Questions on the evaluation of residues, on the evaluation of integrals using the method of residues or in the method's theory.

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0
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1answer
30 views

Calculating the residue of a function

Let $f(z) = \frac{1+z}{1-\cos(z)}$ I wish to calculate the residue of $f$ at $0$, $2\pi$ and $-2\pi$. I believe this can be done by the following since $f$ has simple poles at these points $Res(f, ...
2
votes
3answers
76 views

Compute the integral $\int_{0}^{\infty} \frac{(1 + x + x^2)}{(1+x^4)} dx $ with a residue on suitable contour.

I believe that I could try to compute the same integral with limits from $-\infty$ to $\infty$ using residue on a half circle and then let the radius tend off to infinity, and once I have that value I ...
1
vote
2answers
35 views

Simple Question About Residues/Poles/Zeros/Singularities

I'm having a little bit of trouble with residues. If we have the $f(z)=\left(\frac{\cos(z)-1}{z}\right)^2$ at $z_0=0$, we have a zero of order 2 in the numerator and a zero of order 2 in the ...
0
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2answers
35 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 ...
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. ...
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 = ...
1
vote
3answers
89 views

Finding $\int_0^\infty\frac{\sin^{2}x}{1+x^4}dx$

I am trying to evaluate $$\int_0^\infty\dfrac{\sin^{2}x}{1+x^4}dx$$ and I am stuck on how to start. I am thinking the first step would be to substitute $$\dfrac{(1-e^{2ix})+(1-e^{-2ix})}{4}$$ for ...
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
29 views

Finding $a_{-n}$ where $\cot (\pi z)=\sum_{n=-\infty } ^\infty a_nz^n$

The following is problem 5.11.2 of Berkeley Problems in Mathematics. Let $\cot (\pi z)=\sum_{-\infty} ^\infty a_nz^n$ be the Laurent expansion for $\cot (\pi z)$ on the annulus $1<\vert z \vert ...
0
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1answer
40 views

Integral of a two-valued function with two branch cuts

Is it possible to calculate in closed-form the integral $\int_{-\infty}^{+\infty}\mathrm{sinc}(\sqrt{1+x^4})\,dx$ (sinc being the cardinal sine, $\sin(x)/x$)? The function is everywhere defined (all ...
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 ...
0
votes
1answer
25 views

How to compute the following real integrals using the residue theorem?

How to compute the following real integrals using the residue theorem: $$\int_{-\infty}^{\infty} \frac{1}{(x^2+p^2)(x^2+q^2)} dx$$ $$\int_{0}^{2\pi} \frac{sin^2(\theta)}{5+4cos(\theta)} d\theta$$ ...
2
votes
1answer
53 views

How to choose a contour in order to use the residue theorem to sum up a series from Ryzhik?

I would like to know how to sum up to following series (from the Gradshteyn-Ryzhik tables): ...
3
votes
1answer
58 views

Therem of Residue application

I want to determinate the following integral: $$\int_{\gamma} \frac{e^z}{\cos{(z)}} dz$$ Where $\gamma (t)=\frac{\pi \cos t}{1 +\sin^2 t}(1+i\sin t)$, $0\leq t \leq 2\pi$ So I see that ...
1
vote
1answer
44 views

Find and show that the residues of the meromorphic differential $dx$ for Fermat's curve $C:$ $\xi_1^4+\xi_2^4=\xi_0^4$ is zero

Find the residues of the meromorphic differential $dx$ of Fermat's curve $C:$ $\xi_1^4+\xi_2^4=\xi_0^4$ at its poles. Check that their sum is zero, as it must be. Attempt: Let $\xi_2\not=0$. Then ...
2
votes
3answers
89 views

Why does the residue method not work straight out of the box here?

I'm trying to evaluate the integral $$I = \int_0^{\infty} \frac{\cos(x)-1}{x^2}\,\mathrm{d}x $$ The way I've done this is by rewriting $\frac{\cos(x)-1}{x^2}$ as ...
3
votes
2answers
50 views

How do I use residue theorem to evaluate this improper integral to get a good looking solution?

The problem is $\int_{0}^{\infty} \frac{\sqrt{x}}{x^2+2x+5}dx$ I replace x with z, and did some algebra, but the solution was rather nasty. it contains exponential and arctan such and such. However, ...
0
votes
2answers
26 views

Prove $f|_{U_0}$ is $m$-to-$1$ except at $z_0$.

