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

learn more… | top users | synonyms

2
votes
4answers
78 views

Show that $\operatorname{Res}\left(\frac{1}{f}, z_0\right)=\frac{1}{f'(z_0)}$

I'm learning complex analysis, specifically (Laurent) series and residues, and need help to understand the solution to the following problem: Let $f$ be holomorphic and non-constant on the open ...
1
vote
1answer
46 views

Calculating $\int_0^{\infty } \frac{\ln (x)}{\sqrt{x} \left(a^2+x^2\right)^2} \, \mathrm{d}x$ using contour integration

I can do this integral using the keyhole contour the answer is:$$\int_0^{\infty } \frac{\ln (x)}{\sqrt{x} \left(a^2+x^2\right)^2} \, \mathrm{d}x = -\frac{\pi (-6 \ln (a)+3 \pi +4)}{8 \sqrt{2}a^{7/2}}$...
1
vote
2answers
53 views

Analytic continuation of Euler's reflection formula with the Gamma function

Let $\widetilde\Gamma$ be an analytic continuation of $\Gamma$ on $\mathbb C\setminus(-\mathbb N_0)$. Show that the function $$\widetilde\Gamma(z)\widetilde\Gamma(1-z)-\frac{\pi}{\sin(\pi z)}$$ ...
-1
votes
1answer
47 views

Residue of a trig function multiplied by a polynomial

can somebody help me to find the residue for: I tried to make two series centered at $(z - k\pi)$ for $\sin(z)$ and $1- \cos(2z)$ but I don't know what to do with the $(z+\pi)^2$....and obviously, i ...
0
votes
3answers
32 views

Simple Contour Integral

I have forgotten much of the complex analysis I once knew. How do I go about using the Cauchy Integral Formula / Residue Theorem to solve this contour integral? The region is the unit circle. $$\...
0
votes
1answer
36 views

Calculate the residues of this complex function

Calculate the residues of this complex function $$\frac{1}{z^2\sin(z)}$$ I can notice that we have singularities at $z=n\pi$, where $n=0,1,2,3,\dots$ But, how to find the residues?
0
votes
0answers
31 views

$\lim_{\rho\to0}\int_{\gamma_{\rho}}g(z)e^{iz}dz=-\pi i Res(f,a)$ with a pole $a\in\mathbb{R}$

Let $U$ be an open neighbourhood of $\overline{\mathbb{H}}=\{z\in\mathbb{C}:\Im(z)\ge0\}$ and $g:U\rightarrow\mathbb{C}$ meromorphic with a finite number of poles in $\mathbb{H}=\{z\in\mathbb{C}:\Im(z)...
0
votes
1answer
31 views

How do i prove that $Res(f(z)e^\frac{1}{z};0)=\sum_{n=0}^\infty \frac{a_n}{(n+1)!}$ with $f(z)=\sum_{n=0}^\infty a_nz^n$

$f(z)=\sum_{n=0}^\infty a_nz^n$ (around $0$) I need to prove that $Res(f(z)e^\frac{1}{z};0)=\sum_{n=0}^\infty \frac{a_n}{(n+1)!}$ I know that $Res(z^ne^\frac{1}{z},0)=\frac{1}{(n+1)!}$ but I don'...
1
vote
2answers
102 views

Closed form for $\int_0^1 d u \, \frac{1}{u + \lambda} \ln \left(\frac{1 + u}{1 - u} \right)$

The parameter $\lambda$ is complex and it's not on the real axis. There are some similar cases: Help me evaluate $\int_0^1 \frac{\log(x+1)}{1+x^2} dx$ Evaluate $\int_0^1 \frac{\ln(1+bx)}{1+x} dx $ ...
0
votes
1answer
39 views

How do i prove that $Res(F_n,0)=\frac{1}{(n+1)!}$ with $F_n(z)= z^ne^\frac1z$

I have $F_n(z)= z^ne^\frac1z$ and i've to prove that $Res(F_n,0)=\frac{1}{(n+1)!}$ And i know that $\sum_{n=0}^\infty \frac{1}{n!} \frac{1}{z^n} = e^\frac{1}{z}$ but i don't know how to procede ...
2
votes
3answers
74 views

Trying to solve improper integral

I've been trying to solve this $$ \int_{-\infty}^\infty {\sin(x)\over x+1-i }dx $$ using residue theorem. I've tried using a square contour pi, pi+pii, -pi+pii, pi and half a circle but with the ...
3
votes
1answer
48 views

Improper integral complex analysis $\int_{-\infty}^\infty \frac{e^{ax} \, dx}{\cosh(x)}$

I tried the following problem but I don't think I got the right answer. I checked it by substituting $a=\frac{1}{2}$ into the integral and putting that through Wolfram Alpha but it didn't match the ...
0
votes
3answers
191 views

Evaluate the integral $ \int_0^{+\infty} \frac{\sin(x^2)}{x^4+1} dx $ using the residue method

I have a problem in evaluating the integral above. So far I've proceeded in this way. We have an even function, so: $$ \int_0^{+\infty} \frac{\sin(x^2)}{x^4+1} dx = \frac{1}{2} \int_{-\infty}^{+\...
1
vote
1answer
41 views

