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
Filter by
Sorted by
Tagged with
17 votes
0 answers
2k views

Calculate using residues $\int_0^\infty\int_0^\infty{\cos\frac{\pi}2(nx^2-\frac{y^2}n)\cos\pi xy\over\cosh\pi x\cosh\pi y}dxdy,n\in\mathbb{N}$

Q: Is it possible to calculate the integral $$ \int\limits_0^\infty \int\limits_0^\infty\frac{\cos\frac{\pi}2 \left(nx^2-\frac{y^2}n\right)\cos \pi xy}{\cosh \pi x\cosh \pi y}dxdy,~n\in\mathbb{N}\...
user avatar
  • 2,873
10 votes
0 answers
1k views

Integral of rational function over $\mathbb{H}^4$

Suppose I have a rational function of $8$ coordinates $a,b,c,d,e,f,g,h$ that I want to integrate over $\mathbb{H}^4$: ...
user avatar
10 votes
0 answers
777 views

Contour integration with 2 branch points

I need to compute a quite complicated Fourier transform, but I'm having problems due to the facts that I have two branch points. The integral I need to solve is $$\int_\infty^{-\infty} \frac{dk}{\...
user avatar
  • 415
8 votes
0 answers
324 views

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 ...
user avatar
7 votes
0 answers
163 views

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 ...
user avatar
  • 444
7 votes
0 answers
127 views

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'...
user avatar
7 votes
0 answers
633 views

Complex contour integral: How does the stationary point method used in this case?

I was reading a paper which has the following integral in order to do the inverse Laplace transformation: $$ I=\frac{1}{2\pi i}\int_{-i\infty+\gamma}^{i\infty+\gamma} e^{st}\frac{\Omega^2}{(s^2+4\...
user avatar
7 votes
0 answers
151 views

Integral with contours

I want to evaluate the integral $\displaystyle \int_0^\infty \dfrac{\ln x}{e^x+1}\,{\rm d} x$ using contour integration. At first I though using a rectangular. Problem is that I cannot establish the ...
user avatar
  • 6,560
7 votes
0 answers
289 views

Computing the integral $ \int_0^{\infty} e^{-\phi^2+\phi}\cdot \phi^{2} \ln(1-2x\cos\phi+x^2)\, d\phi. $

Integrate $$ \int_0^{\infty} e^{-\phi^2+\phi}\cdot \phi^{2} \ln(1-2x\cos\phi+x^2) \, d\phi. $$ Something that may help $(1-2x\cos\phi+x^2)=(1-xe^{i\phi})(1-xe^{-i\phi})$. And using the series ...
user avatar
  • 9,452
6 votes
0 answers
155 views

Is it possible to evaluate $\int_{-\infty}^{\infty}e^{-x^2}\operatorname{sech}^2(\frac{x}{2})\,dx$?

I am curious to know if it is possible to evaluate the following integral: $$\int_{-\infty}^{\infty}e^{-x^2}\operatorname{sech}^2\left(\frac{x}{2}\right)\,dx$$ Here is the context: Someone had asked ...
user avatar
  • 3,348
6 votes
0 answers
113 views

How to evaluate $\int_0^\infty\frac{ \sin{x}}{x^2+1}\text{d}x$

For some reason I couldn’t find an answer to this online even though it seems very basic. I am trying to evaluate the following improper integral. $$ \int_0^{\infty} \frac{ \sin{x}}{x^2+1}dx$$ I ...
user avatar
  • 5,246
6 votes
0 answers
320 views

Evaluate two integrals involving $\operatorname{Li}_3,\operatorname{Li}_4$

I need to evaluate $$\int_{1}^{\infty} \frac{\displaystyle{\operatorname{Re}\left ( \operatorname{Li}_3\left ( \frac{1+x}{2} \right ) \right ) \ln^2\left ( \frac{1+x}{2} \right ) }}{x(1+x^2)} \...
user avatar
6 votes
0 answers
212 views

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 ...
user avatar
  • 176
6 votes
0 answers
209 views

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. ...
user avatar
6 votes
0 answers
295 views

On $\sum a^n \tan(n\theta)$

It is well known that $$\sum_{n=0}^{\infty} a^n \cos(n\theta) = \frac{1-a\cos(\theta)}{1-2a\cos(\theta)+a^2}$$ $$\sum_{n=0}^{\infty} a^n \sin(n\theta) = \frac{a\sin(\theta)}{1-2a\cos(\theta)+a^2}$$ ...
user avatar
  • 476
6 votes
0 answers
204 views

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 ...
user avatar
  • 13k
6 votes
0 answers
401 views

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 ...
user avatar
5 votes
1 answer
283 views

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 ...
user avatar
5 votes
1 answer
172 views

A uniform bound on a family of complex integrals

In my research, I have encountered the following integrals, which I want to bound. For $0 < t < 1$, let $C_t$ denote the portion of the complex unit circle consisting of $z$ for which $\Re (z) \...
user avatar
  • 1,011
5 votes
0 answers
119 views

Contour Integral of $\int_{0}^{\infty}\frac{\cos(x)}{(x^2+1)^2}dx$

I had this question $\int_{0}^{\infty} \frac{\cos(x)}{(x^2+1)^2}dx$ on a introductory complex analysis final and really had a poor go of it. The poles are $\pm i$ both of order 2, so I replaced $cos(x)...
user avatar
  • 323
5 votes
0 answers
113 views

Can this closed form for $\sum_{n=1}^\infty \arctan \frac{a^2}{n^2}$ be proved directly by contour integration?

