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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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65 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 ...
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2answers
35 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
5 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
23 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 ...
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0answers
84 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 ...
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3answers
171 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 ...
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34 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 \...
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1answer
36 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 ...
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47 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 ...
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18 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)=\...
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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 ...
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2answers
32 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. ...
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2answers
43 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 ...
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1answer
51 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 ...
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0answers
24 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_{-\...
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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 ...
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1answer
44 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 ...
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1answer
31 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 ...
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1answer
25 views

Calculate $\oint_{\gamma} \tanh(z) dz$ on the curve $|z - \frac {\pi}{4}i|=\frac 12$

Calculate $\oint_{\gamma} \tanh(z) dz$ on the curve $|z - \frac {\pi}{4}i|=\frac 12$. I am not sure if I computed this correctly: I tried to rewrite $\tanh(z)= \frac {\frac {e^z-e^{-z}}{2i}}{\frac {e^...
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0answers
23 views

Correct way to analytically continue a multi-dimensional integral

Consider a multi-dimensional integral \begin{equation} \int dx_1 \int dx_2 ... \int dx_n f(x_1,...,x_n) . \end{equation} where $f$ has simple poles in each of the variables $x_1,...,x_n$. Is it ...
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2answers
37 views

Compute $\oint\frac{e^{2z}+\sin z}{(z^2+1)^3}dz$, over the curve C

Compute $I = \oint\frac{e^{2z}+\sin z}{(z^2+1)^3}dz$, over the curve $C:|z+i|=r, r\neq2$ So what I understood from my classes I have to find what it's poles are and then apply the residual theorem ...
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1answer
17 views

The condition for interchange path integral and integral over the real line

The question rises from a book I'm reading: Let $D$ be a domain, we have a function $g:D\subset \mathbb{C}\to \mathbb{C}$ defined by integration over a complex-valued fucntion $f$, where $f$ is entire ...
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1answer
33 views

Finding contour integral $ \int_\gamma \frac{\mathrm{Im} (z)}{z - \alpha} dz $

I'm trying to find the contour integral $$ \int_\gamma \frac{\mathrm{Im} (z)}{z - \alpha} dz $$ where $ \alpha $ is a complex number such that $ 0 < |\alpha| < 2 $ and $ \gamma $ is the circle ...
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1answer
37 views

Finding value of contour integral, if given values of function

I want to find the value of the contour integral $$ \int_C \frac{[g(z)]^4}{(z-i)^3} \,\mathrm{d} z $$ where $ C $ is the circle centred at origin with radius 2. If I have some values of the function $...
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2answers
137 views

Evaluating the integral $\int_{\left|z\right|=\pi}\frac{\left|z\right|e^{-\left|z\right|}}{z}dz$, where the function is not holomorphic

I need to evaluate on the circle $\left|z\right|=\pi$ the integral $$\int_{\left|z\right|=\pi}\frac{\left|z\right|e^{-\left|z\right|}}{z}dz.$$ The function is not holomorphic there. Anyway, I tried to ...
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0answers
27 views

Can any Gaussian integral with complex limits be written as a (complex) error function?

Can the integral, $$I = \int^{-x}_{-\infty} e^{-at^2}\ \mathrm dt,\ a \in \mathbb{C},\ Re(a) > 0,$$ be written as an error function? I tried, by substitution, $$\int^{-x}_{-\infty} e^{-at^...
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1answer
25 views

evaluate $f(z) $ described in the anticlockwise ( i.e. positive direction)

Evaluate $$\int_{C} \frac{dz}{(z^2+ 4)^2}$$ where $C = \{ z \in \mathbb{C} \mid |z-i| = 2\}$ described in the anticlockwise ( i.e. positive direction) My attempt: I used Cauchy integral formula ...
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1answer
26 views

Countour integral with branching point and pole behaviour

I want to compute this contour integral: \begin{equation} \int\limits_C \! \mathrm{d}z \; \log(\frac{z+1}{z-1}) \frac{e^{tz}}{z-1} \end{equation} Where $C$ is a path going around the branch cut. My ...
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39 views

Integral Using Argument Principle

I think I have the following problem solved, but I'm not completely sure my reasoning is sound: Let $n\in\mathbb{N}$ and let $C$ denote the unit circle with the counterclockwise orientation. ...
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1answer
28 views

Integral of Complex Polynomial with Quotient of Complex Sine

I'm stuck trying to finish the following problem: Evaluate the integral $\int_{C}\frac{2z^{2}-5z+2}{\sin(z)}~dz$, where $C$ is the unit circle (oriented counterclockwise). My work so far: We will ...
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0answers
22 views

problem with a residue at infinity

I have this contour integral, which is just a standard one. It has two simple poles in $z=i$ and $z=4i$ and I already worked out the result for a path that enclose both of the poles which is $$\int_{\...
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votes
3answers
47 views

Compute the integral $ \int_{Re(s)=2}\frac{x^{-s}}{s^3}ds\quad\text{for real}\ x>0. $

Use the residue theorem to compute the integral $$ \int_{Re(s)=2}\frac{x^{-s}}{s^3}ds\quad\text{for real}\ x>0, $$ where the contour is oriented upwards. (Hint: treat the cases of $ x<1 $ ...
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37 views

