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Questions tagged [contour-integration]

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

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25 views

contour integral of 1/sqrt(z(z-1)(z-2)

How to find contour integral of $1/\sqrt{(z(z-1)(z-2)}$ around 0 and 1?
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3answers
59 views

Contour integration problem with sin and cos

so I'm revising contour integration for an upcoming complex analysis exam. I have been asked to integrate $$\int_0^{2\pi}\frac{\sin^2x}{a+b \cos x}dx$$ I thought the sensible thing to do here would ...
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1answer
20 views

Explanation of coefficient when evaluating contour around a branch for fractional version of Cauchy's Integral Formula

I am working on fractional derivatives which are defined by taking the Cauchy Integral formula and letting the order of the derivative be non-integer. Specifically, \begin{equation} f^{(\alpha)}(z)=\...
1
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0answers
72 views

Inv. Laplace $\frac{1}{s}\frac{1}{\frac{\sqrt{sAB}}{A} \sinh \sqrt{sAB} \frac{C}{\sqrt{sCD}} \sinh \sqrt{s CD}+\cosh{\sqrt{sAB}} \cosh {\sqrt{sCD}} }$

What would be the inverse Laplace of the following function? $\frac{1}{s}\frac{1}{\frac{\sqrt{sAB}}{A} \sinh \sqrt{sAB} \frac{C}{\sqrt{sCD}} \sinh \sqrt{s CD}+\cosh{\sqrt{sAB}} \cosh {\sqrt{sCD}} }$ ...
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1answer
35 views

Inverse Laplace transform of $F(s)=\frac{1}{s\sqrt{s+b\sqrt{as}\tanh{(\sqrt{as})}}}$ using complex integration

I want to find the inverse Laplace transform of $$F(s)=\frac{1}{s\sqrt{s+b\sqrt{as}\tanh{(\sqrt{as})}}}$$ I tried to use the Bromwich integral $$f(t)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\...
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1answer
56 views

Meaning of <a,b> symbol

I am reading some lecture notes about image science and active contours, and I come across the following: $J_1(c)=\int_a^b\left|\frac{\partial c}{\partial q}(t,q)\right|^2dq + \int_a^b g^2(|\nabla I(...
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1answer
35 views

Proving that $\int_{-\infty}^{\infty} {\dfrac{\cos \pi x}{z^2-2z+5}}\mathrm{d}x = -\frac{\pi }{2}e^{-2\pi}$

I want to show that (for $x \in \mathbb{R}$ and $z \in \mathbb{C}$) $$ \int_{-\infty}^{\infty} {\dfrac{\cos \pi x}{z^2-2z+5}}\mathrm{d}x = -\frac{\pi }{2}e^{-2\pi}$$ However, I am a little confused ...
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4answers
74 views

$2\pi i\beta = \int\limits_{|z|=r} \frac{dz}{f(z)-z}$

Let $\beta \in \mathbb{C^{*}}$ and $f(z) = z+ z^{k+1} - \beta z^{2k+1}$. Show if if r is small enough then $$2\pi i\beta = \int\limits_{|z|=r} \frac{dz}{f(z)-z}$$ This is my input: I have to ...
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1answer
68 views

Defining the $\arctan(z)$

We wish to define the $\arctan(z)$ to be: $$\arctan(z) := \int_{\gamma_z} \frac{1}{1+w^2}dw$$ Where $\gamma_z$ is a curve connecting $0$ and $z$ I am asked to: (a) find an open set $U \subset \...
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0answers
37 views

Rewriting a state as a field in CFT

I've been working through a textbook and course on conformal field theory recently. However in a section illustrating how to calculate correlators for secondary fields (using the free boson as an ...
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1answer
28 views

Help with Contour Integration with Finite Limits

I need to perform this integral: $$ \int_0^{\omega_{\large s}/\left(4\pi\right)} \frac{\mathrm{i}\omega\,\mathrm{e}^{\mathrm{i}\omega t}} {\left(\mathrm{i}\omega - \omega_{0}\right) \left(\mathrm{i}\...
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0answers
39 views

Show that $\int_{\mathbb \Gamma}\lambda^kd\lambda = 0 , \forall k \ge 0$

Let $\mathbb \Gamma$ denote the boundary of a convex polygon with vertices $w_1, ..., w_n$ in $\mathbb C$. Show that $$\int_{\mathbb \Gamma}\lambda^kd\lambda = 0 , \forall k \ge 0$$ I've found some ...
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2answers
60 views