Let $f$ be analytic on a domain $U$, $z_0\in U$, and $w_0=f(z_0)$. Suppose that $\mbox{ord}_{z_0}(f-w_0)=m\in\mathbb N$. Prove that there is an open set $U_0$ with $z_0\in U_0\subset U$ such that ...
0
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2answers
41 views

Complex Analysis - Calculating Residues

I am told to calculate the residue of $ \frac{ e^{-z} }{ (z-1)^{2} } $ at $ z = 1 $. The answer is supposed to be $ \frac{ 1 }{ e } $. Can someone give me a hint on how to approach this?
4
votes
2answers
118 views

$\int_0^\infty \frac{1}{1+x^ 9} \, dx$

$\int_0^\infty \frac{1}{1+x^9} \, dx$ I tried taking the integral of $\Gamma_R = [0,R] \cup \gamma_R \cup I_R$, where we see that \gamma_R is the circle parametrized by $z = Re^{it}$ with ...
3
votes
1answer
35 views

Residue at essential singularity

I need a little help with the following problem. I've tried many ways, but i didnt succeed. I think there needs to be a trick or something, some transformation. The task is to find the residue of the ...
0
votes
3answers
50 views

Residue Integration

I am attempting to calculate the integral of $\frac{(1+sin(\theta))}{(3+cos(\theta))}$ from $0$ to $2\pi$. I have already changed $sin$ and $cos$ into $\frac{1}{2i(z-z^{-1})}$ and ...
-2
votes
1answer
27 views

Finding integration using Cauchy Residue formula

Compute the residues at all the singularities of the function $tanh(z)$, and compute the integral $\int_C tanh(z)$ where C is the circle of radius 12 centred at $z_0 = 0$. attempt:singularities are ...
1
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1answer
45 views

Residue theorem application [demonstration]

I really don't know how to solve this problem! Consider $F$, an analytic fuction, so that, $$f(z)=F(\frac{1}{z-1})$$ has a pole. Demonstrate that F(z) is a polynomial So, I tried to ...
1
vote
1answer
27 views

Evaluating $\int_\Gamma \frac{2z^2-z+1}{(z-1)^2(z+1)}dz$ along the contour that is shaped by the figure-8 centered at $z=-1$ and $z=1$.

In my answer key, it says this equals $0$, but I get $4 \pi i$. Here's why: $$ \int_\Gamma \frac{2z^2-z+1}{(z-1)^2(z+1)}dz = \int_\Gamma\biggl[\frac{1}{(z-1)^2}+\frac{1}{z-1}+\frac{1}{z+1}\biggr]dz ...
-1
votes
1answer
64 views

Residue theorem application [duplicate]

How could we use the Residue theorem to calculate the following integral: $$\int_0^{2\pi} \frac{1}{1-2p\cos{x} + p^2} dx$$ where $p$ is a real constant, such that $p\in ]0,1[$ Thank you!
0
votes
0answers
35 views

Residue of complex exponential

I need to find the residue of $e^{\frac 1{1-z}}$ using Laurent series. How would I manipulate the function to make it easier? I need to find all singularities and the corresponding residues. I believe ...
0
votes
0answers
25 views

Singularities and Residues [Demonstration]

How could I solve the following problem: "Consider $F$, an analytic function, so that $$f(z)=F(\frac{1}{z-1})$$ has a pole. Demonstrate that F(z) is a polynomial." I know that an analytic function ...
0
votes
2answers
33 views

what is the residue of $\frac{1}{z(1-cos(z))}$ in $0$?

what is the residue of $\frac{1}{z(1-cos(z))}$ in $0$? I tried using the residual formula $Res_{z=a} f(z) = \lim_{z \rightarrow a}(z-a)f(z)$ But then i get infty? Is there something i'm missing ...
1
vote
1answer
31 views

Rational Function Residue Formula Proof [duplicate]

I came across a theorem used to calculate residues of rational functions that states that if f and g are analytic functions at $z_k$ and $g'(z_k)$ isn't 0, then the residue of $f(z)/g(z)$ at $z_k$ is ...
0
votes
0answers
18 views

Residue derivation for integral of sin x/x

I am trying to integrate $e^(iz)/z$ over a rectangle contour from $-R$ to $r$ along the real axis and a small semicircle from -r to r and then goes along the real axis to R and then completes the ...
0
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1answer
68 views

Calculating Residues and singularities

We haven't done many full examples with residues so just wondered how you would answer the following questions for example: Classify the type of singularity for f and determine the residue of f at ...
0
votes
1answer
25 views

How to evaluate $\frac{1}{2\pi i}\int_C \frac{f(z) dz}{(z-z_1)^{m_1}(z-z_2)^{m_2}}$ and the other related contour?

Respected All. I was studying residue theory where I came accross the following problem "If $f$ be analytic in the simply connected domain $D$ and $z_1, z_2$ are two distinct complex point lying in ...
0
votes
1answer
45 views

how to calculate $\frac{1}{2\pi i} \int_{\gamma} \frac{2z}{(z-1)^ 4(z-3)}$

How to calculate $\frac{1}{2\pi i} \int_{\gamma} \frac{2z}{(z-1)^ 4(z-3)}dz$ When $\gamma = C_+(0,4)$ and where $\gamma = C_-(0,2)$. I need to use the residuformula which states that is f is ...
0
votes
0answers
20 views

Find $\text{Res}\left(\dfrac{1}{z(1-e^{-z})},2\pi i\right)$.

