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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2
votes
4answers
62 views

Evaluation of the principal value of $\int\limits_{-\infty}^\infty \frac{\sin 2x}{x^3} \, dx$

I'm trying to evaluate an integral $\int\limits_{-\infty}^\infty \frac{\sin 2x}{x^3}\,dx$ using Cauchy's theorem. Considering an integral from $-R$ to $-\epsilon$, then a semicircular indentation ...
2
votes
0answers
35 views

Compute $\int_{0}^{\infty} \frac{x^{1/3}}{(x^2 + 1)^2} dx$

So I want to compute $\int_{0}^{\infty} \frac{x^{1/3}}{(x^2 + 1)^2} dx$ using complex analysis, Cauchys theorem and the residue theorem. What I did was the following: define $g(z) = e^{1/3(\ln|z| + ...
8
votes
4answers
174 views

How does one integrate $x^2 \frac{e^x}{(e^x+1)^2}$?

How can I show this? $$ \int_{-\infty}^{\infty} x^2 \frac{e^x}{(e^x+1)^2} dx = \pi^2/3$$ I tried applying residuals, but the pole is of infinite(?) order.
0
votes
1answer
21 views

Residue of a non-identically zero function

Assume f(z)is analytic in the complex plane and let f be a complex function which is not identically zero.Then,show that Res(1/f(z^3),0)=0. I know that the residue is calculating for only ...
1
vote
4answers
56 views

Computing the residue of a rational function

The real integral I am trying to compute with residues/contour integration is $\int_{-\infty}^{\infty}\frac{x^2}{(x^2+a^2)^3} \,dx$ For $a$ positive and by using the complex integral ...
0
votes
1answer
28 views

Computation of a Residue (Complex Analysis)

I have attempted to compute this residue: $g(z) = \frac{e^z − 1}{z^3} $ at $z = 0$ and I found that: $\frac{e^z -1}{z^3} =\frac{1}{z^2} + \frac{1}{2! z} + \frac{1}{3!} + \frac{z}{4!} + \ldots $ ...
0
votes
0answers
23 views

Finding a function given its residues

I was given this question : The only singularities in C (the complex set) of the analytic function $f$, are simple poles at z=1 and z=2, with residues at these poles equal to -3 and 7 respectively. If ...
3
votes
0answers
48 views

Use Residue Theorem to Sum Series

Show that $$\sum_{n=- \infty}^{\infty} \frac{1}{(3n-1)^2} = \frac{4 \pi^2}{27}$$ I'm pretty sure I need to use the Residue Theorem to sum the series, but I'm unsure how to apply it. Here's what ...
-1
votes
0answers
18 views

Residues of poles of $\sin(\pi z)$ [duplicate]

I asked the following question earlier, but nobody replied: Finding the residues of the following poles Could you please take a look?
1
vote
1answer
36 views

Finding the residues of the following poles

We have the following function $$\phi(z)=\frac{z^2-4z+4}{\sin(\pi z)}.$$ I know the poles are $z=n$ for $n\in \mathbb{Z}$, but how do I find the residues of them all at once?
0
votes
0answers
22 views

Integral problem with branch point from Physics

The question come from a Summation like this $${ \sum _{ { z=i\omega }_{ n } } { \frac { -\alpha E\pi }{ 4{ z }^{ 3 }\sqrt { -\alpha -z } } } }$$ I can use Cauchy theorem to transform it to a ...
8
votes
3answers
194 views

A Ramanujan-type identity: $11\sum_{n=1}^{\infty}\frac{n^3}{e^{2n\pi}-1}-16\sum_{n=1}^{\infty}\frac{n^3}{e^{4n\pi}-1}=\frac{1}{48}$

Out of curiosity, why it is these sums yield a rational answer? $$11\sum_{n=1}^{\infty}\frac{n^3}{e^{2n\pi}-1}-16\sum_{n=1}^{\infty}\frac{n^3}{e^{4n\pi}-1}=\frac{1}{48}$$ I found this identity ...
0
votes
2answers
23 views

Finding the residue of a function at a point

Find the residue at $\pi$ for the function defined by $$\dfrac{z^2+\sin\left(z\right)}{\left(z-\pi\right)^4}$$ I thought I could do this using the 'gh rule' however this gives ...
1
vote
0answers
35 views

