Questions tagged [complex-integration]

This is for questions about integration methods that use results from complex analysis and their applications. The theory of complex integration is elegant, powerful, and a useful tool for physicists and engineers. It also connects widely with other branches of mathematics.

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Finding unknown from the given complex integral.

Find real number a such that $\oint_c \frac{dz}{z^2-z+a}=π $ where c is the closed contour |z-i|=1 taken in the counter clockwise direction. This is a question that has been asked in the 2021 NBHM ...
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Evaluate $PV \int_{-\infty}^{\infty} \frac{1-e^{iax}}{x^2}dx. a>0$

Evaluate $PV \int_{-\infty}^{\infty} \frac{1-e^{iax}}{x^2}dx. a>0$ using residues. So I have a theory how to calculate $PV \int_{-\infty}^{\infty} f(x)e^{iax}dx$ a>0, but I don’t know how to ...
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Unsure how to resolve two contradicting identities in finding the residue of a complex function

In my complex analysis class I was solving the integral: $$I=\int_{-\infty}^\infty\frac{\sin(2x)}{x^2+x+1}dx$$ using contour integration. I initially tried the integral over the semicircle of radius $...
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Evaluating an integral using the saddle point approximation

The following integral appears in certain physical problem: $$ \lambda_m = \int_{0}^{2\pi} \frac{d\phi}{2\pi} e^{K\cos \phi + im\phi} $$ where $m$ is an integer and $K$ is a large real number. I'm ...
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Evaluating a complex integral of two variables

In https://mathoverflow.net/questions/423124/expectation-of-complex-random-variable?noredirect=1#comment1087394_423124, I got a clue that the following integral could be computed with a suitable ...
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How to find one vector from scalar product of two vectors

According to the dxiv's comment, to make it clear what I'm asking, I'll add a few things. I am not interested in the cross-product term in the description below. That part was clarified by the ...
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Contour Integral with square root

I'm a master degree theoretical physics student and while working on my thesis I've encountered the following guy: $$\int_0^{\infty}dx\frac{e^{-ax^2+ibx}}{\sqrt{x}}$$ with $a,b>0$. I wanted to ask ...
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show $\int_\mathbb{R} \frac{1}{\pi(1+x^2)}dx=1$

I am trying to show that density function of standard Cauchy distribution is well defined: that $$\int_\mathbb{R} \frac{1}{\pi(1+x^2)}dx=1.$$ I tried the following calculation: $$\int_{-N}^N \frac{1}{\...
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Contour integration of $\int_{0}^{+\infty}\frac{x^2\cos x}{\cosh x} \,{\rm d}x$

Using contour integration, find $$\int_{0}^{+\infty}\frac{x^2\cos x}{\cosh x} \,{\rm d}x$$ How to calculate it? I never worked with integrals of this type.
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Some questions on index and single-valued property of Riemann-Hilbert Problems

Let $D \subseteq \mathbb{C} $ be bounded and simply connected domain, $\Gamma := \partial D \in C^2 $, $ g \in C^{0,\alpha}(\Gamma) $ be a complex-valued and nowhere vanishing function defined on the ...
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Evaluate $\int_0^{\infty}\frac{\log( x)}{x^2+a^2} \,dx$ using contour integration; Re a > 0

Evaluate $\int_0^{\infty}\frac{\log( x)}{x^2+a^2} \,dx$ using contour integration; $Re (a) > 0$ I found two questions where a > 0 but in my case I have the following condition: Re a > 0 (It ...
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Convert Equation (API in function of Density to Density in function of API.)

I have an equation I want to convert. However my math capabilities are too limited to do this. So hopefully you can help me with this. :D The Equation I have is: Density = (0.1415x999.012)/(131.5+API)...
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Residue of $\frac{z^3+z^2}{\sin(z)^3}$

I want to compute the residue for $f(z)=\frac{z^3+z^2}{\sin(z)^3}$ in all isolated singularities. For $0$ this is easy, since $0$ is a pole or order 1. But for $k \pi, k \in \mathbb{Z}: k \neq 0$ I am ...
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Winding number of an ellipse $\gamma$ at $0$ using $\omega(\gamma,0)=\frac{1}{2i\pi}\oint_{\gamma}\frac{1}{z}dz$?

