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Questions tagged [complex-integration]

For questions about integration methods that use results from complex analysis and their applications

0
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2answers
25 views

Integrate $\int_{|z|=1}z^{3}e^{1/z}dz$ - verification

I integrate over a circular path centered at 0 with radius 1 $\int_{|z|=1}z^{3}e^{1/z}dz=\int_{|z|=1}z^{3}\sum\limits_{n=0}^{\infty}\frac{1}{n!z^{n}}dz=\int_{|z|=1}\sum\limits_{n=0}^{\infty}\frac{1}{...
0
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3answers
21 views

Integrate $\int_{|z|=1/2}\frac{e^{1-z}}{z^{3}(1-z)}dz$ verification

I integrate over the edge of a circle $K$ with radius 1/2 $\int_{|z|=1/2}\frac{e^{1-z}}{z^{3}(1-z)}dz=\int_{|z|=1/2}-\frac{e^{1-z}}{z^{3}}\frac{1}{(z-1)}dz$ By the Cauchy Integral form $f(w)=\frac{...
1
vote
0answers
14 views

How the line element change in a complex change of variables?

So I'm learning conformal field theory and having a hard time to prove the conformal Ward identity. From the lectures notes from John Cardy, he express the integral $$ \delta S = \frac{1}{2\pi} \...
1
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0answers
37 views

Why is the limit of $\int {\frac{1}{z-z_0-x} - \frac{1}{z-z_0+x}}dz$ on a vertical segment as $x$ approaches $0$ not $0$?

A solution to the problem below is given by parametrization, converting the integral from $dz$ to $dt$. The result would be $ -2\pi i$, which would be correct. My problem is why is it not $0$, which ...
1
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1answer
45 views

How to determine $\gamma$ in Fox H-function

In the following Fox H-function the contour $L$ is either $L_{-\infty}$, $L_{+\infty}$ or $L_{i\gamma\infty}$. $$ H_{p,q}^{\,m,n} \!\left[ z \left| \begin{matrix} ( a_1 , A_1 ) & ( a_2 , A_2 ) &...
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1answer
27 views

Integral of conjugate complex numbers [on hold]

I know contour integral but I don't know what to do with this??? $$\int_{0}^{1+i} ({z^{*}}) ^{2}dz\\$$ along the line $$\\x=3y\\$$ Thanks in advance!
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0answers
16 views

solving integral with real exponent and real pole with residue theorem

I'm trying to solve this integral: $ \int_{-\infty}^{\infty} \frac{x^3 e^{- \alpha x^2}}{\beta - x} dx$ It looks similar to a complex integral with a pole but notice a few subtleties: The exponent ...
1
vote
1answer
29 views

prove that for any $f \in C^1(U)$

Guys, I need help with this question. I honestly don't even my take on this one yet because I do not know where or how to start. Please help/direct me. Thanks in advance
1
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1answer
49 views

Integral of Complex Gaussian: $\int_{-\infty}^{\infty} e^{-(2\pi x +i\omega)^2}dx$.

I wonder if the integral $\int_{-\infty}^{{\infty}}e^{-\alpha x^2}=\sqrt{\frac{\pi}{\alpha}}$, for $\alpha\neq 0$, how could the integral $\int_{-\infty}^{\infty} e^{-(2\pi x +i\omega)^2}dx$ be ...
2
votes
2answers
66 views

Integral of real part of $z$ around the unit circle

What is the result of integrating the real part of z (a complex number) anti clockwise around the unit circle? At first glance, I couldn't identify any points within the circle where analyticity ...
0
votes
2answers
65 views

Calculate the integral using complex analysis $\int_0^{\infty}\frac{\sin(\sqrt s)}{(1+s)^{2}} ds$

The task is to calculate this integral using complex analysis: I know some methods, which involve calculating the integral around half of the circle, also it would be wise to add here $i cos(s^{\frac{...
0
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0answers
38 views

Integrate $\int_{-\infty}^{\infty} e^{itx}\frac{1}{\sqrt{2 \pi}} e^{-x^2/2} dx?$

This is the characteristic function of normal distribution. My instructor showed that we can evaluate this with differentiation under integral trick (to obtain and solve a differential function). He ...
4
votes
5answers
110 views

Quick way of solving the contour integral $\oint \frac{1}{1+z^5} dz$

Consider the contour integral in the complex plane: $$\oint \frac{1}{1+z^5} dz$$ Here the contour is a circle with radius $3$ with centre in the origin. If we look at the poles, they need to satisfy $...
0
votes
1answer
18 views