The general closed form for $$\sum_{n=1}^\infty \arctan \frac{a^2}{n^2}= \arctan \left( \frac{\tan \frac{\pi a}{\sqrt{2}}-\tanh \frac{\pi a}{\sqrt{2}}}{\tan \frac{\pi a}{\sqrt{2}}+\tanh \frac{\pi a}{\...
user avatar
  • 30.4k
5 votes
0 answers
146 views

The choice of contour in the definition of Meijer G

It appears that when the Meijer G function is discontinuous on the unit circle, the integrals over the left and the right loops can exist but differ. For $G_{1,1}^{0,1}\left(z\,\middle|\begin{array}{c}...
user avatar
  • 10.3k
5 votes
0 answers
291 views

Deriving a Series Representation of the Bessel Function of the First Kind

I've tried to use an integral representation of the Bessel Function of the First Kind $J_n(x)$ to derive a series representation of the function. My end result is pretty close to the answer that it ...
user avatar
  • 3,229
5 votes
0 answers
356 views

Proof of Sophomore's Dream using Contour Integration

Sophomore's dream is a relatively common identity, that states $$ \int _0^1 x^{-x} dx = \sum_{n = 1}^\infty n^{-n}$$ The common proof is found using the series expansion for $ e^{- x \log x} $ and ...
user avatar
  • 61
5 votes
0 answers
330 views

inverse Laplace transform by finding residues of essential singularities

I want to find the inverse Laplace transform of $$F(s)=\exp\Big(-\sqrt{2s}\tanh(\sqrt{2s})\Big).$$ Despite the square roots, $F$ doesn't have any branch points since $$\sqrt{2s}\tanh(\sqrt{2s})=\frac{\...
user avatar
5 votes
0 answers
163 views

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: ...
user avatar
  • 6,957
5 votes
1 answer
182 views

Computing an integral using residues

I am trying to find an integral: $$\int_{-\infty}^{+\infty}\frac{e^{-\sqrt{(x^2 + 1)}}}{(x^2 + 1)^2}\,\mathrm dx$$ I went about applying contour integral over a semicircle with diameter along $ x = +...
user avatar
  • 73
5 votes
0 answers
370 views

How to integrate $\int_{-\infty}^{\infty}dp \ p e^{ipx}e^{-it\sqrt{p^2+m^2}}$?

In Lancaster & Blundell's QFT book they show that \begin{equation}A:= \int_{-\infty}^{\infty}dp \ p e^{ipx}e^{-it\sqrt{p^2+m^2}}\end{equation} returns a nonzero value for $x$, $t$ and $m$ ...
user avatar
5 votes
1 answer
115 views

using complex or real analysis solve $\int_{0}^{\pi/2}\frac{x^m}{\sin x}dx$

closed form for $$\int_{0}^{\frac{\pi}{2}}\frac{x^m}{\sin x}\ dx$$ I slove it for some m but in general i failed. I tried by part , by substitution,by using $\sin x =\frac{e^{ix}-e^{-ix}}{2i}$ . I ...
user avatar
  • 5,675
5 votes
0 answers
116 views

Solving an integral (using Cauchy contour integral?)

I need to solve this integral: \begin{equation} f(t)=\int_0^\infty x^2 \sqrt x \left( e^{a x} -1\right)^{-1/2} \frac{e^{i(b-x)t}-1}{b-x} dx \end{equation} where $a$ and $b$ are real, positive ...
user avatar
  • 123
5 votes
0 answers
94 views

$ \int_\gamma e^{\frac{1}{z^2-1}}\sin{(\pi z)}dz $ on a closed curve of index $N$ with respect to the point $1$.

Let $\gamma$ be a closed curve in the right half plane that has index $N$ with respect to the point $1$. Find $$ \int_\gamma e^{\frac{1}{z^2-1}}\sin{(\pi z)}dz $$ This is a problem from an old ...
user avatar
  • 1,274
5 votes
0 answers
153 views

Ugly-nice double series

I'm trying to evaluate the following ugly double sum (presented in raw notation as used in my calculations): $\sum _{m=1}^{\infty } \sum _{n=1}^{\infty } \frac{4 m \cos \left(\frac{2 \pi m x}{T_x}\...
user avatar
  • 101
5 votes
0 answers
474 views

Where's my mistake applying Perron's Formula?