In this exercise, we are supposed to firstly find a path that parametrizes the following ellipse: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ for $a,b \in \mathbb{R}$ $\textit{I have found the following path: ...
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Complex Integral $\int^\pi_0 x\cdot \overline{\sin(nx)}dx$ [closed]

I was solving the following problem: $$\langle f, e_n \rangle = \int^\pi_0 f\cdot \overline{e_n}\ dx=\int^\pi_0 x\cdot \overline{\sqrt{\frac{2}{\pi}}\sin(nx)}\ dx$$ $$\sqrt{\frac{2}{\pi}}\int^\pi_0 x\...
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Why do we use exponentials while integrating trigonometric functions in complex analysis

Let p(x) be some polynomial function. Now, we have an integral of the form : $$I=\int_{-\infty}^{\infty} \frac{\cos(x)}{p(x)}dx$$ What is usually done is that, we define this integral as : $$I'=\int_{-...
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Convergence conditions of the integral $\int_{-\infty}^{\infty} \frac{e^{cx}}{1+e^{x}} dx$.

I want to find the conditions such that $\int_{-\infty}^{\infty} \frac{e^{cx}}{1+e^{x}} dx$ converges. I am considering as contour the rectangle with vertices $\pm R$, $\pm R + 2\pi i$. \begin{align*}...
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Laurent series of $e^z$

find the Laurent series centered at $z=1$ $$ f(z)=\frac{e^z}{(z-1)^2} $$ I thought that the denominator part is safe by our center and the expansion is just about the exponential which is a Taylor ...
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Laurent series of this function

Find the Laurent series of the function$$ f(z) = \frac{z+1}{z(z-4)} $$ in the annulus $0<|z-4|<4$. My approach: $$ \begin{aligned} f(z) &= \frac{z+1}{z(z-4)} =\left(\frac{-1}{4}\frac{1}{z}+\...
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Can I apply Fubini's theorem to prove that this function is holomorphic?

Let $\phi:U\times [a,b]\rightarrow\mathbb{C}$ be a continous function such that for each fixed $t\in[a,b]$ the function $z\mapsto \phi(z,t) $ is holomorphic. Define $F(z)=\int_{a}^{b} \phi(z,t) dt $. ...
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Is the Cauchy's Theorem valid for a path with multiple points?

Let $f$ be analytic in a finite simply-connected region $\mathcal{R}$ described by the inside of a closed curve $C$ (that can contain multiple points) and on its boundary $C$. Then $$ \oint_C f \...
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Is this relation correct? How do I get there?

It's about the following relation: $$ \biggl|\int_{-T/2}^{T/2}e^{-it(E_p-E_k)/\hbar}\,dt\biggl|^2=2\pi\hbar T\delta(E_p-E_k) \textrm{, for $T\rightarrow\infty$ } $$ This is out of a QM course ...
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contour integrals of trigonometric functions

∫γ cos(z/2) where C : γ(t) := {t + i √ (π 2 − t 2), −π ≤ t ≤ π}. This integral is confusing me I don't understand what to do to start i thought it might be using Cauchy's residue theorem but I'm ...
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Inner product in the Segal-Bargmann space: how to integrate?

I am studying the Segal-Bargmann representation of quantum states (i.e. in terms of holomorphic functions). One central definition is the inner product in the Segal-Bargmann space: \begin{align} \...
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Integrate $\frac{x^b\ln x}{1+x}$

Suppose $0<b<1$ and we are asked to evaluate $$I=\int^\infty_{0}\dfrac{x^b\ln x}{1+x}\,dx$$ Initially, I thought of using the contour in this link. But then the problem comes when I am trying to ...
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surface integrals in the complex plane in polar form

if i am not mistaken the definition of a 1d integral in the complex plane of some analytical $f(z)=u+i v$ where $z=x+iy=re^{i\theta}$ is $\int\limits_\gamma f(z)dz=\int\limits_\gamma udx-vdy+i\int\...
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Apparent complex integration paradox