Integration by transforming to complex

Evaluate the following by transforming it into a complex integral: $$\int_{-\infty}^{\infty} \frac{\cos 4x}{x^4+5x^2+4}dx.$$ Could someone show me where to start? This is not homework, it's a study ...
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2answers
41 views

Inverse Laplace transform of $f(s)={\frac{1}{s^{3/2}}}$ using complex integration

I want to find the inverse Laplace transform of $$f(s)={\frac{1}{s^{3/2}}}$$ Refer to the Laplace transform table, and I found that the result is $$F(t)=2\sqrt{\frac{t}{\pi}}$$ But I do not know how ...
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1answer
35 views

Calculating $\oint_{\gamma}\bar z^ndz$ for a known path

I wish to calculate the following integral: $$\oint_{\gamma}\bar z^ndz$$ $\gamma$ is a triangle with vertices at $0,1,i$, in the positive direction and $n\in \mathbb Z$. Since $\bar z^n$ is not ...
5
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1answer
163 views

contour integration - definite integral!!

I solved this question and had try as shown ... i got similar form ... but not getting correct answer.. here is the question: Here is my try : Can anyone tell me where is my mistake ... i will be ...
2
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2answers
55 views

Inverse Laplace transform of $\frac{π\cosh(\sqrt s)}{2 s^{3/2} \sinh(\sqrt s)}$ using complex integration

I want to find the inverse Laplace transform of $$F(s)=\frac{π\cosh(\sqrt s)}{2 s^{3/2} \sinh(\sqrt s)}$$ using the Bromwich integral $$f(t)= \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} \frac{π\...
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1answer
15 views

contour integration - Estimation Lemma

Let $C_n$ denote the boundary of the square formed by the lines x= $\pm$ $N$$\pi$ and y = $\pm$ $N$$\pi$ where $N$$\in$ $\mathbb{N}$, and let the orientation of $C_n$ be counterclockwise. Show ...
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1answer
28 views

Help with complex valued integral

I'm given that $f(z)=z \bar{z}=x^{2}+y^2 $ with $z=x+iy$ and $r(t)= <cost,sint>,\quad 0 \leq t \leq 2\pi$. I was able to evaluate the line integral by parametrization of f using r(t). I am ...
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0answers
43 views

Formulations of Cauchy's theorem that don't seem consistent

So I am reading through the book Complex Analysis by Lars Ahlfors, and there is one point that is causing some confusion for me: In chapter 4.2 - Cauchy's integral formula, we first encounter the ...
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0answers
57 views

Evaluation of the integral $ \int \frac{x^{\frac{1}{3}}}{1+x^3 } dx $ [closed]

I'm looking to solve this integral right here: $ \int \frac{x^{\frac{1}{3}}}{1+x^3 } dx $ I would like to know what approaches I could take to solve this using complex analysis.
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0answers
13 views

Analytic continuation and complex integration of one variable of a multivariate function

Consider a function $f:(x_0,x_1,...,x_n)\in\mathbb{R}^n\rightarrow f(x_0,...,x_n)\in\mathbb{R}$. $f$ is a ratio of polynomials in $x_0,...,x_n$ which only has simple poles in the variable $x_0$, whose ...
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2answers
65 views

Finding the infinite sum involving $\coth$ function using contour integration

I am looking to show: $\sum_{n=1}^∞ \frac{\coth(nπ)}{n^3} = \frac{7π^3}{180}$ There is a hint earlier that you are supposed to be using the function $f(z)=\frac{\cot z\coth z}{z^3}$. I have ...
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1answer
47 views

Computing $\int_{|z-i|=\frac{3}{2}}\frac{e^{\frac{1}{z^2}}}{z^2+1}$

Compute the integral using residues: $\int_{|z-i|=\frac{3}{2}}\frac{e^{\frac{1}{z^2}}}{z^2+1}$ Inside the circumference there are the following singular points $-i$ which is a pole of order 1 and $0$ ...
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1answer
49 views

$\int_{0}^{\infty} \ln{(1+a x)}{x^{-b-1}} dx$ difficult integral with two branch cuts

$$\int_{0}^{\infty} \ln{(1+a x)}{x^{-b-1}} dx$$ I defined two branch cuts along the real axis: $[-\infty ,-\frac{1}{a}]$ & $[0,\infty]$ with the following contour: I defined the $arg{(z)} =0$ ...
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0answers
31 views

Integrating $\int_0^{\infty} \frac{(\log x)}{(x + a)^2 + b^2} \operatorname d\!x$