Sum over all residues is zero

Let $f$ be a rational function mapping to $\mathbb{C}$, $$f(z)=\frac{P(z)}{Q(z)}$$ with $\deg P\leq \deg Q -2$. I want to show that the sum over all residues is zero. What am I asked to show? I think:...
1
vote
0answers
12 views

Relationship between the residues $Res(g\circ\varphi,z_0)$ and $Res(g,w_0)$

Let $\varphi:U\rightarrow\mathbb{C}$ be holomorphic with $\varphi'(z_0)\neq 0$ for some $z_0\in U$. Let $g$ be another function having a pole of order $1$ in $w_0=\varphi(z_0)$. What is the ...
0
votes
1answer
43 views

Find and classify isolated singularities of $f(z) = \frac{z}{1-e^{z^2}}$ and calculate residues on them

Obviously, all isolated singularities will be of the form $z=\sqrt{2 \pi i k}$ for $k \in \mathbb{Z}$ but I don't know how to classify. I tried expanding $\frac{1}{f} = \frac{1-e^{z^2}}{z}$ to $- \...
0
votes
0answers
25 views

Calculating the residue with series that has negative powers

I want to calculate the residue of $f(z)=(z-sin(z))^{-1}=\frac{1}{z^3}+ \frac{3}{10z}+..$ at 0. We get that expansion using binomial theorem using provided $z$ small the residue of $f$ at $a$ is $\...
0
votes
1answer
25 views

3d-Integral with residue method

How to deal with angular dependency $\theta_{kr}$ in the attempt to calculate this integral: $$\int \frac{\mathrm {d^3} k}{(2 \pi)^3} \frac{e^{i \vec{k}\cdot\vec{r}}}{a-k^2+ib}$$ by residue method?
0
votes
1answer
61 views

Determine this real integral with the Residue-theorem.

$$\int_{-\infty}^{\infty}{\frac{\sin x}{x^4-6x^2+10}\,\mathrm dx}$$ I get that when I evaluate the $\frac{\sin x}{x}$ one, I work with $\frac{e^{ix} - e^{-ix}}{2ix}$, I create a huge semicircle and a ...
2
votes
1answer
64 views

Show that $\int_0^\infty\frac{1}{1+x^n}\,\mathrm dx = \frac{\pi/n}{\sin(\pi/n)}$ for $\mathbb{N}\ni n\geq 2$

Show that $$\int_0^\infty\frac{1}{1+x^n}\,\mathrm dx = \frac{\pi/n}{\sin(\pi/n)}$$ for $\mathbb{N}\ni n\geq 2$. Let $S=\{r\mathrm e^{\mathrm i\varphi}\in\mathbb{C} \mid 0\leq r\leq R,0\leq \varphi\...
1
vote
3answers
40 views

complex analysis

Find the residue of the function $f(z) = z/ (8-z^3)$ at $z = 2.$ I tried with the formula $\lim_{z\rightarrow a} \frac{1}{(m-1)!} \frac{ d^{m-1}}{ dz^{m-1}} (z-a)^m f(z)$. but it becomes very tedious. ...
1
vote
3answers
78 views

principal value of $\int_{-\infty}^{\infty}\frac{\sin^2(x)}{x^2}\mathrm{d}x$

I know the answer is $\pi$ there is a proof here. Now looking to my textbook (textbook image) the result should be $0$. Using the last equation on the right hand page we have: $$ i\pi(\sin^2(x))'|_{x=...
3
votes
4answers
223 views

How to find $L = \int_0^1 \frac{dx}{1+{x^8}}$

Let $L = \displaystyle \int_0^1 \frac{dx}{1+{x^8}}$ . Then $L < 1$ $L > 1$ $L < \frac{\pi}{4}$ $L > \frac{\pi}{4}$ I got some idea from this video link. But got stuck while evaluating ...
-2
votes
2answers
71 views

Solution Ahlfors 4.5.3 3(f) [closed]

Using the Redidue Theorem, find $$\int_0^\infty \frac{x \text{sin}(x)}{x^2+a^2} dx$$ I don't know how to proceed.
2
votes
4answers
75 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| + i\...
8
votes
5answers
248 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
22 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
58 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 $$\int_{C_R}\...
0
votes
1answer
29 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
56 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 I ...
1
vote
1answer
38 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
25 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
201 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
27 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 $$\dfrac{\pi^2+\sin\...
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
37 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
18 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
113 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 z+\...
1
vote
1answer
52 views

Problem in residue calculation in case of essential singularity

The function I'm dealing with is expressed as follows $$\psi\left(v\right)=\frac{1}{\left(v-\beta\right)\left(v+\beta\right)\left(v-\gamma_1\right)\left(v+\gamma_2\right)}\mbox{exp}\left[-\frac{A_1}{...
0
votes
0answers
26 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
50 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
52 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}} {e^z+1}...
5
votes
1answer
42 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 the ...
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} \theta}{2+\cos(\...
2
votes
0answers
47 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 $p > 0$ ...