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

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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
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0answers
41 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
46 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
101 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.
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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 \...
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0answers
40 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
49 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{...
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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
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
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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
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0answers
43 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
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
107 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
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4answers
250 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
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1answer
107 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
288 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
123 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
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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
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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
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2answers
195 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
71 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
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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 = \...
3
votes
2answers
76 views

Calculating residues of function with branch cut

Show that $$I= \int_{0}^{\infty} \frac{\ln x}{x^\frac{3}{4} (1+x)} dx = -\sqrt{2} \pi^2$$ I used a keyhole contour, with branch point at $z=0.$ Around $\Gamma$, $|zf(z)|$ tends to $0$ as $z$ tends ...
1
vote
1answer
91 views

Representing the function $f\left ( x \right )=\frac{1}{e^{2}e^{\cos\left ( x \right )}-1}$ in terms of Fourier series

The function is periodic with main period of $2\pi$, and it is even. So only the coefficients of the cosine terms remain. Wolfram alpha gives the result for $a_{0}$ as follows: I guess it is only ...
3
votes
1answer
72 views

Fourier series of $\frac{1}{5+4 \cos x}$ using contour integration

The function $$f(x)=\frac{1}{5+4 \cos x}$$ is periodic with the main period being $T=2\pi$. The graph is easily obtained, but here is a graph from Desmos as it looks better: The function is even, ...
0
votes
1answer
41 views

What is the definition of this symbol $ \int_{\sigma-i\infty}^{\sigma+i\infty} f(s) \, ds, \quad \sigma>0.$

What is the definition of this symbol $$ \int_{\sigma-i\infty}^{\sigma+i\infty} f(s) \, ds, \quad \sigma>0.$$ Thank you in advance
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0answers
49 views

how to evaluate this contour integral

how to evaluate $\int_0^\infty \! e^{ipx} \, \mathrm{d}x$, I know I can take the contour in the first quadrant, but why how does the integral over the arc vanish as R goes infinity, as it does not ...
5
votes
2answers
91 views

Computing alternating sum using contour integration

By considering the integral of: $$\left(\frac{\sin\alpha z}{\alpha z}\right)^2 \frac{\pi}{\sin \pi z},\quad \alpha<\frac{\pi}{2}$$ around a circle of large radius, prove that: $$\sum\...
6
votes
2answers
130 views

Computing $\sum\limits_{n=1}^\infty\frac{\sin n}{n}$ with residues

I'm running into some error in computing the sum. Since $\dfrac{\sin n}{n}$ is even, I'm considering the function $f(z)=\dfrac{\pi\sin z\cot\pi z}{z}$ and the contour integral $$\oint_\gamma \frac{\pi\...
1
vote
1answer
55 views

Express $\int_{\sin nx}^{\sin(n+1)x}\sin t^2dt$ in terms of $x$ and $n$

Please help me to express $$\int_{\sin nx}^{\sin(n+1)x}\sin t^2\,dt$$ in terms of $x$ and $n$. If it is not possible please help to establish bounds on the integral again in terms of $x$ and $n$. The ...
3
votes
3answers
102 views

Contour integral with a logarithm squared

The integral I'd like to evaluate is $\int_0^\infty \frac{\log^2 x \, dx}{(1+x)^2}$. I consider the function $f(z) = \frac{\text{Log}^2 z}{(1+z)^2}$, which has a pole of order 2 at $z=-1$ and has a ...
3
votes
1answer
80 views

Violating Cauchy's Integral Theorem

With regards to utilizing Cauchy's Integral Theorem for integration over closed contours: https://en.wikipedia.org/wiki/Cauchy%27s_integral_theorem In particular the result that $\int_\gamma f(z)\,...
2
votes
2answers
238 views

Finding Cauchy Principal Value for $\int_0^\infty \frac{\ln^2x }{(1-x)^2\sqrt{x}(x-4)} dx$

