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

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3
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2answers
67 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
votes
0answers
35 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
vote
0answers
58 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 |...
6
votes
1answer
190 views

Another beta integral due to Cauchy.

I have the following identity which I want to prove: $$C(x,y):= \int_{-\infty}^{\infty} \frac{dt}{(1+it)^x(1-it)^y} = \frac{\pi \cdot 2^{2-x-y}\Gamma(x+y-1)}{\Gamma(x)\Gamma(y)}$$ where $\Re(x+y)>...
2
votes
0answers
117 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: $$\int_0^\...
2
votes
1answer
47 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 > \frac{1}{...
1
vote
1answer
37 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
votes
1answer
50 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
vote
1answer
34 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 integral,...
2
votes
0answers
48 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. ...
1
vote
2answers
30 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 ...
0
votes
1answer
25 views

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
votes
2answers
93 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 $z=-1$ For ...
0
votes
0answers
16 views

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 ...
0
votes
0answers
31 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
84 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 = \frac{5\pi}{16\sqrt{2}...
2
votes
1answer
31 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 ...
3
votes
3answers
153 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
votes
1answer
39 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 = \frac{\pi}{4}e^{-4a}\{...
0
votes
3answers
75 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
votes
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 )
0
votes
1answer
20 views

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
votes
2answers
75 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 }}\int_{...
1
vote
0answers
38 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
vote
1answer
45 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
99 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 $\int_{0}^{\infty}\frac{\...
3
votes
3answers
112 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.
0
votes
0answers
12 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 \...
0
votes
0answers
39 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
votes
2answers
175 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
votes
3answers
55 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 $\frac{z}{1+z^...
2
votes
1answer
48 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} \frac{...
0
votes
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
vote
1answer
84 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
votes
2answers
91 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
votes
1answer
22 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
votes
1answer
31 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, i\...
0
votes
0answers
42 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
votes
1answer
53 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
106 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 \infty}e^{st}F(...
7
votes
4answers
238 views

How does contour integral work?

It might be a vague question but I can't help 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
106 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
269 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 ...
5
votes
2answers
119 views

What is the Fourier transform of $\exp(2 \pi i / x)$?

The Fourier transform of $e^{2 \pi i / x}$ makes sense as a distribution, I believe. Does it have a nice expression in terms of functions and well-known distributions (e.g. Dirac delta)?
2
votes
1answer
42 views

Integrating secans over the imaginary axis using the residue theorem

I am trying to integrate $\sec(z)$ over the whole imaginary axis using the residue theorem. i.e., I want to calculate the integral $$\int_{\Gamma} \frac{dz}{\cos{z}}$$ where $\Gamma$ is the (open) ...
0
votes
0answers
37 views

Indefinite integral - tricks for expressing solution concisely

Consider the following indefinite integral: $I_n (b) = \int \mathrm{d}x \frac{\sin(nx) \sin(x)}{\cos^2(x)+b^2}$ where $b$ is a constant and $n = 1, 2, 3...$ Is it possible to write the solution ...
0
votes
0answers
37 views

Can any contour integral be calculated by direct methods?

Or is there ever an integral that can only be evaluated using integral theorems? Does substituting the parameterized contour into the equation and evaluating it as a Riemann integral always work?
6
votes
2answers
189 views

Confusion about contour integration of constant function: intuition vs. Residue Theorem

Let's say we have the holomorphic function $$f(z) = 1.$$ Because $f(z)$ has no poles, according the Residue Theorem we have $$\oint_\gamma f(z)\,dz = 0$$ for any closed counterclockwise path $\gamma$. ...
2
votes
2answers
70 views

$\int_0^{2\pi} \log |1-w e^{i\theta}| \, d\theta$

Let $w$ be a complex number such that $|w| < 1$. Evaluate the integral $$\int_0^{2\pi} \log |1-w e^{i\theta}| \, d\theta.$$ I am having a hard time moving forward on this question. I tried ...
1
vote
0answers
35 views

Evaluating real trigonometric integral using contour, with pole order n

Use the residue theorem to compute the real integral: $$I = \int_{0}^{2\pi} \sin^{2n}\theta d\theta$$ I have considered a contour around a unit circle C, and used the substitutions: $sin\theta = \...