I'm trying to show that $\int_0^{\infty} \frac{(\log x)}{(x + a)^2 + b^2} \operatorname d\!x = \frac{1}{b}\arctan \sqrt{a^2 + b^2}.$ However, I am a bit confused applying the key hole "method." I ...
3
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1answer
56 views

Construct $\varphi (z)$ such that $\int_{|z|=1} \frac{\varphi (z)}{z-w} dz =0$

I have this problem to complex analisis. Construct $\varphi (z)$ a continous function nonzero in $S^{1}$ such that $$\int_{|z|=1} \frac{\varphi (z)}{z-w} dz =0$$ for $|w|<1$. I have the idea to ...
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1answer
77 views

$\int_{-\infty}^\infty \frac{e^{pz}}{e^z-1}dz$ Cauchy principal value

$$\int_{-\infty}^\infty \frac{e^{pz}}{e^z-1}dz$$ I started by defining the following contour: rectangular contour It is easy to show that the integrals along the 2 vertical sides of the rectangle ...
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1answer
45 views

Evaluate $\int _{-\infty }^{+\infty }\frac{\left(x-1\right)\cos(2x)}{x^2-4x+5}\,dx$

Consider $$f\left(z\right)=\frac{\left(z-1\right)e^{2iz}}{z^2-4z+5}$$ The poles are $$z=2\pm i$$ $$\int _{\gamma }f(z)\,dz=2i\pi \operatorname{Res}\left(f,2+i\right)$$ $$\operatorname{Res}\left(...
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2answers
158 views

Evaluate $\int _0^{2\pi }\frac{\cos (n\theta) }{a+\cos\theta}\,d\theta$ with $a>1$, $n\in \mathbb{N}-\left\{0\right\}$

Evaluate $$\int _0^{2\pi }\frac{\cos (n\theta) }{a+\cos\theta}\,d\theta,\quad\,a>1$$ I wrote $$f\left(z\right)=\frac{\frac{1}{2}\left(z^n+z^{-n}\right)}{\frac{iz^2}{2}+aiz+\frac{i}{2}}$$ The ...
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1answer
28 views

How to solve the following integral by method of residue? [duplicate]

$$ \int_{-\infty}^{\infty} \frac{dz}{(z^2+a^2)^2}$$ Now this has the poles at $$(z^2+a^2) = 0 $$ if we consider the contour from -R to +R extending from $-\infty$ to $+\infty$ i get the roots $z = ...
3
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1answer
80 views

Evaluating $\int_{-2}^{-1} \int_{-2}^{-1} \int_{-2}^{-1}\frac{x^2}{x^2+y^2+z^2} \,dx \,dy \,dz$

I am trying to solve this triple integral problem , but I am having some issues. $$\int_{-2}^{-1} \int_{-2}^{-1} \int_{-2}^{-1}\frac{x^2}{x^2+y^2+z^2} dx dy dz$$ I tried with the 2 different ...
2
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2answers
48 views

Need help with an Improper integral

I need to evaluate $$\int_0^{2\pi}\frac{d\theta}{a+\sin^2\theta}.$$ I immediately noticed how this is an integral of the form $\int_0^{2\pi} f(\cos(\theta),\sin(\theta))d\theta$. I first tried to ...
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1answer
7 views

Please clarify scenario about contour integral over closed path

Contour integral: If function $f:\Bbb C-\{0\}\to\Bbb C$ continuous function on punctured domain.Suppose anti derivative for $f$ exists everywhere except $\{0\}$ then on a closed contour $C$, which ...
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1answer
24 views

Integrating a complex function over the unit circle

We need to integrate the following: $$\oint_C \frac{1}{z-a} \ dz$$ Where $|a| < 1$, and $C$ the unit circle ($e^{it} \ | \ t \in [0,2\pi] $) . My idea was to find a geometric series, but I don't ...
5
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1answer
151 views

How do you solve this $\int\limits ^{\infty }_{0}\frac{\cos( x)}{x^{n} +1} dx,\ n >0$

Me and my friend have tried a wedge,a triangle, and we even tried Feynman's technique. None of these things got us an answer to the integral $\int\limits ^{\infty }_{0}\frac{\cos( x)}{x^{n} +1} dx,\ n ...
9
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3answers
195 views

Evaluate the definite integral $\int^{\infty }_{0}\frac{x \,dx}{e^{x} -1}$ using contour integration

My friend and I have been trying weeks to evaluate the integral $$\int^{\infty }_{0}\frac{x \,dx}{e^{x} -1} .$$ We have together tried 23 contours, and all have failed. We already know how to ...
0
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0answers
36 views

How to solve this contour integral?