Need help evaluating the integral $\int_{-\pi}^{\pi} \dfrac{x \sin x\ dx}{1- 2a \cos x + a^2}$ where $0 < a < 1$. [duplicate]

Need help evaluating the complex integral $\int_{-\pi}^{\pi} \dfrac{x \sin x\ dx}{1- 2a \cos x + a^2}$ where $0 < a < 1$.
0
votes
1answer
30 views

Residue of a function sin(1/z)/(1-z)

How do I find the residue of a function $\sin(1/z)/(1-z)$ at $z=0$ ? $z=1$ is a simple pole and residue of this function is $-sin1$. When at I write Laurent series of $\sin(1/z),$ $z=0$ appears to ...
1
vote
1answer
15 views

Find the isolated singularities of the function and calculate their residues

Find the isolated singularities of $\dfrac{3z}{z^3+1}$ and calculate their residues using the 'g/h rule' I understand that $z=-1$ is a singularity but I'm not sure about the others or how to apply ...
0
votes
0answers
105 views

Finding the singularities and residues of a Gamma/Riemann Zeta function.

The function I have is $f(z)=\zeta(z)\Gamma(z − 1)\sin(\pi z)$ and I need to find all singularities and their residues so I can evaluate a clockwise contour integral for the contour $\left\lvert ...
1
vote
1answer
51 views

Problem in residue calculation in case of essential singularity

The function I'm dealing with is expressed as follows ...
0
votes
0answers
25 views

Improper integral using residue calculus and contours [duplicate]

Can someone please show me how to solve this improper integral "using residue calculus and appropriate contours"? $$\int_0^{\infty} \frac{1}{x^4+1}dx$$
0
votes
1answer
19 views

Complex contour integral Problem

Show that $$\oint_{|z|=1} \dfrac {(z+w)(z^{n-1})} {z-w}dz=0$$ using Residue calculus, where $n<0$ and $|w|<1$.
2
votes
1answer
48 views

Evaluate $I(x)=\int_{-\infty}^{\infty}\frac{e^{xv}}{1+e^v}dv$

This is the last part of a multistage evaluation of $I(x)=\Gamma(x)\Gamma(1-x)$. Through various substitutions we get the integral $$I(x)=\int_{-\infty}^{\infty}\frac{e^{xv}}{1+e^v}dv$$ We're also ...
1
vote
1answer
24 views

Locate poles and find residues

Locate the poles and the find the residues of the following function: $\dfrac {e^{5z}}{\left( z+i\pi \right) ^{3}}$ From my understand there should be a pole at $-i\pi$ so y the residue theorem we ...
2
votes
2answers
51 views

Residue of $\frac {e^{sz}} {e^z+1}$

I'm usually okay with calculating residues but in this case the denominator isn't a polynomial and I'm not sure what to do. Can someone tell me how to calculate the residue of $\dfrac {e^{sz}} ...
5
votes
1answer
40 views

Evaluate the improper integral with residues.

$$\int_0^{\infty} \frac{x^2+1}{x^4+1}dx$$ What i've found are the singularities at: $e^{\pi/4+\pi/2n}$ for $n=0,1,2,3$. But i'm unsure how to calculate the integral since I don't want to include ...
1
vote
1answer
23 views

Prove that if a function has an isolated singularity at $z_0$, then its derivative also has an isolated singularity at $z_0$

Prove that if a function has an isolated singularity at $z_0$, then its derivative also has an isolated singularity at $z_0$. Find $Res(f',z_0)$. My approach: Suppose $f$ has an isolated singularity ...
0
votes
1answer
15 views

Residue of $\dfrac{1}{z^2+4z+1}$. Laurent series.

I want to calculate : $$ \int_0^{2\pi} \dfrac{\mathrm{d} \theta}{2+\cos(\theta)} $$ I use $z=\mathrm{e}^{\mathrm{i} \theta}$ and residue theorem : $$\int_0^{2\pi} \dfrac{\mathrm{d} ...
2
votes
0answers
46 views

Evaluating Improper Integrals with Residues - don't think I'm calculating the residues properly

I have to evaluate the integrals $\displaystyle \int_{-\infty}^{\infty}\frac{dx}{x^{2}+p^{2}}$, for $p > 0$, and $\displaystyle \int_{-\infty}^{\infty} \frac{dx}{(x^{2}+p^{2})^{2}}$, for $p > 0$ ...
2
votes
1answer
54 views

Finding residues at a point $a$ where $a$ is a pole.