I applied Perron's Formula to Riemann Zeta Function and got a weird result. First, I started with a simple definition of Riemann Zeta Function, $$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}$$ where $\...
user avatar
  • 567
5 votes
0 answers
355 views

difficult integral involving arcsin(x)

I have a difficult integral to compute. I know the result by guessing the answer, but need to know the method of calculation. The integral is $$ \int_{a}^{b}{\rm d}p\,{p \over p^{2} - 2\mu}\, \...
user avatar
4 votes
0 answers
138 views

Integrating $\int^{\infty}_0\frac{x^n}{e^x-1}\text{ d}x$ using contour integration only

I have the integral $$I=\int^{\infty}_0\frac{x^n}{e^x-1}\text{ d}x$$ for real $n>0$ that I want to evaluate only with contour integration. (I already know the identity that $\Gamma(s)\zeta(s)=\int^{...
user avatar
  • 1,776
4 votes
0 answers
54 views

Integral of the Laurent series with Bessel function coefficients

I'm trying to integrate a function of the following form, where $A$ and $B$ are both positive: $$\int_0^{\infty}\exp\left(\frac{A}{2}\left(\frac{B}{1+x^2}-\frac{1+x^2}{B}\right)\right)dx.$$ My first ...
user avatar
4 votes
0 answers
162 views

The direction of the steepest descent path at the saddle point (Picard-Lefschetz theory)

I am having to perform oscillatory integrations like $e^{iS}$ using Picard-Lefschetz theory. One can write this as $e^{h+is}$ where $h(x,y)=-{\rm Im}(S(x,y))$ is the Morse function. To perform these ...
user avatar
4 votes
0 answers
71 views

Writing the Real Part of Complex Integrand

I'm having a hard time to understand how's Eq. $(6.73)$ become Eq. $(6.75)$. It's taken from Numerical Methods for Laplace Transform Inversion by Cohen. Here's the problem: [...]. The basis of their ...
user avatar
  • 2,087
4 votes
0 answers
249 views

How do I calculate the double integral of a Hankel function?

I would like to know how to show the following: $$ \int_{0}^{2\pi}\int_{0}^{2\pi}H_{0}^{(1)}\left(\left|a+be^{i\theta}+ce^{i\phi}\right|\right)e^{im\theta}e^{in\phi}d\theta d\phi=4\pi^{2}(-1)^{m+n}e^{...
user avatar
  • 357
4 votes
0 answers
111 views

$\int_0^\infty\frac{x^2+3}{x^4+1}\,dx$ via contour integration?

I want to calculate $\displaystyle I=\int_0^\infty\frac{x^2+3}{x^4+1}\;dx$ using contour integration. I've obtained the correct answer of $I=\sqrt{2}\pi$ below, but I would appreciate if someone ...
user avatar
4 votes
0 answers
139 views

Contour Integral involving Zeta function

I'm trying to compute the contour integral $$\frac{1}{2 \pi i} \int_{c - i \infty}^{c + i \infty} \zeta^2(\omega) \frac{8^\omega}{\omega} \ d \omega$$ where $c > 1$, $\zeta(s)$ is the Riemann zeta ...
user avatar
4 votes
0 answers
169 views

Inverse Laplace transform of $1/\sqrt{P(s)}$ with $P(s)$ a polynomial of degree 4

I am trying to compute the inverse Laplace transform of $$F(s)=\frac{1}{\sqrt{[(s+\lambda_3)^2-(\lambda_1-\lambda_2)^2][(s-\lambda_3)^2-(\lambda_1+\lambda_2)^2]}},$$ where we can assume that $\...
user avatar
  • 161
4 votes
0 answers
279 views

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 ...
user avatar
  • 141
4 votes
0 answers
159 views

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: \...
user avatar
  • 711
4 votes
0 answers
97 views

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 $$\lim_{z\to \infty}f(z)=0.$$ ...
user avatar
4 votes
0 answers
105 views

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 ...
user avatar
  • 1,540
4 votes
0 answers
145 views

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 ...
user avatar
  • 9,042
4 votes
1 answer
234 views

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 ...
user avatar
  • 41
4 votes
0 answers
275 views

Asymptotic form of an integral to an power law decaying function

$$ f(x)=\frac{1}{2}+\frac{1-x^2}{4x}\ln\left|\frac{1+x}{1-x}\right| $$ This function is not analytic at $x=1$. The plot is shown: The integral is: $$ I=\int_0^\infty g(x) \sin(2b rx) dx $$ where $...
user avatar
4 votes
0 answers
284 views

Integration over a variety

If $ M $ is a differentiable manifold equipped with an Atlas $ \mathcal{A} = ( U_i , \varphi_i )_{ i \in I} $, we can then calculate the integral of a differential form $ \omega $ over $ M $ with the ...
user avatar
  • 804

1
2 3 4 5
17