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

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1answer
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Complex integration on upper-half plane

In order to prove the normalisation property of a Lorentzian function, $L = \dfrac{1}{\pi}\displaystyle \int_{-\infty}^\infty \dfrac{b}{(z-a)^2+b^2} dz = 1$ we take a closed contour on the ...
6
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2answers
134 views

Computation of an iterated integral

I want to prove $$\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty\frac{\sin(x^2+y^2)}{x^2+y^2}dxdy=\frac{\pi^2}{2}.$$ Since the function $(x,y)\mapsto\sin(x^2+y^2)/(x^2+y^2)$ is not ...
2
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3answers
36 views

Compute the integrals using the residue theorem

Compute the following integrals: $I:=\int_{|z|=2}\frac{1}{(z-3)(z^{13}-1)}dz$ $J:=\int_{|z|=10}\frac{z^3}{z^4-1}dz$ I do not know where to begin. I know I am supposed to use the substitution ...
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3answers
40 views

Contour Integration: non-convergent integral

The question is $$I=\int_{-\infty}^{\infty} \frac{\sin^2{x}}{x^2} dx$$ My attempt: $$I=-\frac{1}{4}\int_{-\infty}^{\infty} \frac{e^{2ix}-2+e^{-2ix}}{x^2} dx$$ $$I=-\frac{1}{4} \Big[ ...
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1answer
31 views

Contour Integration with pole on contour

I have come across an example I don't understand.. So, here is the problematic part: Consider the integrals: $ I = \int_C \frac{e^{iz}}{z} dz $ $ J = \int_C \frac{e^{-iz}}{z} dz $ Where $C,C_-, ...
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1answer
34 views

Generalisation of an already generalised integral

Inspired by these two questions: Closed form for $\int_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$ Interesting integral formula I ask whether the following integral has a closed form: ...
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0answers
36 views

Integral with logarithmic residue (Residue general method)

I was reading about this post, about Integration in complex plane with logarithmic residue: Integral with logarithm - residue And I thought about: what if $R(x)$ function is inside the logarithm? ...
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2answers
141 views

How to solve $\displaystyle\int_0^{\frac{\pi}{2}} \ln{(x^2 + \ln^2{(\cos{x})})} \mathrm{d}x$ [duplicate]

$\displaystyle\int_0^{\frac{\pi}{2}} \ln{(x^2 + \ln^2{(\cos{x})})} \mathrm{d}x$ I was given this integral yesterday by someone on a forum and after a few hours of having a go at it I didn't really ...
3
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2answers
47 views

Convolution Integral to Evaluate Fourier Transform

According to Mathematica with Fourier transform convention $$\widehat{f}(\xi)=(2\pi)^{-1/2}\int_{-\infty}^{\infty}f(x)e^{i\pi x}dx$$ The Fourier transform of the function $f(x):=|x|^{-1/2}e^{-|x|}$ ...
0
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0answers
23 views

Limit of complex integral with no primitive

I'm having trouble trying to calculate the following limit. I know the answer is not 0, but after several attempts I am stuck on reducing it. We have $z_0$ as a constant complex number and a fixed ...
1
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0answers
39 views

Integral of complex logarithm on a disk in the plane

Let $a$ be a complex number and $D$ the disk centered around $0$ and of radius $R$. I would like to compute the integral I=$\int_D \log(|z-a|)d^2z$. I am interested in particular in the case $R\gg ...
2
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0answers
89 views

Integration: $\int_0^\infty \frac{\sin(at)}{2\cosh 2\pi t-1} \text{ d}t$

I am trying to calculate the following integral: $$\int_0^\infty \frac{\sin(at)}{2\cosh 2\pi t-1} \text{ d}t$$ The inspiration for it was that a similar integral carries a closed form: ...
2
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1answer
36 views

express as contour integral $ f(x) = \int_0^\infty dt \; e^{-t/g} \; \frac{1}{\sqrt{1 - 2 t x}} $

Let $0 < x < 1$, I have to compute this Laplace transform: $$ f(x) = \int_0^\infty dt \; e^{-t/g} \; \frac{1}{\sqrt{1 - 2 t x}} $$ I am not 100% this interal is defined. If $t > ...
1
vote
1answer
30 views

