Questions tagged [contour-integration]
Questions on the evaluation of integrals along a locus in the complex plane.
4,070 questions
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Is Abel summability the same as contour integration?
I'm dealing with the following integral $$\int_{-\infty}^\infty \frac{ke^{ikx}}{\sqrt{k^2+m^2}}dk$$
where $m,x$ are some real positive fixed constants. I asked a question about the calculation of this ...
- 2,354
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Help in filling in the gaps from a contour integration in QFT [duplicate]
In a QFT class, we were calculating the following integral $$\int_{-\infty}^{\infty}dk \frac{ke^{ikx}}{\sqrt{k^2+m^2}}$$
and we decided to do a contour calculation. We chose a branch cut at negative $...
- 2,354
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How can the integral $\int_{0}^{\infty} \left(\frac{\sin x}{x}\right)^3dx$ be calculated? [duplicate]
The integral
$$\int_0^\infty \left(\dfrac{\sin x}{x}\right)^3dx$$
can be computed by contour integration. Using the function
$$f(z)=\dfrac{e^{3iz}-3\,e^{iz}+2}{z^3}$$
and an upper half-plane ...
user1713570
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How to deform contour in the complex plane correctly
I have a physics related problem where I need to calculate integrals of following type $\displaystyle\int_{-\infty}^{\infty} d\epsilon\, A(\epsilon) g^R(\epsilon + \omega/2)g^A(\epsilon - \omega/2)$, ...
- 83
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Contour integral with four branch points on the unit circle
I want to compute the contour integral
$$
\oint_{|z|=2} z \sqrt{z^4-1}\text{d}z,
$$
where the path is positively oriented (it is the blue one below).
It is non-zero thanks to the four branch-points $\...
- 31
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Contour integration inconsistency
I am trying to evaluate the following integral
$$I = \int_1^{\infty}\frac{e^{i\alpha z-\epsilon |z|}}{z^2}(z-1)^{i\alpha}dz$$
where $\alpha \in \mathbb{R}^+$ and $\epsilon$ is very small. I have ...
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Closed form of $\int_0^{\infty} \frac{\sin (\tan x) \cos ^{2n-1} x}{x} d x?$
Being attracted by the answer in the post
$$\int_0^{\infty} \frac{\sin (\tan x) }{x} d x = \frac{\pi}{2}\left(1- \frac 1e \right) , $$
I started to investigate and surprisingly found that
$$
\int_0^{\...
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How to show that the argument principle with $f'/f$ gives the change of argument of $f$
Let $f$ be an analytic function on a domain $D$ and say $C$ is a simple closed curve, counterclockwise oriented contained in $D$. Suppose $f$ does not have a zero or pole on $C$.
I understand that
$$
\...
- 3,147
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Contour integration $ \int_0^\infty \ln(1+2\cos(x)+x^2) \frac{dx}{1+x^2} $
Evaluate $$ \int_0^\infty \ln(1+2\cos(x)+x^2) \frac{dx}{1+x^2} $$
I tried using contour integration by evaluating the function over a semicircular contour in the upper half-plane. The residue at $z=...
user1707952
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answers
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Show that $\lim_{s\to -2}\Gamma(s)\eta(s)=7\frac{\zeta(3)}{8\pi^2}$
Consider the Henkel contour and $\eta(x)=\frac1{\Gamma(s)}\int_0^\infty\frac{x^{s-1}}{e^x+1}dx.$ Then, by Residues theorem and $\Re s<0$,
$$(1-e^{2\pi is})\Gamma (s)\eta(s)=2\pi i\sum_{k\in\Bbb Z}((...
- 16.4k
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Zeros of quaternion functions
do you know if it's possible to find the number of zeros on an open set of quaternions via one integral?
There's a way in the set of complex numbers. For f being holomorphic in G and behind its ...
- 91
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Closed form of $\operatorname{Ei}(-t) \theta(t) \star \operatorname{Ei}(t) \theta(-t)$
Question: Is there a (simpler) closed form of $\color{blue}{C(t) = \operatorname{Ei}(-t) \theta(t) \star \operatorname{Ei}(t) \theta(-t)}$?
Definitions:
Exponential Integral:
$$\operatorname{Ei}(x) = \...
- 2,365
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Contour integration for $\int_0^\infty \cos(x^2) e^{-ax^2} dx$ leads to a divergent integral?