Placement of singularities in the residue theorem

Why do the singularities in Cauchy's residue theorem have to be within the contour, and why do they still count if they're not on the path of integration, like I'd suspect for real integrals? Sorry if ...
2
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1answer
74 views

Estimating complex integral

I am reading chapter 12 of "Lectures on the Riemann Zeta function" by H. Iwaniec. I am stuck at understanding the computation done at the beginning of page 45. We want to estimate the following ...
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0answers
32 views

How to calculate $\int_0^\infty \frac{\sin(a+2n)x-\sin{ax}}{(1+x^2)\sin x}\mathrm dx,~a>-1,~n\in \mathbb{N^+}$ using the residue theorem?

The integral is $$\int_0^\infty \frac{\sin(a+2n)x-\sin{ax}}{(1+x^2)\sin x}\mathrm dx,~a>-1,~n\in \mathbb{N^+}$$ I tried to follow the common way of calculating integrals involving trigonometric ...
0
votes
1answer
58 views

Evaluate $\int_\gamma \frac{1+z}{1-\cos z} dz$ where $\gamma$ is origin centered circle with radius 7

I am beginner at this topic, i need help to solve the following question for my exam preparation. Evaluate $$\int_\gamma \frac{1+z}{1-\cos z} dz$$ where $\gamma$ is origin centered circle with radius ...
2
votes
2answers
70 views

A weird value obtained by using Cauchy Principal Value on $\int_{-\infty}^{\infty}\frac{1}{x^2}dx$

so I'm trying to evaluate the integral in the title, $$\int_{-\infty}^{\infty}\frac{1}{x^2}dx$$ by using complex plane integration. I've chosen my contour to be a infinte half circle with it's ...
0
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1answer
37 views

Different ( Is it equivalent?) hypothesis for stronger version of Goursat's theorem

Ahlfors, states that Theorem: Let $f(z)$ be analytic on the set $R'$ obtained from a rectangle $R$ (interior and on the boundary of the rectangle) by omitting a finite number of interior points $a_i$...
1
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0answers
39 views

1-loop integral from QFT

This question has no detail answers and I would like to ask the similar question. I deal with the following integral: $$\int_{0}^{1}dx\,x(1-x) \ln\frac{M^2-k^2x(1-x)}{\lambda^2},$$ where one can ...
0
votes
0answers
19 views

Value of integral of Omnes function

I want to get the value of this integral: $$\int_s^\infty \left(\frac{a}{z(z-b)} \right) dz$$ where a and b denote real constant , and b > s so there is a singularity and I don't know how to get the ...
1
vote
1answer
24 views

Simplify integral considering only the real part

I happened to stumble on the following simplification of an integral: $$ \frac{1}{\pi} \int_{0}^{\infty} dx \ e^{-ax} \cdot \cos(kx) = \frac{1}{\pi} Re \left[ \int_{0}^{\infty} dx \ e^{x (ik - a)} ...
1
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0answers
41 views

Show $\int _ { 0 } ^ { \infty } \frac { x \sin x } { \left( x ^ { 2 } + 1 \right) ^ { 2 } } dx = \frac { \pi } { 4 e }$ with residue theorem [duplicate]

Problem: Compute $\int _ { 0 } ^ { \infty } \frac { x \sin x } { \left( x ^ { 2 } + 1 \right) ^ { 2 } } dx = \frac { \pi } { 4 e }$ First we can study $\int_{\Gamma_R}g(z)dz$ where $\Gamma_R$ is ...
0
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0answers
20 views

Laplace inverse of functions involved with square roots

Let $ F(s) = \sqrt(s+ia)g(s) $, where $s$ is a complex number with $\Re(s) >0 $. I want Laplace inverse of $F(s) $. I have tried contour integration and convolution theorem but couldn't come up ...
1
vote
1answer
38 views

Laurent serie of $ \frac { \cos z}{ \sin z + \sinh z - 2z}$

I'm working on an example given in my book of complex analysis: $$ \frac { \cos z}{ \sin z + \sinh z - 2z}$$ but I can't figure out how he finded the residue in 0. The few steps he is showing make ...
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0answers
36 views

Riemann bilinear relations and meromorphic abelian differentials

I am getting quite confused with Riemann bilinear relations. Let $\Sigma$ be a compact Riemann surface of genus $g$, with a canonical homology basis $a_1,b_1,\dots,a_g,b_g$, with associated ...
1
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0answers
32 views