Normalisation of Lorentzian function proof

According to WolframMathworld, when we normalise the Lorentzian function, we get the following result: $\dfrac{1}{\pi}\displaystyle \int_{-\infty}^\infty \dfrac{b}{(z-a)^2+b^2} dz = 1$ I'm trying to ...
0
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1answer
22 views

Contour integral and primitive

Given $$f(x,y)=\frac{2}{i(1-y)-x}$$ I have to integrate $f$ over the origin-centered circle of radius 4. I see that $$f(x,y)=-\frac{2x+i2(1-y)}{x^2+(1-y)^2}$$ There is a singularity in $-i$ so I ...
1
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1answer
29 views

how to choose the path for complex line integrals? (Explicit question)

I have a small question about complex line integrals. I give an example to demonstrate what my question is. Consider the integral $$\int_{|z|=2}\frac{1}{z-1}dz.$$ If I want to calculate this ...
2
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0answers
33 views

Integral calculation by using Mellin Transform

I want to use the Mellin Transform (MT) to calculate the integral: $\int_0^{1 } \exp(-2\rho^2) J_0(\pi \rho r)\rho \, d\rho$ in which $r>=0$ and real. I have calculated it by numerical methods. ...
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2answers
25 views

Given $\vec{F}=(y,x)$, calculate contour integral $\int_C\vec{F}•\vec{dr}$, on length of closed path $C:\vec{r}(t)=(\cos t, \sin t), 0\leq t\leq2\pi$

I'm studying in preparation for a Mathematical Analysis II examination and I'm solving past exam exercises. If it's any indicator of difficulty, the exercise is Exercise 4 of 4, part $c$ and graded ...
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1answer
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Let $f$ be an arithmetic function of class $C^2$ with $f_{xx}+f_{yy}=0$. Prove that $\oint\vec{F}•\vec{dr}=0$ where $\vec{F}=(f_y,-f_x)$

I'm studying in preparation for a Mathematical Analysis II examination and I'm solving past exam exercises. If it's any indicator of difficulty, the exercise is Exercise 4 of 4, part $b$ and graded ...
2
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2answers
64 views

Using contour integrals to evaluate sum - Problem calculating residues

"Compute $$\sum_1^{\infty} \frac{1}{{n^2}{(n+1)^2}}$$ using contour integration" I have used the function $F(z) = \frac {\pi cot\pi z}{z^2(z+1)^2}$ Which has double poles at $z=0$ and $z=-1$ ...
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0answers
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What exactly is $p$ in numerical contour integration method by Lyness and Delves

http://www.ams.org/journals/mcom/1967-21-100/S0025-5718-1967-0229388-0/S0025-5718-1967-0229388-0.pdf: As far as my reading goes, I do not see how they set $p$ - is it the free variable that can be ...
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0answers
24 views

Numerical contour integration of a function by Lyness and Delves

In this link, which is one of the first papers dealing with numerical contour integration, in page $12$ right after $(5.12)$, they say: The factor of $4^{-(p^2)}$ in this expression is a number ...
2
votes
2answers
78 views

Contour integration around a closed loop

By considering the contour integral $$\int z^{5/4}(z-1)^{-1/4} dz$$ around a closed loop C that encircles the real interval $[0,1]$, show that $$\int^1_0 x^{5/4}(x-1)^{-1/4} dx = ...
2
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1answer
28 views

Residue of pole

I am trying to integrate $$\frac{1}{2\pi} \int^\infty_{-\infty} \frac{6e^{-ipt}}{(p+1)^2 +9} dp$$ I am using a D-contour and I am trying to calculate the residue at the pole $p = -1$. I am trying to ...
2
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2answers
76 views

I need to solve $\phi (x,y) = \frac{2V}{\pi} \int_{0}^{\infty} \frac{\sin(kx)\cosh(ky) dk}{k\cosh(ka)}$