I'm trying to solve this integral
$$\int_0^\infty \cos(x^2) e^{-ax^2} dx \quad (\text{for } a > 0)$$
First I did $\cos(x^2) = \text{Re}(e^{ix^2})$, so the problem becomes:
$$I = \text{Re} \left( \...
user1707952
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2
answers
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Tricky integral $\int_{0}^\infty \frac{\ln(x)}{x^2-2ix-2} dx $
I'm trying to solve the following integral using a keyhole contour, but I'm stumped.
$$I = \int_{0}^\infty \frac{\ln(x)}{x^2-2ix-2} dx$$
I'm pretty sure I've set up the contour correctly, but my ...
user1707952
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votes
2
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Contour integration with poles that coincide with branch points
A version of the following was posted at MathOverFlow three weeks ago; it has not yet been answered there.
Suppose that $n \geq 2$. Compute the following using contour integrals:
$$I_n = \int_{-\...
- 4,048
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Asymptotic expansion of $\oint_{C}\frac{z^{n-1}\log\Gamma(z)}{(z-1)^{n+1}} dz$
I need to find the asymptotic expansion of $$I_n:= \oint_{C}\frac{z^{n-1}\log \Gamma(z)}{(z-1)^{n+1}} dz $$ as $n\to\infty$, where C is a closed rectangle around $z=1$ in positive direction from $2+2i\...
- 1,059
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Ambiguity of the analytic continuation expression for the Beta function
The standard definition of the analytic-continuation of the Beta function by way of a Pochhammer contour integral is
$$
(1-e^{2\pi i\alpha})(1-e^{2\pi i\beta})\text{Beta}(\alpha,\beta)=\int_{P} z^{\...
- 143
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Evaluating $\int_0^\infty \frac{(\ln{x})^2}{1+x^2}dx$ by different methods
I'm working from Arfken-Weber's mathematical methods for physics, the chapter on calculus of residues. The problem asks to compute
$$
I = \int_0^\infty \frac{(\ln{x})^2}{1+x^2}dx
$$
and to do it in ...
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Question in the proof of Hankel's integral representation of the Bessel function of the first kind
I am trying to understand the proof of Hankel's integral representation of $J_\alpha(x)$:
$$ J_\alpha(x) = \frac{(x/2)^\alpha}{2\pi i} \int_{c-i\infty}^{c+i\infty} t^{-\alpha -1} \exp\left(t-\frac{x^2}...
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Extending holomorphic function on two components
This is exercise 1, page 101 of Michael Taylor's Introduction to Complex Analysis, which you may find here.
Statement: Let Ω ⊂ C be a connected domain. Suppose γ is a smooth curve in Ω, and Ω \ γ has
...
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Alternative to Formal Contour Integration to Evaluate Residues
***** Edited to correct some bad notation *****
I've been evaluating the residue at an essential singularity $z$ along the negative real axis of the complex plane by performing an actual formal ...
- 1,298
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Contour integral with higher order poles over several variables
Suppose I have an integral of the form
$$
\oint_C d^n\vec{z}\ \frac{\mathcal{I}(\vec{z})}{\prod_{i=1}^k P_i(\vec{z})},
$$
where the $P_i$ are polynomials in the $z_i$ variables determining a system of ...
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How to solve the second derivative of the following $_3F_2$ hypergeometric function
This problem originated from the solution of following integral:
$\displaystyle \begin{aligned}\int_{[0,π]^2}^{}{\ln^2\left(a-\cos x-\cos y\right)\ \mathrm{d}x\ \mathrm{d}y}\end{aligned}$
let $\...
- 437
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Question about (the limits of) differentiation under the integral sign (Feynman's trick)
I am trying to do the following integral:
$$I= -\int \frac{dk}{(2\pi)^2}\int\frac{d\omega}{2\pi}\frac{\omega^2}{(\omega^2-v^2k\cdot k)(\omega-q\cdot k)^2}\left[1-\cos(k\cdot(x-x') -\omega(t-t'))\right]...
- 21
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Rigorous Proof of Argument Principle with Winding Number
Looking to get a rigorous proof for the Argument Principle, as stated below.
Argument Principle:
Let $f$ be a function analytic on a simply-connected region $R$. Let $ \Gamma $ be a simple closed ...
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What is the contour integral around a branch cut?
For $f(z)=\sqrt{(z-a)(z-b)}$ with a<b, suppose we have a contour which enclose the segment [a,b], what is the contour integral around this closed loop?