I am faced with the following problem: (a) Find $\displaystyle res_{a} \frac{\varphi(z)}{(z-a)^{n}}$ where $\varphi$ is a given function analytic at $a$, $\varphi(a) \neq 0$, and $n$ is a positive ...
0
votes
1answer
35 views

Residue of $1/(\sin(1/z))$ defined at $z=0$? Trying to derive Laurent Series of $\csc (1/z)$ to find it.

This question is related to this one. I was able to figure out on my own that the residue of $\displaystyle \sin \left(\frac{1}{z} \right)$ is defined at $z=0$ by finding the Laurent Series of ...
2
votes
1answer
27 views

Evaluate $\int_{C[0,3]} \frac{\exp(z)}{(z+2)^2\sin(z)} dz$ Using Residue Theorem

$\displaystyle \int_{C[0,3]} \frac{\exp(z)}{(z+2)^2\sin(z)} \, dz$ using Residue Theorem. I have found singularities within $C[0,3]$, which are $-2$ and $0$. For $z=-2$, it is a pole with degree $2$. ...
0
votes
2answers
31 views

Residue of $\cot z $ at $z= 0$

I need to calculate the Residue of $\cot z$ at the pole $0$ . Now, I think that the order of the pole is $1$. But I do not know how to show it. So, assuming the order of pole to be $1$, I calculated ...
0
votes
1answer
68 views

Integrating $\int_{-\infty}^{\infty}\frac{e^{ax}}{1+e^x}dx$ for $0<a<1$.

I'm supposed to find the value of $\int_{-\infty}^{\infty}\frac{e^{ax}}{1+e^x}dx$ for $0<a<1$. I wanted to integrate over the upper semicircle of radius $R$, and take the limit as $R\to\infty$. ...
0
votes
0answers
36 views

Using the $\cot (\pi z)$ to find $\sum \frac{1}{n^2}$ [duplicate]

I'm trying to prove the result that $$\sum_1^\infty \frac{1}{n^2}=\pi^2/6$$ using cotangents and residue theory. I know that $\sum f(n)=-$Sum of residues of $\pi \cot (\pi z)f(z)$ at the poles of ...
6
votes
3answers
105 views

Calculate $\int_0^{\infty}\frac{1}{(x+1)(x-2)}dx$ using residues

I'm supposed to calculate $$\int_0^{\infty}\frac{1}{(x+1)(x-2)}dx$$ using residues. The typical procedure on a problem like this would be to integrate a contour going around an upper-half ...
-2
votes
1answer
36 views

Integrate $\int_0^{2 \pi } \frac{1}{(a+b \cos^2 (x))^2} \, \mathrm{d}x$ [closed]

I'm having a trouble with this integral expression: $$\int_0^{2 \pi } \frac{1}{(a+b \cos^2 (x))^2} \, \mathrm{d}x$$ I want to solve to using residue but it seems hard.
2
votes
0answers
66 views

Sum of Gamma Function Residues

I was exploring Cauchy's residue theorem with the gamma function and came across an interesting identity. Consider $$\int_{C_R} \Gamma(z) \, dz $$ Over the complex plane where $C_R$ is the curve ...
2
votes
0answers
39 views

$\frac{\omega}{2\pi}\int_0^{\frac{2\pi}{\omega}}\frac{\sin^2\theta \cos^2\omega t}{(1+\beta\cos\theta \sin\omega t)^5} dt$

I'm going to write out the whole problem as it is given to me (bad grammar and all) even though some of the info may be irrelevant to finding a solution. A charge $e$ moving along a straight line ...
0
votes
1answer
20 views

Using Residue Theorem/Laurent Series to evaluate $\int_{c}$ $\frac{z^m}{2-\frac{1}{Z}} dz$