I start with a integral in complex plane $$\oint_c \frac{e^{izx} e^{zy} dz}{z\cosh(za)}$$ where $c$ is a countour starting in $z = -R$ along the real axis and jumping the pole at origin and continuing ...
2
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1answer
28 views

Contour integral around 'D-contour'

By considering the contour integral aroun a 'D-contour' of the function $$f(z) = \frac{ze^{iaz}}{z^2 + 6 + 25}$$ for $a>0$, show that $$\int\frac{x\sin(ax)}{x^2 +6x +25}dx = ...
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3answers
71 views

How to evaluate the integral $\int_{0}^{\infty}\frac{\cos {(ax)}-\cos{(b x)}}{x^2 }dx$?

I'm wondering how to integrate the so-called integral using Residue theorem,as it has a pole of second order on the real axis(not simple) so we cannot use $\pi i Res(@ z=0)$.Would you please give me a ...
0
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1answer
60 views

easier way to find an integral?

Is there an easy way for : $$\int \frac{e^{\frac{1}{z}-2}}{z+2}dz$$ on$|z|=3$ ? thanks for any hint. (I try it by series expansion , but someone told me there is an easy way to find it )
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1answer
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Contour integral to real integral: find suitable change of variables

There's probably simple solution but... I have a contour integral of the form $\int _{-i \infty}^{+i \infty} f(t) \ dt$. I want to make a transformation $t = g(s)$ so that the integral is real and of ...
3
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2answers
63 views

Fourier transform of a Lévy density $\frac{1}{\sqrt{2\pi }}\int_{0}^{\infty} e^{ikx-\frac{1}{2x}}x^{-\frac{3}{2}}dx$

A Lévy density is defined as $$q(x;1/2,1)=\frac{1}{\sqrt{2\pi }}e^{-\frac{1}{2x}}x^{-\frac{3}{2}}$$ for $x>0$ I am looking for it's Fourier transform: $$g(k;1/2,1)=\frac{1}{\sqrt{2\pi ...
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0answers
27 views

Convolution of complex functions (Laplace Domain)

Convolution of functions in the time domain is equivalent to multiplication in the frequency domain. However, I am interested in multiplication of functions in the time domain, which is convolution in ...
1
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1answer
43 views

Proof that $ \int_0^\infty x^{d-4}\sin x\, dx = \cos \frac{\pi d}{2} \Gamma(d-3)$, for $2<\Re(d)<4 $?

Can one prove that $$ \int_0^\infty x^{d-4}\sin x\, dx = \cos \frac{\pi d}{2} \Gamma(d-3),\text{ for }2<\Re(d)<4? $$ I would prefer using the methods of contour integration.
3
votes
1answer
71 views

How to calculate $\int_{-\infty}^{\infty}\frac{x^2}{\cosh(x)}\mathrm{d}x$ [duplicate]

I know the poles are $z=i\pi/2+i n\pi$ and therefor I got an rectangular contour for the integration which wasn't so useful. I also know with change of variables I can get to ...
3
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3answers
99 views

How to show that $\int_0^{\infty} dx \frac{\log{x}}{1+x^2}$ is zero using complex analysis

I want to show this using contour integration, the appropriate contour is a keyhole I think.
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0answers
10 views

Contour integral of multivalued vector field

I would like the find the contour integral given by \begin{align} \oint_C d(\vec{u} \cdot \vec{u}) \end{align} where $C$ is the wedge shaped contour defined by $0 \leq r \leq R$ and $0 \leq \theta ...
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0answers
33 views

Is there anything wrong with the following work on the Argument Principle?