Szegő's method of finding the generating function of the Jacobi polynomials

In Orthogonal Polynomials (4th ed., Amer. Math. Soc. Colloq. Publ., vol. 23, 1975), Szegő starts off section 4.4 by giving the following integral representation of the Jacobi polynomials: $$P_n^{(\...
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0answers
17 views

Modified Error Function Integral

I was given an identity recently without any stated derivation, so I am attempting a derivation on my own. The identity in question is $$\int_0^\infty\frac{e^{-s^2t}\sin(sy)}{s}\,ds=erf(\frac{y}{2\...
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0answers
57 views

Applying the residue theorem to $\log^2(|1+hz|)$

I have to calculate an integral of the following form:$$\oint_{|z|=1} \log^2\left(\frac{|1+hz|^2}{(1-y2)^2}\right)\left(\frac{1}{z-r^{-1}}\right)\,\mathrm dz,$$ where $h,r$ and $y2$ are constants with ...
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1answer
17 views

Using Contour Integration to solve an integral that holds for all p.

I have been practicing contour integrals and I have come across the following integral and I have been trying to solve it via contour: $$ \int_0^\infty \frac{dx}{x^p+1} = \frac{\pi}{p} \frac{1}{\text{...
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0answers
27 views

Using the Cauchy-Goursat theorem to prove a statement

For $C$ a simple closed contour in the counterclockwise direction and $C_1$, $C_2$, $C_3$, $C_4$ are subsets in $C$ all in the counterclockwise direction, use the Cauchy-Goursat theorem to prove that: ...
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1answer
28 views

Sign convention for Fourier transform and contour integration - example

I was wondering about one (probably trivial) fact during computing the Fourier transform while using contour integral. As an example I have following function: $$f(x)={{1}\over{x^2+a^2}}$$ and its ...
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30 views

Branch cut of $\int_0^{\infty} \frac{x^{s-1}}{1+x}$

Why do I need to use a branch cut from $0$ to $\infty$ when evaluating $\int_0^{\infty} \frac{x^{s-1}}{1+x}$? I know there is a pole at $x=-1$ I can't seem to understand the purpose of the branch ...
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2answers
88 views

Proving $g'(z)=\frac{1}{2\pi{}i}\int_{C}\frac{g(u)du}{(u-z)^2}$ for $g(z)$ holomorphic in and on contour $C$, and $z$ in $C$'s interior

Prove that if $g(z)$ is holomorphic everywhere inside and on a simple closed contour $C$, taken in a positive sense, and $z$ is any point interior to $C$, then $$g'(z)=\dfrac{1}{2\pi{}i}\int_{C}\...
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10 views

the end point contribution by saddle point method

I am solving this equation and i get confused when considering to the end point contribution because the integral ranges between finite values $$\int_{\phi_0-\pi/2}^{\phi_0+\pi/2} \cos(\phi'-\phi_0)...
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2answers
54 views

Taking Residues of Infinity of Square Roots.

I am looking for worked out exercises of real valued integrals with square roots where ideas of residues at infinity are used. I was hoping that from $$\int_0^1 \sqrt{x} \thinspace dx $$ I could ...
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0answers
26 views

Using the Cauchy Principal Value along a differentiable contour?

I am looking at the following problem from Marsden and Hoffman: Let $ f ( z ) $ be analytic inside and on a simple closed contour $ \gamma $. For $ z_0 $ on $ \gamma $, and $ \gamma $ differentiable ...
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0answers
14 views

Converse of ML inequality for contour integrals

If the ML inequality estimates a region of an integral in the complex plane to be zero then that's the actual value of the integral, and I've been using this for evaluating some integrals along the ...
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0answers
43 views

Change of variables integration for all real p>1?

Originally I have the following problem to show it holds for any real $p>1$, $$\int_0^\infty \frac{1}{x^p+1}\, \mathrm{d}x = \frac{\pi}{p}\sin\left(\frac{\pi}{p}\right).$$ However, since there are ...
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2answers
61 views

Please compute $\int_0^{\infty}e^{-(x+ai)^2}dx$, where, $i^2=-1$.

I'm trying to do the following Fourier (Hankel?) transform for a cylindrincally symmetric function: $$ \int_0^{\infty} \!\int_0^{\infty} \! \left( Are^{-(r^2+z^2)/\delta^2}\right)J_1(k_r r) e^{-ik_z ...
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votes
3answers
108 views

Complex Contour Integral Evaluation.

I am having troubles evaluating this integral: $$\int_{-\infty}^{\infty} \frac{\cos x-1}{x^2(x^2+4)}dx$$ Well, actually, I'm quite sured about the numerical part: the value of this integral is likely ...
2
votes
2answers
32 views

Does there exist a branch of $(a^2 - z^2)^{1/2}$ holomorphic on $\mathbb{C}\setminus [-a,a]$?

To compute the integral \begin{align*} \int_{-a}^a (a^2-x^2)^{1/2} \, dx \end{align*} (and practice contour integration) I am trying to define a branch of the integrand with branch cut $[-a,a]$, and ...
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votes
1answer
50 views

Counting numbers of Fixed point of Zeta function by Argument Principle

This is my first post about this topic and now I am trying to evaluate the integral, $$N=\frac{1}{2\pi i}\oint_{|z-1|=1}\frac{\zeta'(z)-1}{\zeta(z)-z}dz+1$$ $\zeta-$is the Riemann Zeta function. I am ...