Let $m$ be an integer and $C$ be the circle $C(0; 1)$ traversed in the counterclockwise direction. What is the value of $\int_{c}$ $\frac{z^m}{2-\frac{1}{Z}} dz$ a) when $m\ge-1$? b) when ...
-1
votes
1answer
24 views

Pole on the Real Axis of Complex Integral

Save me! SOS!. Please see image. I'm completely baffled about how to go about solving this. Please can you explain the how you get -pi*i from the 4th line? Can you please explain by this paragraph as ...
0
votes
2answers
68 views

Complex Numbers - Equation manipulation

Please see link, I'm wondering how you end with RHS of equation on second line of image. Please can explain the step and algebra manipulation used Thanks
0
votes
0answers
39 views

Residue of $f(z)=e^{-1/z^2}$ using it's pole order

Find the residue of $f(z)=e^{-1/z^2}$ at $z=0$ Can somebody check both methods? I wrote out the series of $$e^z=1+z+\frac{z^2}{2}!+...$$ and subbed in $-1/z^2$ instead of $z$ to get: ...
0
votes
1answer
30 views

How to find and classify the singularities of $\frac{e^{-z} \sin(2(z-1)^2)}{(z^2-4)(z-1)^2}$?

How to find and classify the singularities of $$\frac{e^{-z} \sin(2(z-1)^2)}{(z^2-4)(z-1)^2}$$ Here is what I have: I think the singularities are all isolated and are located at $z=1,\pm2$. I have ...
3
votes
2answers
49 views

Contour Integration of $\sin^2(x)/(1+x^2)$

How should I calculate this integral $$\int\limits_{-\infty}^\infty\frac{\sin^2x}{(1+x^2)}\,dx\quad?$$ I have tried forming an indented semicircle in the upper half complex plane using the residue ...
3
votes
1answer
44 views

Using residue theorem to integrate from $-\infty$ to $\infty$

I'm trying to integrate $$\int_{-\infty}^{\infty} {x^2 \over {(x^2 + 1)}^2(x^2 + 2x + 2)} $$ given that the function $$f(z) = {z^2 \over {(z^2 + 1)}^2(z^2+2z+2)} $$ has residues $${9i - 12 \over ...
0
votes
0answers
13 views

Cauchy integral with matrix and imposed contour

Let's say I have a matrix $H$ (that is non hermitian), and a real number $E$. The matrix $H$ is a function of $k$, and $H(k)$ varies smoothly. I want to compute the following integral: ...
1
vote
1answer
70 views

Solve $\int_{0}^{\infty} \frac{x^{1/3}}{1+x^2}dx$ using residue calculus [duplicate]

I need to find the value of the following integral using residue calculus - $$I = \int_{0}^{\infty} \frac{x^{1/3}}{1+x^2}dx$$ I have used substitution to start with. Let $x = t^3$. Then $I $ changes ...
0
votes
1answer
23 views

How to determine the residues of $\frac{z}{\sinh(kz)}$?

What methods can I use to determine the residues of $\frac{z}{\sinh(kz)}$? Singularities occur at $z=\frac{i n \pi}{k}$ for $k \neq 0$ and $n$ an element of the integers. I've attempted a series ...
3
votes
2answers
41 views

Answer not matching for calculating the value of an integral

Consider the following integral $$\int_ {-\infty}^{\infty} \frac{x^2 - x + 2 }{x^4 + 10x^2 + 9} dx \!$$ I need to find its value using residue calculus. So, I considered the complex function $$\int_ ...
1
vote
1answer
46 views

Calculating residue of $f(z)$ around $a=\infty$

Let $f(z) := \frac{3\sin 4z}{z^4}$. Find $$\mbox{res}\left [f(z),\infty\right ]$$ For $a=\infty$, the residue is given as $-\frac{1}{2i\pi}\int_{|z|=R} f(z)\mbox{d}z$. $$\int_{|z|=R} \frac{3\sin ...
3
votes
1answer
22 views

How to choose the point at which we want to calculate residue?

Suppose I have to calculate the following integral using residue calculus - $$\int_ {-\infty}^{\infty} \! \frac{e^{-ix}}{x^2 + 1 } \, \mathrm{d}x. $$ Now my approach is to construct a contour in the ...