The Argument Principle states that : $$\oint_C {d\over dz}(log (f(z))) \, dz = 2\pi i(N-P)$$ Let $g(z)={d\over dz}\log(f(z))$ If $f: C \to C$ is a continuous function on a directed smooth curve, ...
2
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2answers
169 views

how to calculate $\int_{0}^{\infty}\frac{x}{\sqrt{e^x-1}}\mathrm{d}x$

I was trying to solve another integral when then I reached this, I've no idea of how to select the contour for the integration.
3
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3answers
37 views

integration using residue

I am solving the following integral: $$\int_0^\infty \frac{x}{1+x^3}dx$$ I need to solve it using integration and residue theorems. I tried to convert it to complex function, for example ...
2
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1answer
38 views

Trigonometric contour integral

I cannot figure out what I'm doing wrong: $$\int_0^{2\pi} \frac{1}{a+b\sin\theta} d\theta\quad a>b>0$$ $$\int_{|z|=1} \frac{1}{a+\frac{b}{2i}(z-z^{-1})} \frac{dz}{iz}$$ $$\int_{|z|=1} ...
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0answers
26 views

Can somebody check whether I have calculated this contour integral correctly?

$$\int_{|z-\frac{1}{2}|=1}\frac{e^{-iz}}{z(z-1)(z^2-1)} dz$$ I used the Residue Theorem and got this answer: $2i\pi-\pi e^i -\frac{3}{2}\pi i e^i$ Is there also some software that can compute these ...
1
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1answer
80 views

Can this integral be evaluated/approximated?

I've been trying to evaluate this integral without much success: $\displaystyle \int_{-\infty}^\infty dx\, e^{iax} \frac{1- e^{-c\sinh^2 bx}}{\sinh^2 bx}$ I've tried contour integration. There are no ...
2
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2answers
86 views

How do I evaluate $\int_{0}^{\infty} u^{z-1}(e^{iu}-1) \, du$?

I am trying to evaluate the following integral that shows up in this paper http://arxiv.org/pdf/1103.4306v1.pdf $I=\int_{0}^{\infty} u^{z-1}(e^{iu}-1)du= \Gamma(z)e^{\frac{iz\pi}{2}}$ for ...
2
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1answer
20 views

Computing this contour integral on the line $\mathbb{R} - 10 i$?

Let $$ \int_{\Gamma} dz \frac{e^{iz}}{1 + z^2} $$ be a contour integral. Now we have two cases. First $\Gamma$ is the real line $\mathbb{R}$ (i.e. the real axis), and second, where $\Gamma$ is the ...
0
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1answer
30 views

$1/\sinh^2z$ near real infinity

I was looking at a contour integration where the claim had been made that the following function $1/\sinh^2 z $ goes to zero along the following lines in complex plane $(-\infty, 0)$ to $(-\infty, ...
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0answers
38 views

Contour integration of a non-single valued complex function

Let $\xi>0$ and be real, $0<\alpha < 1$ and be real and $m\in\mathbb{N}$. Consider the integral on the complex plane z, $$\mathcal{F}(u) = \int_{\mathcal{C}}\exp(-(z\xi)^{\alpha})(u-z)^mdz$$ ...
0
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1answer
51 views

Is this Complex Integration correct?

I want to integrate $\displaystyle \int_{-\infty}^\infty dx \, e^{iax}\frac{1-e^{-bx^2}}{x^2}$ for a>0. I am going to try and do this using the method of contour integration. I will choose a ...
6
votes
1answer
74 views

Inverse Laplace transform of $1/\sqrt{s^2-a^2}$ using complex integration

I want to find the inverse Laplace transform of $$F(s) = \frac{1}{\sqrt{s^2-a^2}}$$ preferably using the Bromwich integral: $$f(t) = \frac{1}{2\pi i}\int_{\beta -I \infty}^{\beta +i ...
7
votes
4answers
139 views

How does contour integral work?

It might be a vague question but I can't help but asking what is so powerful in contour integral that makes it possible to compute certain improper real integrals that is seemingly very difficult to ...
4
votes
1answer
70 views

Using a contour integral about a branch cut to compute $\int \limits ^\infty _0 \frac {\ln x} {x^a (x+1)} dx$

Find the value of $I = \int \limits ^\infty _0 \frac {\ln x} {x^a (x+1)} dx$ for $a \in (0,1)$, placing the branch cut of the logarithm on the positive real axis. You can use the result that $\int ...
6
votes
2answers
159 views

What is the integral of 1/(z-i) over the unit circle?

At present there is a simple pole on the closed contour, so the Residue Theorem appears to be inapplicable. But I want to claim that we can enlarge this circle to make sure that it encloses the ...