I encountered the following integral : $$ \int_{\gamma} \frac{dz}{z-1-i}, $$ which has to be integrated along two straight-line contours : $\gamma_{1} : 2i$ to $3$ $\gamma_{2} : 3$ to $0$ and then ...
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1 answer
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Evaluate the following Contour Integral: $\int_C\frac{dz}{z^4+1}$ where $C$ is the circle $x^2+y^2=2x$

Question: Evaluate assuming the closed contour is traversed in the positive direction: $\int_C\frac{dz}{z^4+1}$ where $C$ is the circle $x^2+y^2=2x$ My Thoughts: Considering the circle $C:=x^2+y^2=2x$,...
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How the following equality true?

I'm reading a lemma's proof, and the proof starts with: Let $\alpha \in \text{arg}(\int^b_af(t)dt)$: $$ \Bigg|\int^b_af(t)dt\Bigg|=e^{-i \alpha}\int^b_af(t)dt $$ I know that $z=r\cdot e^{i\alpha}$ ...
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1 answer
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How to understand complex (line) integral?

I am taking a complex analysis class, but I have serious troubles with understanding the complex integral. We defined it as $$ \int_{\Gamma} f(z) dz = \int_{\alpha}^{\beta} f(z(t))\dot{z}(t)dt $$, ...
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Integrate $\int^\infty_0 \dfrac{\sin x}{x^3}\,dx$

By using this method we can evaluate $\displaystyle\int^\infty_0\dfrac{x-\sin x}{x^3}\,dz=\dfrac{\pi}{4}$ and I intended to solve $\displaystyle\int^\infty_0 \dfrac{\sin x}{x^3}\,dx=\displaystyle\int^\...
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Why I can not calculate the residual of $\frac{\cos(x)}{x^2+1}$ like this to solve this integral?

I have the following integral to solve: $$I =\int_{-\infty}^{+\infty}\frac{\cos(x)}{x^2+1}$$ and after applying the big circle lemma and the residue theorem, I run into this expression: $$I = \pi i \...
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A complex integral I don't understand

I was reading H. O. Fattorini's book Second Order Linear Differential Equations in Banach Spaces, in page 9 of Chapter 1, in the proof of Hille-Yosida theorem (3.1), I found this complex integral $$\...
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4 answers
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Question on contour integrals. $\int_C \frac{1}{1+z^2} dz$ without using ML estimation!

NOTE: I went though every every previous post on contour integration (here and on youtube) looking for a resolution to this question, so please do NOT mark this question as already answered. Let $C$ ...
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2 votes
1 answer
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How to calculate the correct residue?

I am trying to calculate the residue of the function $$f(z)=\frac{2}{3z^2+8iz-3}$$ so as to evaluate the integral $$I=\int_{0}^{2\pi}\frac{1}{3\sin(\theta)+4}d\theta$$ I have found that $f$ has ...
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Is this a valid application of Jordan's lemma?

Suppose I'm trying to evaluate the following integral along a large semicircle in the upper half plane: $$ \lim_{R\to \infty} \int_{C_R} e^{iaz} \, f(z) \, , \qquad \text{where } f(z) = \frac{1}{z} \, ...
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Showing $\arcsin(z)-i\operatorname{Log}(-iz)-i\log 2 \in H^2(\mathbb{H})$

In order to prove a result about Hilbert transforms, I need to show the complex function $F(z) = \arcsin(z)-i\operatorname{Log}(-iz)-i\log 2$ lies in $H^2(\mathbb{H})$, the Hardy Space for the upper ...
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Complex Integration $\int^\infty_0\frac{(\ln x)^2}{1+x^2}\,dx$ using theory of Residue and Branch Cut [duplicate]

The question asks to compute $$\displaystyle\int^\infty_0\frac{(\ln x)^2}{1+x^2}\,dx$$ but I have no idea how to do that. I have thought about using the following branch cut but doesn't seem to work. ...
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Intuition of contour integration of |z| [duplicate]

Given the following contour integral: $$ I=\int\limits_{-1}^1 |z| \ \mathrm{dz}, $$ with path of integration being the upper half of unit circle, it can be parameterized to give: $$ \int\limits_{-1}^1 ...
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Let $C$ be the ellipse $x^2/4 + y^2/9 = 1$ traversed once in the positive direction, $G''(−i)$.