How to find Residue of $\dfrac{1}{z(1-e^{-z})}$ at $z = 2\pi i$?
3
votes
1answer
60 views

Integral with logarithm - residue

Let $R(x)$ be rational function. It is any general method to calculate $\int_{0}^{\infty}R(x) \log(x)dx$ ? I can do it in special cases, but I am looking for a general method. What should be a minimal ...
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 ...
3
votes
2answers
44 views

Computing Residue for a General, Multiple-Poled function?

I'm trying to compute the residue of the following function at $a$. I'm having a little trouble seeing which poles are relevant: Compute $\,Res_f(a)$ for the following function: $$f(z) = ...
8
votes
1answer
117 views

Partial fraction expansion for non-rational functions

With regard to this answer to an inverse Laplace transform question, I derived the following result: $$\frac1{i 2 \pi} \int_{c-i \infty}^{c+i \infty} ds \, e^{s t} \Gamma(s)^2 = 2 K_0 \left ( 2 ...
5
votes
1answer
55 views

Integrate by the method of residue

I want to integrate $$\int_{0}^{\infty}\frac{1}{(1+x)^5}dx$$ by the method of residue, but I have done only problems of simple poles, but this is a problem of fifth order pole. So I am stuck in it. ...
4
votes
1answer
55 views

Evaluating $\frac{1}{2\pi j}\int_{c-j\infty}^{c+j\infty}x^{-s}\sigma ^{ms-m} [ \frac{\Gamma ( s )}{\Gamma ( s+2)}]^{m}ds$

I have been trying to solve the problem for $m=3$: $$f(x)=\frac{1}{2\pi j}\int_{c-j\infty}^{c+j\infty}x^{-s}\sigma ^{ms-m}\left [ \frac{\Gamma \left ( s \right )}{\Gamma \left ( s+2 \right )} \right ...
3
votes
3answers
101 views

Residue of $\Gamma^{2}$ and $\Gamma^{3}$

Based on wiki, the residues of $\Gamma$ at non positive integers are given by: $$\text{Res}\left ( \Gamma(z),z=-n \right )=\frac{(-1)^{n}}{n!}.$$ I have been trying to find residue for $\Gamma^{2}$ ...
3
votes
0answers
52 views

Finding residues of rational functions with extremely large powers

$h(z)=\frac{5z^{2015} + 7z^{2010} - 38z^5 + z^4 - 2z^3 + 5}{7z^{2016} + 2z^{2013} - 6z^8 + 3z^7 + z^5 - 4z^2 - z + 111}$ Find the sum of the residues of h at its poles in $C$ How do I find the ...
2
votes
1answer
47 views

Prove that for a holomorphic function $f$ on $\mathbb{C}$, if $-f(z) = f(\frac{1}{z})$, then the residue of $f$ at 0 is 0.

Prove that for a holomorphic function $f$ on $\mathbb{C}$, if $-f(z) = f(\frac{1}{z})$, then the residue of $f$ at 0 is 0. Note: $f$ isn't defined on 0. I'm having a bit of trouble getting this ...
0
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0answers
27 views

An application of Rouches Theorem

Let $f$ be an entire function on the complex plane, with Taylor's expansion around zero as $f(z) = \sum_{k=0}^{\infty}c_{k}z^{k}$. Let $N(r)$ be the number of zeroes of $f$ in $D(0, r)$. Show that for ...
2
votes
2answers
52 views

Calculate integrals concerning a branch

i am trying to calculate this integral: $$\int_{|z|=5}\frac{dz}{\sqrt{z^2+11}}$$ Using the branch that gives : $\sqrt{36} = -6$ The function has 2 poles at $|z| < 5$, lets call them $\alpha$ and ...
1
vote
1answer
45 views

Finding the limit of $a_n$ for $\tan{\frac{\pi z}{2}}$

i need to understand some issues with the prove of the following exercise: Let $f$ be defined as: $$f(z) = \tan{\frac{\pi z}{2}}$$ Its not hard to show that its Taylor series around $z=0$ is from ...
6
votes
2answers
265 views

Help with the integral $\int_{0}^{\infty}\log\left(1+\frac{s^{2}}{4\pi^{2}} \log(1+ix)\right ) e^{-2\pi nx}dx$

We have the integral : $$\int_{0}^{\infty}\log\left(1+\frac{s^{2}}{4\pi^{2}} \log(1+ix)\right ) e^{-2\pi nx}dx$$ Where $s$ is a complex parameter, and $n$ is a positive integer. The integral ...
2
votes
0answers
18 views

Residue of function (two functions multiplied)

I need to know if I'm doing this exercise correctly: $f(z)$ has only one singularity, which is at $z=0$, and it's a pole of order 7 $f(z)=-f(-z)$ $f(z)$ is analytic in $z=\infty$ $g(z)$ is analytic ...
2
votes
0answers
60 views

Example of an analytic continuation for a function in integral form

Given $f(z) = \int_{-\infty}^\infty \frac{exp(-t^2)}{z-t}\,dt$, where $Im(z)>0$. Find an analytic continuation to the region $Im(z)<0$. Firstly the solution said that there is a branch cut on ...