I am trying to find Cauchy Principal value for $$\int_0^\infty \frac{\ln^2x }{(1-x)^2\sqrt{x}(x-4)} dx$$ Can you please suggest me where to start? Any help would be appreciated. Thanks!
3
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1answer
71 views

evaluate the path integral around a circle in complex plane

Let $a \in \mathbb{C}$ with $|a|>1$. I need to evaluate the path integral around the unit circle in $\mathbb{C}$: $$\int_{|z|=1}\frac{|dz|}{|az-1|^2}$$ where $|dz|$ represents integration with ...
2
votes
1answer
62 views

Laurent expansion of $\operatorname{sech}(z)$ centred at $\pi i/2$

I have found that the roots of the $\cosh(z)=0$ occur at $\frac{(2k+1)\pi i}{2}$ where $k \in \mathbb{N}\cup{0}$. But I want to find the order the poles of $\operatorname{sech}(z)$ so I'm trying to ...
16
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2answers
411 views

How to prove that $\int_0^\infty\frac{\left(x^2+x+\frac{1}{12}\right)e^{-x}}{\left(x^2+x+\frac{3}{4}\right)^3\sqrt{x}}\ dx=\frac{2\sqrt{\pi}}{9}$?

A friend gave me this integral as a challenge $$ \int_0^\infty\frac{\left(x^2+x+\frac{1}{12}\right)e^{-x}}{\left(x^2+x+\frac{3}{4}\right)^3\sqrt{x}}\ dx=\frac{2\sqrt{\pi}}{9}. $$ This integral can be ...
2
votes
3answers
51 views

Two Indefinite Integrals

Looking for some hints to evaluate the following integrals (with complex analysis or otherwise): $$\int_0^\infty\frac{x^{p-1}}{x+1}\,dx,\;\;\;\; 0<p<1,$$ $$\int_{-\infty}^\infty e^{-s^2+isz}\,ds,...
5
votes
2answers
113 views

How do I get $ \int_0^1 \frac{dz}{\sqrt{z(z - 1\,)(z+1\,)}} = \frac{\sqrt{\pi}}{2} \frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{9}{4})}$?

While reading physics papers I found a very interesting integral so I decided to write it down. Let $p(z) = z^ 3 - 3\Lambda^ 2 z$ where $\Lambda$ could be any number. If you want $\Lambda = 1$ and $...
2
votes
1answer
38 views

contour integration problem.. [closed]

how can we find $$\int_C e^{2z} 9^{z-2} dz,$$ where $C$ is the the contour from $z = 0$ to $z = 1 − i$
0
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0answers
38 views

Matsubara sum with general exponent

Matsubara sums of the form $$\sum_{i\omega}\frac{1}{(i\omega-\xi_1)^a}\frac{1}{(i\omega-\xi_2)^a} $$ have closed-form solutions for $a=1,2$. See Wikipedia. Are there also closed-form solutions for ...
1
vote
1answer
51 views

If $\lim\inf_{r\to 0}{r}\cdot \max_{|z|=r}|{f(z)}|=0$ then $0$ is removable singularity.

$\lim\inf_{r\to 0}{r}\cdot \max_{|z|=r}|{f(z)}|$ show $0$ is removable singularity, given $f$ is analytic in a punctured neighborhood of $z=0$. What makes it difficult for me is the fact that the ...
1
vote
1answer
148 views

Integrating $\sin(x)/x$, how to treat the pole at the origin? [duplicate]

I want to use residue theory to integrate $$\int_{-\infty}^{\infty}\frac{\sin(x)}{x}dx$$ What would be a good contour to use? I plan to take the imaginary part of this integral: $$\int \frac {e^{...
0
votes
0answers
46 views

Is the integral of any even complex function equal to $0$ on any contour?

Is it true that: $\oint _{C(5i+1,8\sqrt3)} \frac {z}{sh(z)} dz = \oint _{C(i,\sqrt{10})} \frac {z^2}{(1-cos2z)^4}dz = \oint _{C(\pi + i,4)} \frac {z}{tan(z)} dz = 0$ The problem is that i lost my ...