I suggest this should be zero, since the ...
- 1,168
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Issue when solving $\int_{0}^{\infty}\frac{\ln\left(x\right)x}{\left(x^{3}-1\right)}dx$ via keyhole contour.
I wanted to prove that $$I=\int_{0}^{\infty}\frac{\log\left(x\right)x}{\left(x^{3}-1\right)}dx=\frac{4 \pi^{2}}{27},$$
using a contour integral of $$f(z)=\int_{0}^{\infty}\frac{\log\left(z\right)^{2}z}...
- 1,376
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How to integrate $\int_0^\infty \frac{\arctan5x-\arctan3x}{x}dx$
I came across the following integration:
$$\int_0^\infty \frac{\arctan5x-\arctan3x}{x}dx$$
I tried using the arctan formula for the difference of angles, but it led nowhere. I suspect it uses Feynman'...
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The adjoint of $\partial^{a + ib}$ on $H^k([0, \infty))$ for functions that do not vanish at $x = 0$.
Let's say I have a pseudodifferential operator $\partial^{a + ib}$ for $a + ib \in \mathbb C$ that is defined on the Sobolev space $H^k([0, \infty))$ for a sufficiently large $k$. (In fact, I really ...
- 1,348
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Clarification of integrating $\frac{x}{(x+1)^{\frac{5}{2}}}$ using contour integration
Currently I am learning to solve improper integrals with techniques from complex analysis. There are a lot of nice example here and on Youtube to learn from. But I wondered if it is possible to ...
- 13
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Error in integral $\int_0^\infty \frac{\cos(ax^n)}{x^n+b^n}dx = \frac{\pi}{nb^{n-1}} \frac{\cos\left(ab^{n}\right)}{\sin\left(\frac{\pi}{n}\right)}$
This integral is a slight modification of that in this post. The result below,
$$
I = I_1+iI_2 = \int_0^\infty \frac{\cos(ax^n)+i\sin(ax^n)}{x^n+b^n}dx
\\
= \frac{\pi}{nb^{n-1}} \frac{\cos\left(ab^{n}...
- 453
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Integrating $\int_0^\infty \frac{\ln{x}}{{({1+x^2})}^2}dx$
$$\int_0^\infty \frac{\ln{x}}{{\left({1+x^2}\right)}^2}dx$$
This integral evaluates to $-\frac{\pi}{4}$. However trying to show this is difficult.
Integrating the function along a keyhole contour ...
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Complex integral of $1/z$
I am now studying complex analysis in the university and this integral came up on my last exam. Since the correct answers were not posted, we tried to compare the solutions of us, but they did not ...
- 39
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votes
1
answer
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Evaluate $\int_{-\infty}^{+\infty} \frac{dx}{(\cos(x)+2)(1+x^2)}$ as an infinite series using residues
I am attempting a problem out of Weinberger's PDEs book, that is to solve the following integral
$$\int_{-\infty}^{+\infty} \frac{dx}{(\cos(x)+2)(1+x^2)}$$
as an infinite series via the residue ...
- 145
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votes
1
answer
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Complex Integral $\int_{0}^{\infty} \Gamma(1+it) (it)^{-it} dt$
For context, my original attempt was at the integral $\int_0^\infty \frac{x!}{x^x}dx$. However, I ended up having to evaluate the integral in the title.
Let's begin by naming our complex function $f(z)...
- 453
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votes
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answer
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Evaluate $\int_0^{\infty} \frac{\ln^n(x)}{(p^a+x^a)^b}dx$
I've been working on this generalized problem using complex analysis, but I can't quite figure out how to disentangle my solution.
$$
I = \int_0^{\infty} \frac{\ln^n(x)}{(p^a+x^a)^b}
$$
where $a,b,p&...
- 453
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votes
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Calculating $\Im\int_0^\infty \Gamma(a+it)dt$
WolframAlpha is a beast. After some numerical experiments, I considered the integral
$$I(a)=\int_0^\infty \Gamma(a+it)dt,$$
where $a>0$. From a deleted answer of mine, I know that $\Re\, I(a)=\frac\...
- 16.4k
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Integrating along a contour containing branch points
Let $f(z)=\sqrt{z^2-1}$. It suffices to cut $[-1,1]$ to make $f$ single-valued. Now let $C$ be a closed contour “containing” $[-1,1]$ “inside” it. For example, take $C$ to be the circle centered at ...