My answer is different from the one chegg. Let C be the ellipse $\frac{x^2}{4} + \frac{y^2}{9} = 1$ traversed once in the positive direction, and define $$ G(z) :=\int_C \frac{\zeta^2-\zeta+2}{\zeta-z}...
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integration over a circle using residues

We are asked to solve the following integral using residue theorem $\int_{|z|=1} \frac{1}{z\sin^2z} dz$. I was able to show that it has one pole of order 3 inside $|z|=1$ given by $z=0$. We know that $...
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Is there any generalization of Darboux's Formula for an analytic weight?

Quick Reference for the formula I mean For any analytic function $f(z)$ and polynomial $\phi(w)$ it is known that: $$(z-a)^{(n+1)}\int_0^1 \phi (t) f^{(n+1)}(a+t(z-a)) d t= \sum_{k+l=n} (-1)^{n+k} (z-...
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2 answers
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Integrating complex exponentials

I want to compute the following integral involving complex exponentials but my approach is leading me to problems with infinities. $$ I = {a \over 2} \int_{-\infty}^\infty (e^{-bt^2+i \omega t} + e^{-...
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Confusion about the asymptotic behavior of a Fourier integral

I would like to consider the following (inverse) Fourier transform: $$I(t) = \int_{-\infty}^{\infty} d\omega \, e^{i\omega t} (\omega + i \varepsilon)^{-1-i\alpha} \, ,$$ where the pole at $\omega = 0$...
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Louville theorem and identity principle application on concrete examples

I have a two part question about Louville theorem, and identitiy principle: Loville theorem states that if function $f$ is holomorphic on $\mathbb{C}$ and $|f|$ is bounded $\Rightarrow$ $f$ is ...
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Calculating the integral $I(w)=\int_\mathbb{R}\exp(-\pi(x+w)^2)dx$ for $w\in\mathbb{C}$

Suppose $I(w)=\int_\mathbb{R}\exp(-\pi(x+w))^2dx$ with $w\in\mathbb{C}$. There are two parts to this problem : i) I have to show that $I(w)$ converges uniformly on compact sets $K\subset\mathbb{C}$ ii)...
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How to calculate $\int_{\partial D(0;a)} \frac{dz}{(z-a)(z-b)}$

How to calculate $\displaystyle\int_{\partial D(0;r)} \frac{dz}{(z-a)(z-b)}$ when $|a|<r<|b|$?. My first idea: I tried to separate by partial fractions, so for some $A,B$ \begin{equation} \...
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A question about convergence to infinity of a sequence of integrals of functions with complex values

I have a fixed real number $t\neq 0$ and a function with complex values but defined in $\mathbb{R}$: $$\ell: \mathbb{R} \to \mathbb{C}, \quad \ell(x) = e^{itx} - 1$$ For each $n \in \mathbb{N}$, I ...
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2 answers
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compute $\int_0^{\infty}\frac{\cos x}{x^2+a^2}dx$ in complex plane without using the residue

compute $\int_0^{\infty}\frac{\cos x}{x^2+a^2}dx$ in complex plane without using the residue I know that one way is to calculate this integral $\int_{c(r)}\dfrac{e^{iz}}{z^2+a^2}$ over half a circle, (...
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Why is WolframAlpha saying that two equal expressions are not equal?

I am trying to solve the following integral: \begin{equation}\label{1} \int \sin(\alpha t)\exp(-i\alpha t)\, \text{dt}. \end{equation} Using the exponential form of $\sin(\alpha t)$, the integral ...
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