I was reading this where I encountered the following contour integral as given in equation (2.4) of the same. $$S = -i\int_{-\infty}^{+\infty} d\omega \log(\omega^2 + m^2 + E)$$ where $m,E \in \...
0
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1answer
37 views

Path dependency of complex integrals

This is a basic question. But it’s my first time really dealing with this topic. When we have a integral along the real line we have: $$\int^{b}_{a} f(x)dx$$ Which obviously has no path dependency. ...
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1answer
21 views

Contour integration with the contour $\sigma=[0,1]+[1,i]$

$\sigma=[0,1]+[1,i]$ is a contour. I am asked to sketch the contour, and evaluate $\int_\sigma Re(z)$. Firstly, I am not sure how to visualise this contour, since there are two parts. What does it ...
2
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0answers
54 views

Questions regarding the complex integral $\int_{\gamma} \frac{1}{(z-a)(z-b)} dz$

I don't know Cauchy's integral formula or branch cuts yet, and the book I'm learning complex analysis from asks to prove $\int_{\gamma} \frac{1}{(z-a)(z-b)} dz = 2\pi i$ where $\gamma$ is a circle ...
0
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0answers
34 views

Understanding Rademachers Contour/Hardy-Ramanujan's circle method

I'm trying to understand how to derive formulas for coefficients of certain functions, and in order to do this I need to be able to understand the so called "circle method". So I know that if $f(x)=\...
0
votes
0answers
38 views

why Cauchy - Goursat theorem needs the function to be analytic on the contour?

I can not understand why Cauchy - Goursat theorem needs the function to be analytic on the contour. Can anyone please make me understand what is the need for this? The following is an excerpt from ...
0
votes
2answers
34 views

Evaluate $\int_{C}(z-i) \,dz$ where $C$ is the parabolic segment: $z(t) = t + it^2, −1 \le t \le 1$

Evaluate $\int_{C}(z-i) \,dz$ where $C$ is the parabolic segment: $$z(t) = t + it^2, −1 \le t \le 1$$ by integrating along the straight line from $−1+i$ to $1+i$ and applying the Closed Curve Theorem. ...
1
vote
2answers
44 views

Why is $\int_{\gamma}f(z)\,dz\neq\int_a^bf(\gamma(x))\,dx$, and what is $\int_a^bf(\gamma(x))\,dx$

Let $\gamma:[-\pi,\pi]\mapsto\mathbb{C}$ be defined by $\gamma(x)=e^{ix}$. What is the geometric significance of $$\int_{-\pi}^{\pi}f(e^{ix})\,dx$$ versus $$\oint_{\gamma}f(z)\,dz.$$ As somebody who ...
1
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1answer
52 views

how $\int_a ^ b |f'(x)|$ gives the length of the arc of the contour $f$ : $(f(x) : x \in [a , b])$

I got to know that $\int_a ^ b |f'(x)|$ gives the length of any contour. Where $f(x)$ is a piece-wise differentiable function from $[a,b]$ to $\mathbb R^2$. I was reading complex integral . Can anyone ...
0
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0answers
28 views

Cauchy Principal Value for Complex Oscillating Function

I'm not particularly well-versed in how to compute Cauchy principal values, so any help here would be appreciated. I am trying to evaluate the PV for the following integral: \begin{align} \int_{-\...
1
vote
2answers
65 views

An integral over $\mathbb C$:

I would like to know is that this equality is just: $$\int_{\mathbb C}e^{-z}dz=\int_{\mathbb R}e^{-x}dx \times \int_{\mathbb R}e^{-iy}dy$$ for all $z=x+iy\in \mathbb C$. Or in general: $$\int_{\mathbb ...
0
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1answer
48 views

Integration (similar to Laplace transform) & limit for large argument

I am faced with the following integral, that looks like essentially finding a Laplace transform, and would like to know how to extract the asymptotic behaviour for large argument. The integral in ...
0
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1answer
36 views

Computation of integral $\int_{\rho} \frac{dz}{(z-a)(z-b)}$ [duplicate]

Let $a,b$ be complex number and $|a| < r < |b|$, compute. $\int_{\rho} \frac{dz}{(z-a)(z-b)}$ where $\rho$ is the circle with radius $r$ and the usual orientation. I've tried the common path ...