Computing a potential with a complex integration

I'm trying to understand the link between $ \mathbb R^2 $ and $ \mathbb C $. I know how to compute potentials when I know the gradient of it : $$ f(x,y) = \left( \frac {-x}{(x^2+y^2)^2 } , \frac {-y}...
1
vote
1answer
21 views

Is $\int\limits_0^R e^{-(1+i)r^2}dr$ the same as $\int\limits_0^R e^{-({1\over2}+{i\over2})r^2}dr$

In a corrected exercise there is this equality $\Gamma_R^3$ is the line $\{re^{i\pi/8}~: ~r\in[0,R]\}$ $-\sqrt2(re^{i\pi/8})^2=-\sqrt2(r^2e^{i\pi/4})=-\sqrt2r^2({\sqrt2\over2}+i{\sqrt2\over2})$ so I ...
2
votes
2answers
98 views

Solving a complex integral $\oint_L\frac{e^{1/(z-a)}}zdz$ using Cauchy's formula

I am practicing complex integration using Cauchy's formula, and I ran into a problem. The following integral: $$\oint_{L}{\frac{e^{\frac{1}{z-a}}}{z}}dz$$ where $$L=\{z\in\mathbb{C}:|z|=r\}$$ for ...
1
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0answers
32 views

Complex integral $\log(z)/((z-2)^2)z^n)$ [closed]

I'm studying complex through Conway and I'm obviously struggling with some key concepts. I'm trying to integrate $\int_{\gamma} \frac{\log(z)}{(z-2)^2z^n},\; n\in \Bbb{N}$ where $\gamma$ is ${[3-...
2
votes
1answer
55 views

result of this integral? $\int_{|z|=3} \frac{e^z}{(z-1)(z-2)}$

I think I can solve it by taking: $z(t)=3(\cos t + i \sin t)=3e^{it}$ and by using $$\int_c f(z)dz = \int_c \frac{dz}{dt}\cdot f(z(t))$$ we have: $\displaystyle\int_{|z|=3} \frac{e^{3e^{it}}}{(3e^{...
2
votes
1answer
73 views

Why is $\lim\limits_{r\rightarrow\infty}\int\limits_{C_r}f(z)dz=0$ where $C_r$ is the half circle $\{|z|=r,~\text{Im}(z)<0\}$

Let $$f(z)=\frac{\cos (z)e^{-2iz}}{z^2+2z+2}$$ Why is $\lim\limits_{r\rightarrow\infty}\int\limits_{C_r}f(z)dz=0$ where $C_r$ is the half circle $\{|z|=r,~\text{Im}(z)<0\}$? I was trying to ...
0
votes
2answers
56 views

If $\int\limits_\gamma f(z)dz\in\Bbb R$ does it imply $f(z)\in \Bbb R~\forall z\in\gamma$?

If $\int\limits_\gamma f(z)dz\in\Bbb R$ does it imply $f(z)\in \Bbb R~\forall z\in\gamma$? This is from a proof where $\gamma$ is a circle and we have $1={1\over 2\pi}\int\limits_\gamma f(z)dz={1\...
0
votes
1answer
43 views

Given $f$ is complete and $\lim_{z \rightarrow \infty} \frac{f(z)}{z} = 0$, prove that $f$ is constant [closed]

Does anyone know how to solve the following question without zeros poles and Macloren series? Given $f$ is complete and $\lim_{z \rightarrow \infty} \frac{f(z)}{z} = 0$, > prove that $f$ is ...
5
votes
1answer
237 views

Contour for $\int_0^\infty \arctan(z) e^{-z^2}\,dz$ or some variant

I'm trying to practice my contour integration skills and got interested in the following integral: $$\int_0^\infty \arctan(z) e^{-z^2}\,dz$$ I know that the usual way to calculate integrals on $[0,\...
1
vote
1answer
48 views

Integration of a multivalued function

The integral is: $$I=\int_1^2\frac{\sqrt{(x-1)(2-x)}}{x^2}dx$$ To solve this problem I integrate over a path $C$ that surrounds clockwise the branch cut, so the integral becomes: $$I=\frac{1}{2}\oint_{...
1
vote
1answer
63 views

Integration of $x^{\alpha}(1-x)^{1-\alpha}dx$

I'm trying to solve a definite integral $$\int^{1}_{0} x^{\alpha}(1-x)^{1-\alpha}dx $$ where $-1<\Re\alpha<2$ with a beta function: $$B(2-\alpha, \alpha + 1) = \frac{\Gamma(\alpha+1)\Gamma(2-\...
1
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0answers
52 views

