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

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2
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0answers
24 views

Integral representation of Bessel function K

Does someone have an idea how to connect the following function (appearing in the quantization of a real scalar field in a uniformely accelerated frame) : $$ K(x,y) = \int_{0}^{\infty} \frac{dt}{t} ...
0
votes
2answers
88 views

Integrating a contour of circle with singularities.

This question has $2$ singularities at $+i$ and $-i$. I am not sure how to solve.
4
votes
1answer
203 views

Evaluating $\int_0^{\infty}e^{-\alpha x^2 \cos \beta} \cos(\alpha x^2 \sin \beta) dx$

Q: Suppose $\alpha>0$ and $|\beta|<\pi/2$, show that \begin{align*} \textbf{(1)} \; \int_0^{\infty}e^{-\alpha x^2 \cos \beta} \cos(\alpha x^2 \sin \beta) dx &= \frac 1 2 ...
3
votes
1answer
81 views

Computing $\int_{0}^\infty\frac{t^a}{1+t^2}dt$ for $-1<a<0$

I am integrating the following integral $$\int_{0}^\infty\frac{t^a}{1+t^2}dt$$ for $-1<a<0$. by computing residues inside some contour. But I'm not sure what contour to use here, since ...
0
votes
1answer
16 views

Evaluating this contour integral

Let $R$ be the rectangle with vertices at $-1$, $1$, $1+2i$, $-1+2i$. Compute $$\int_{\partial R} \frac{(z^2 +i)\sin(z)}{z^2+1}dz$$where the boundary of $R$ is traversed counterclockwise. Here is ...
0
votes
0answers
44 views

Applying contour integration to $\int_{0}^{\pi}dx\frac{cos(x)}{\sqrt{x^2 + x_0^2}}$

Is it possible to apply contour integration to find the value of following integral $$\int_{0}^{\pi}dx\frac{cos(x)}{\sqrt{x^2 + x_0^2}}$$
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0answers
23 views

Compute the integrals $\int_{C_1}f(z)dz$ and $\int_{C_2}f(z)\,dz$.

Given two smooth contours, $C_1$ and $C_2$, that respectively lie on the upper and lower half plane compute $\int_{C_1}f(z)dz$ and $\int_{C_2}f(z)\,dz$. Let $a$ be a fixed real positive number. ...
0
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2answers
36 views

Compute the following integral, where $C$ is the circle $|z|=3$

Evaluate:$$\int_{C} (1 + z + z^2)(e^\frac{1}{z}+e^\frac{1}{z-1}+e^\frac{1}{z-2}) dz $$ where $ C$ is a circle $|z|=3$ and $z \ \epsilon \ \mathbb{C}$ The function that is being integrated has ...
1
vote
2answers
4k views

Calculate the Fourier transform of ${\rm b}\left(x\right) = 1/\left(x^{2} +a^{2}\right)$

I need help to calculate the Fourier transform of this funcion $${\rm b}\left(x\right)=\frac{1}{x^{2} + a^{2}}\,,\qquad a > 0$$ Thanks.
1
vote
2answers
80 views

$\int\limits_{\gamma} \frac{z}{(z-1)(z-2)}dz$, $\gamma(\theta) = re^{i\theta}$, $2 < r < \infty$

For $0 < r < 2$, we can use Cauchy's integral formula and choose our holomorphic function to be $f(z) = \frac{z}{z - 2}$ since $z = 1$ is the only pole, but if $r > 2$, then both poles $z = ...
1
vote
3answers
79 views

Evaluating an integral using Gamma function [closed]

For $r \in (0,2)$, I would like to evaluate the integral $$\frac{2}{r} \int_0^{\infty} \frac{\sin(u)}{u^r} du.$$ The answer should be $$\frac{\pi \cdot \mathrm{cosec}{\frac{r\pi}{2}} ...
1
vote
1answer
33 views

Show that there exists an entire function $h$ such that $\lim_{n\to\infty}{h(nz)}=0$ for all $z\ne0$

Show that there exists an entire function $h$ such that $\lim_{n\to\infty}{h(nz)}=0$ for all $z\ne0$. The following construction is in Walter Rudin's Real and Complex Analysis Chapter 16, Exercise 11. ...
3
votes
1answer
41 views

$\frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}\frac{1}{s^2}e^{s(t - \frac{1}{2}x^2)}ds$ - different answers depending on value of $t$?