- 53
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Why do Cauchy-Goursat Theorem requires holomorphicity inside the entire countour of integration?
A common proof of the Cauchy-Goursat theorem that I have seen around is to first prove it for a triangular contour (but to be more concrete, I'm following the proof presented in Complex Variables by ...
4
votes
1
answer
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How does contour integration work for split complex numbers?
The complex split numbers are defined as $w\in\mathbb{D}:=\{x+jy\,|\;j^2=1,j\neq\pm1\;\; x,y\in \mathbb{R} \}$ now say i want to integrate the function $f(w)=\frac1{w-2}$ and I want to integrate it ...
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Inverse Laplace Transform of $\frac{1}{(s^2+a^2)^n}$
I am interested in finding the following inverse Laplace Transform
$$\mathscr{L}^{-1} \biggl\{ \frac{1}{(s^2+a^2)^n} \biggl\} \hspace{1cm} n \in \mathbb{Z}^+, \hspace{2mm} a>0$$
Once, during a car ...
- 453
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answers
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$\int_0^\infty \frac{\psi^{(1)}\left(\tfrac{1 - i u}{2}\right) - \psi^{(1)}\left(\tfrac{1 + i u}{2}\right)}{\cosh\left(\tfrac{\pi u}{2}\right)}\,du$
I have been attempting to solve
$$
\operatorname{Im}\int_{0}^{\infty}\frac{\psi^{\left(1\right)}\left(\frac{1-iu}{2}\right)-\psi^{\left(1\right)}\left(\frac{iu+1}{2}\right)}{\cosh\left(\frac{\pi u}{2}\...
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Contour for integrating a function which also has branch points at infinity
I was wondering what contour to take to integrate a function $f(z)$ along a real line from $x=0$ ($x = Re[z])$ to $x=\infty,$ where the $f(z)$ has branch points at $x=+1$, $x=-1$, and $z=\infty$ (in ...
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votes
5
answers
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$\int_{1}^{\infty}\frac{2\operatorname{arccosh}\left(x\right)}{\left(x^{2}+4\right)^{3}}dx$ with Contour Integration
$$\int_{1}^{\infty} \frac{2\operatorname{arccosh}(x)}{(x^{2} + 4)^{3}} \, dx$$
Question:
How may I evaluate this integral with complex analysis? I’m especially curious which kind of contours and ...
- 182
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votes
1
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An integral or a sum of Bessel functions
I am trying to compute the integral
$$
\alpha(t) = \int\limits_0^t e^{-i\Delta t'}\exp\bigg[-ix \sin\frac{2\pi t'}{\tau}\bigg]dt',
$$
where $x>0$, $0\le t \le \tau$, and $\Delta>0$. I haven't ...
- 1,469
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votes
1
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188
views
How to calculate the improper integral of $\int_{-\infty}^{\infty}\sin(e^{-x^2})dx$ by contour integration
I was playing around with geogebra and came across this function:
$$\sin(e^{-x^2}).$$
I was wondering if it's possible to calculate its area using contour integration. I plotted the function here ...
1
vote
0
answers
93
views
Tracking the 'phase' of complex contour integrals
Consider the integral $$I = \int_{-1}^{1}{(x+1)^{\frac{1}{3}}(1-x)^{\frac{2}{3}} dx}$$
A common method of evaluating the integral is to consider $f(z) = {(z+1)^{\frac{1}{3}}(z-1)^{\frac{2}{3}}}$ with ...
- 351
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1
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98
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How to quickly show divergent part in contour integral
I came across an integral and want to show the following step is divergent in real part, but vanish in imaginary part, as $r\rightarrow 0$.
$$I=\int_0^\pi \frac{e^{i\pi r^2 e^{2i\theta}}2rie^{i\theta}}...
- 21.6k
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How to solve $\int_{0}^{\infty} \ln{x} \frac{1-x}{1-x^4} \,dx$ using Contour Integration?
So I tried it by making a keyhole contour but I did not know what to do about the singularities at -1 and -i.
Also, can we make a keyhole contour above the real axis? How will it look like then and is ...
- 690
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votes
4
answers
648
views
Residue theorem and piecewise contour confusion
I'm trying to evaluate the integral
$$
\oint_\gamma \frac{1}{z(z - 2)} \, dz
$$
where the contour $ \displaystyle \gamma$ is defined piecewise as:
$$
\gamma(\theta) =
\begin{cases}
e^{2i\theta}, &...