Showing $\sin(z)$ is elliptic via conformal maps

One could use Euler's identity to show that $\sin(z)$ is an elliptic function, however, if we define the $\arcsin(z)$ as $$\arcsin(z) = \int_0^z \frac{1}{\sqrt{1-w^2}}dw$$ we would like to show ...
0
votes
1answer
40 views

Find upper bound of an integral using the ML inequality. $\left| \int_{\Gamma} \ln (z+3) \right|$

I need to find an upper bound of the following integral: $$\left| \int_{\Gamma} \ln (z+3) \right|$$ Where $\Gamma$ is the line segment from $(-1+3i)$ to $(4+3i)$. We have: $$\left| \int_{\...
0
votes
1answer
48 views

When would I use Cauchy's Integral Formula over Residue

Just a quick question I've been wondering about. When would I use Cauchy's Integral Formula over the Residue Theorem to solve complex integration problems with poles? To me it seems that Residue ...
0
votes
1answer
52 views

Computing a gaussian integral involving both real and imaginary coefficients in a stochastic system

I am stuck with the integral: $$\int_{-\infty}^{\infty} \frac{\exp[-a(x-b)^2]}{1+cx^2} dx$$ where $a,c$ are real and $b$ is purely imaginary. I tried to solve it by contour integration but the ...
0
votes
1answer
33 views

Find the complex integral over a function [closed]

I need to find $$ \int_\gamma{\Im zdz} $$ where $\gamma = \{x, y: y=2x^2, 0\le x\le1\}$. I have no clue how to do this.
0
votes
2answers
43 views

Calculate the complex integral

I have $$ \int{\frac{dz}{z^2+9}} $$ Also I'm given 2 different conditions. First is $|z|=\pi$, second is $|z-2i|=2$. Okay, so for the integral i have $\int{\frac{dz}{(z+3i)(z-3i)}}$. For the first ...
0
votes
1answer
40 views

$\int_{\Gamma} \dfrac{z^2+z^{-2}}{(z^*-r_1)(r_2-z^*)}dz$

Problem:: $\int_{\Gamma} \dfrac{z^2+z^{-2}}{(z^*-r_1)(r_2-z^*)}dz$ where $\Gamma= \{ z: |z|=r \}$ ($r_2>r>r_1>0$). Question: How to solve this integral? My attempt: My first idea is to use ...
1
vote
1answer
77 views

Given $f(z)$ is entire and a bound on f at each $ z \in \mathbb{C}$, prove that the integral is 0

The Problem: It's given that $f(z)$ is an entire function on $\mathbb{C}$ and that there exist $M > 0$ and $A > 0$ such that $|f(x+iy)|\le\frac{Ae^{2\pi M|y|}}{1+x^2}$ for all $x,y \in \mathbb{...
1
vote
1answer
32 views

Evaluate $\int_c (z - z^2) dz$

In an exercise, it's given that $C$ is a circle such that $|z-2|=3$ Now I know that, a circle with radius $r$ can be represented by $z=re^{i\theta}$, hence, $$ z-2=3e^{i\theta} \implies z=2+3e^{i\...
0
votes
2answers
19 views

Expressing coefficients of a complex function using integrals

Let $f(z)$ be the function defined by: \begin{equation} f(z)=a_{-3}z^{-3}+a_{-2}z^{-2}+a_{-1}z^{-1}+a_0+a_1z+a_2z^2+a_3z^3 \end{equation} How can we express the coefficients $a_i$ using integrals? ...
2
votes
2answers
37 views

Complex Integral $\int_{C} \frac{e^{iz}}{z^3} dz$ using Cauchy Integral Formula of Derivatives

I am trying to find $\int_{C} \frac{e^{iz}}{z^3} dz$ on circle of $|z| = 2$ traversing once on positive direction. My approach was using Cauchy derivative formula $f^{(n)}(a) = \frac{n!}{2\pi i} \...
-1
votes
1answer
59 views

Identity with regards to error of sinc approximation

I have this issue that I'm kind of clueless about, it is peripheral to what I typically do. I will state all the assumptions meticulously, even though I suspect they are not all needed. It is problem ...
2
votes
2answers
45 views

Compute complex integral inside an open curve

I need to compute this complex integral: $$ \int_\gamma \frac{1}{(z-i)(z-2i)} dz $$ $\gamma$ is defined as: $$ \gamma (t) = t + i(3e^t\cos^2(t)) $$ The parameter t belongs to the following ...