After taking an inverse Laplace transform I have the following - $$y = \frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}\frac{1}{s^2}e^{s(t - \frac{1}{2}x^2)}ds$$ In my notes it says if $t ...
0
votes
1answer
52 views

How can I solve this integral with complex number?

$n$ here is a complex number such that $n=n_r+in_i$ How can I solve this integral? $$\int_{0}^{\infty}\frac{x^4}{|x^2-n^2|^2} d x=? $$
3
votes
2answers
34 views

Computing $\int_{\gamma}e^zdz$, where $\gamma$ is a particular semicircle

How can I compute $\int_{\gamma}e^zdz$, if $\gamma$ is the semicircular arc depicted below? So, $\gamma=3e^{i\theta(t)}$, with $0\le\theta(t)\le\pi$, and then ...
8
votes
1answer
75 views

The inverse Laplace transform of $ s^{3/2}-a-bs \over s^{3/2}+a+bs$

How can I solve the inverse Laplace transform as below: $$\mathscr{L}^{-1}\left( s^{3/2}-a-bs \over s^{3/2}+a+bs \right) $$ where a and b are constants. Hint: we can consider $${ s^{3/2}-a-bs ...
2
votes
3answers
28 views

Evaluation of real trigonometric integrals using the Cauchy Residue Theorem

$I = \int^{2\pi}_0 \dfrac{d\theta}{2 - \cos \theta}$ This is straight from a book I'm reading, which suggests to convert $\cos\theta$ into $0.5(z+1/z)$ and then solve the integral on the unit circle. ...
6
votes
2answers
238 views

An intuitive definition of contour integration.

Recently I have been trying to learn the method of contour integration, but the Wikipedia article and others don't really help. Is there some resource which provides a definition which can be followed ...
2
votes
0answers
49 views

Computing an integral using residues

I am trying to find an integral: $$\int_{-\infty}^{+\infty}\frac{e^{-\sqrt{(x^2 + 1)}}}{(x^2 + 1)^2}\,\mathrm dx$$ I went about applying contour integral over a semicircle with diameter along $ x = ...
7
votes
1answer
983 views

Dogbone contour integral/branch cuts/residue at infinity

I am trying to compute: $$\int_0^1 \frac{\sqrt{x-x^2}}{x+2} dx$$ by contour integration. I define $f(z) = \sqrt{z-z^2}$ with a branch cut on $[0,1]$ in such a way that $f(-1)=\sqrt{2}i$, then define ...
0
votes
0answers
19 views

Integrate function with 2 branch points

Every example I see in textbooks so far has not shown me cases like this, so please help with the following question. I wish to integrate a function $f(z)$ around the contour shown below. $f(z)$ has ...
4
votes
3answers
121 views

Compute $\int_{0}^{\infty}\frac{x \log(x)}{(1+x^2)^2}dx$

Given $$\int_{0}^{\infty}\frac{x \log(x)}{(1+x^2)^2}dx$$ I couldn't evaluate this integral. My only idea here was evaluating this as integration by parts. \begin{align} \int\frac{x ...
0
votes
1answer
27 views

How to use Cauchy's integral formula with more than one pole?

$\int\limits_{\gamma} \frac{z^2}{z(z-2)}$ $\gamma(\theta) = 3e^{i\theta}$, $0 \leq \theta \leq 2\pi$ Cauchy's integral formula is given by: $$\int\limits_{\gamma} \frac{f(z)}{(z-a)^{n+1}} = ...
6
votes
4answers
188 views

Integral by residue - “dog bone”

Let $I=\int_{-1}^{1}\frac{x^2 dx}{\sqrt[3]{(1-x)(1+x)^2}}$. I used complex function $f(z)=\frac{z^2}{\sqrt[3]{(z-1)(z+1)^2}}$, which we can define such that it is holomorphic on ...
1
vote
1answer
39 views

Why is $\int\limits_{\gamma} \frac{1}{z-1} \neq 2\pi i$, $\gamma = \{z : \lvert z \rvert = 1\}$?

