $$\int_0^{2\pi} dx \int_0^{2\pi} dy \frac{1 - e^{iax} e^{iby} }{1 - \cos x \cos y}$$ I am assuming that $a$ and $b$ are nonnegative integers and that $a+b$ is even. (This ...
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### A contour written as the sum of finitely many simple smooth curves

A smooth curve $\gamma:[a,b] \rightarrow \mathbb{C}$ is a continuously differentiable map such that $\gamma'(t) \not= 0$ for all $t \in [a,b]$. A contour is a curve that is equivalent (up to ...
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### Integral Representation for the Fox-H function on several variables

I have a problem that involves the H-function of several variables, and I have noticed that the implementation of such function when the number of variables are relatively high (greater than 5) must ...
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