$\int\limits_{\gamma} \frac{1}{z-1}$ $\gamma = \{z : \lvert z \rvert = 1\}$ I use Cauchy's integral formula, which says $$\int\limits_{\gamma} \frac{f(z)}{(z-a)^{n+1}} = \frac{2\pi i}{n!} ...
0
votes
3answers
27 views

$\int\limits_{\gamma} \frac{1}{z-1}$, $\gamma(\theta) = 2e^{i\theta}$, $0 \leq \theta \leq \frac{\pi}{2}$

$\gamma(\theta) = 2e^{i\theta}$ is a circle centered at $(0,0)$ with radius $2$, so $z = 1$ is inside this path and thus we have to use Cauchy's integral formula for $\int\limits_{\gamma} ...
0
votes
1answer
59 views

integral of $ \int_{\gamma}e^{1/z}dz$ [closed]

How do you find the integral of $$ \int_{\gamma}e^{1/z}dz$$ in the domain $ D= \{Re z >0\}$
0
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0answers
34 views

Contour integration with a branch cut. Parameterizing f(z) properly

I have a contour integral of a function of the form $(z^6-P)^\alpha z^\beta$ Here $\alpha\in R$, $\beta\in N$ and $P$ is some constant. I therefore have branch points at the sixth roots of $P$. The ...
0
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0answers
14 views

Contour Integration example check

I have this question and have solved that the residue is zero? hence the integral is zero by the residue theorem? could someone confirm this please?? Also would the answer to this integral be ...
0
votes
2answers
31 views

Contour integration example question

I'm currently trying to solve this however I get to the point where I have, $$\int_{0}^{2\pi} \frac{ie^{\exp(it)}}{\exp(it)+3}.dt$$ am I on the right tracks? if yes could you help with the ...
5
votes
1answer
102 views

Can $\oint_{|z|=2}z^3 \bar {z} e^\frac{1}{(z-1)} dz$ be solved?

How we can calculate the result of following Integral? $$\oint_{|z|=2}z^3 \bar {z} e^\frac{1}{z-1} \mathrm{d}z$$
2
votes
1answer
66 views

how to calculate $\int_{0}^{\infty} \frac{\cos(x)}{(1+x^2)^2} dx$

$$\int_{0}^{\infty} \frac{\cos(x)}{(1+x^2)^2} dx$$ The main problem here is to choose the smart contour integral, but i don't see how. I think i am supposed to do this: note our integral is: $$0.5 ...
0
votes
0answers
39 views

Fourier transform of a tough composite function (sinc, sqrt, polynomial…)

Is it possible to compute the Fourier transform of $\mathrm{sinc}(\sqrt{1+x^4})$ in closed form? It appears the problem to be suited for contour integration, and I started to tackle the mere ...
1
vote
1answer
53 views

The Poisson Integral is harmonic

We have proved that for $h(e^{\mathcal{i}\theta})$ continuous on the unit circle, the Poisson Integral of $h$ defined by ...
2
votes
2answers
92 views

Using Complex Analysis to Compute $\int_0 ^\infty \frac{dx}{x^{1/2}(x^2+1)}$

I am aware that there is a theorem which states that for $0<a<2$ we have $$\int_0^\infty\frac{x^{a-1}}{x^2+1}dx=\frac{\pi \cos\big(\frac{a\pi }{2}\big)}{\sin (a\pi) }$$ but I prefer to evaluate ...
2
votes
1answer
42 views

Simple Residue calculation

$$\int_{\gamma(0;2)}\frac{e^{i\pi z/2}}{z^2-1} \, dz$$ Using the residue calculus i got $$-2\pi$$But the answer is $$=i$$ I must be wrong at this, but shouldn't the answer have $\pi$ at least since ...
6
votes
0answers
124 views

Complex contour integral: How does the stationary point method used in this case?

I was reading a paper which has the following integral in order to do the inverse Laplace transformation: $$ I=\frac{1}{2\pi i}\int_{-i\infty+\gamma}^{i\infty+\gamma} ...
1
vote
2answers
88 views

$\int_{0}^\pi \frac{\sin(nx)}{\sin x} dx$

How do I integrate :$\int_{0}^\pi \frac{\sin(n\theta)}{\sin \theta} d\theta $ I did the following: $\int_{0}^\pi \frac{\sin(n \theta)}{\sin \theta}d\theta = \mbox{Im} \int_{0}^{\pi} \frac{e^{i n ...
0
votes
1answer
223 views

Complex integration over a general ellipse

I'm having trouble evaluating the complex integral over an ellipse : $\int_C{\dfrac{1}{z^{4} + 1}} dz$ where C is the ellipse given by $x^{2} - xy + y^{2} + x + y = 0$. How should I go about it?
1
vote
3answers
117 views

Complex definite integrals/residue

I am trying to evaluate this integral: $$ \int_0^\pi \frac{d\theta}{(a + b \cos \theta)^2} $$ for $0 < b < a$. I feel like the way I'm trying to solve [using $\cos t=(e^{it}+e^{-it})/2$ and ...
0
votes
0answers
23 views

Calculate the integral $f(z)=\frac{e^{iz}}{z(z-\pi/2)^2}$ over $|z+1|=2$

Calculate the integral $f(z)=\frac{e^{iz}}{z(z-\pi/2)^2}$ over $|z+1|=2$. Since the singularity at $z=0$ is in the given contour, I integrated using Cauchy's theorem to get $$2\pi i \left[ ...
-1
votes
0answers
38 views

Calculate the integral of $f(z)= 2Re(z) + 3Im(z)$ over the contour $|z|=4$

I know to parametrize $z$ but really need step by step help with contour integration.
1
vote
1answer
106 views

Contour Integral and Complex Identity

While studying I came across this problem: (a) For $z=x+iy$, show that $$|\cos \pi z|^2=\frac{1}{2}(\cos(2\pi x)+\cosh(2\pi y))$$ (b) For a positive integer let $\gamma N$ be the square connecting ...
2
votes
1answer
139 views

Use the residue theorem to evaluate

$$ \int _ {|z|=2} \frac { dz} {(z-4)(z^3-1)} $$ What I've done now is the following. $f$ has isolated singularities at $z=4$, $1$, $\exp(\pi i/3)$, $\exp(-\pi i / 3)$ $$ \int _ {|z|=2} \frac { dz} ...
1
vote
1answer
722 views

Contour Integral on Square in complex plane.

If we define $f(z)=\pi \exp(\pi\bar{z})$ and define the contour $\gamma$ to be the boundary of the square with vertices $0$, $1$, $i$ and $1+i$ traversed in the positive direction. I'm trying to ...
2
votes
2answers
75 views

Contour Integral of $\log(z)/(1+z^a)$ where $a\gt1$

I am asked to prove that: $$ \int_{0}^{+\infty}\frac{\log z}{1+z^{\alpha}}\,dz = -\frac{\pi^2}{\alpha^2}\cdot\frac{\cos\frac{\pi}{\alpha}}{\sin^2\frac{\pi}{\alpha}},$$ provided that $\alpha > 1$, ...
1
vote
2answers
146 views

$\int_0^\infty \frac{x^2}{(x^2-4)(x^2-9)}\,\text dx$

I am trying to compute the following contour integration but am quite stuck I have to evaluate it analytically, by extending it to the complex plane and solving an appropriate integral involving a ...
4
votes
1answer
86 views

A bessel function integral

$$\int_{-\infty}^{\infty} dy \frac{J_1 \left ( \pi\sqrt{x^2+y^2} \right )}{\sqrt{x^2+y^2}} = \frac{2 \sin{\pi x}}{\pi x} $$ How do I show this?
1
vote
1answer
56 views

Change the order of integrals:$\int_0^1dx\int_0^{1-x}dy\int_0^{x+y}f(x,y,z)dz$

$$\int_0^1dx\int_0^{1-x}dy\int_0^{x+y}f(x,y,z)dz$$ From this it is obvious that $x\in[0,1],y\in[0,1-x],z\in[0,x+y]$. For it asks for the order to be in $$\int dz\int dx\int f(x,y,z)dy$$ . My method ...
1
vote
1answer
62 views

Show the length of a contour, given by traversing once round a circle radius r, is 2πr

I have tried this problem using the definition for length of a contour $$ L(\gamma) = \int |\gamma'(t)| dt $$ Along the contour $\gamma =Z +re^{it}$ But I cannot get it to work out at $2\pi r$.