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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13 views

Argument principle for meromorphic forms on Riemann surfaces

Let $X$ be a compact Riemann surface and $D \subset X$ a compact domain with boundary $\partial D$. Let $\omega$ be a meromorphic $1$-form in a neighborhood of $D$ which does not have neither zeros, ...
1
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1answer
60 views

Help on line integral $\int\limits_\gamma \frac{1}{(z + 1)(z + 2) \cdot \ldots \cdot (z + r)} dz$

I need help on the following line integral: $$\int\limits_\gamma f dz = \int\limits_\gamma \frac{1}{g} dz = \int\limits_\gamma \frac{1}{(z + 1)(z + 2) \cdot \ldots \cdot (z + r)} dz$$ For a fix $r \in ...
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2answers
42 views

Complex Frequency Shifting in Fourier Transform

When dealing with Fourier transforms, it is often useful to take advantage of the following property in order to simplify work: $$\mathcal{F}(e^{i\omega_0t}f(t))=G(\omega-\omega_0)$$ where $G(\omega)...
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1answer
38 views

Laurent expansion - Faster technique

I'm currently preparing for an exam in complex analysis. There is a type of exercise, where I need to compute Laurent expansions about different places. However, my ...
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0answers
49 views

Integral Calculus with non integer power using Cauchy integral's theorem or Residue theorem

I want to calculate the following integral: $$I(\beta)=\int_0^\infty \frac{\sin(x)}{ x^{\beta+1}}dx$$ for $ 0 \leqslant\beta<1$ I used comparison test for improper integrals to be sure that these ...
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0answers
36 views

Proving that an infinite series equal a finite series

Suppose we have a function $f(z)$, which has $m$ isolated singularities, which are non-integers (say, $z_1$, $z_2$,...,$z_m$). Define $H(z):=\frac{\pi f(z)}{\sin(\pi z)}$. Assume that there exists a ...
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0answers
18 views

Residue at all integers of complex function involving sines

Given $H(z)=\frac{\pi f(z)}{\sin(\pi z)}$, where $f(z)$ is some function which has isolated singularities only at non-integers, is it correct to calculate its residue at $z=k$ (for $k\in\mathbb{Z}$) ...
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0answers
24 views

calculating curvilinear integral by residue theorem

Calculate the following integral by transposing to a curve integral and then using the residue theorem: $\displaystyle \int_{0}^{2\pi}{\frac{e^{int}}{C-e^{it}}dt}, \qquad |C|\ne1, n\in \mathbb N$. ...
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2answers
490 views

$\int_{-\infty}^\infty \frac{\sin (t) \, dt}{t^4+1}$ must be zero and it isn't

I'm trying to evaluate the integral $$\int_{-\infty}^\infty \frac{\sin (t) \, dt}{t^4+1}$$ using residue and complex plane integration theory. Let $f(t):=\frac{\sin (t)}{t^4+1}$, $f(z):= \frac{\sin (...
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1answer
37 views

Evaluating real integral by complex contour method

Please let me know where my mistake could be. I've verified the integral $$\int_{-\infty}^\infty \frac{dt}{(t^2+1)(t^2+4)}$$ to be equal to $\frac{\pi}{6}$ with a computer math system. However, I'm ...
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0answers
27 views

Proof that if $z_0$ is a zero of order $k$ of $f$ then $\exists$ analytic $g$ saisfying this expression

Let $f:S\to \mathbb{C}$ be analytic, $z_0 \in S$ be of order $k$ of $f$. Prove that there exists an analytic function $g$ satisfying $$\frac{f'(z)}{f(z)}=\frac{k}{z-z_0}+g(z)$$ $\forall z$ in $\...
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1answer
56 views

residue theorem, integral of inverse quadratic not working

$$\int_{-\infty}^\infty\frac{dx}{3x^2+0.4x+10}$$ roots of $3x^2+0.4x+10$ $= z_1,z_2=-\frac{1}{15}\pm 1.825i$ Using $\mathrm{res} = \frac{1}{z_1-z_2}=-0.274i$ ans: $2\pi i\cdot -0.274i=1.72$ the ...
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0answers
76 views

Evaluate $\int _{ 0 }^{ \infty }{ \frac { { x }^{ n }-1 }{ \ln { x }}} dx$ using residue theorem.

$$\int _{ 0 }^{ \infty }{ \frac { { x }^{ n }-1 }{ \ln { x }}} dx$$ I couldn't solve this problem using the residue theorem. Can anyone help me get the answer? I know the steps like taking the ...
3
votes
1answer
63 views

A Contour Integral

I'm interested in computing the integral: $$ - \frac{1}{2 \pi} \int_{- \infty}^{\infty} dE \; \frac{e^{-iEt}}{E^2 - \omega^2 + i\epsilon}. $$ I have two small queries: How does one choose the ...
3
votes
3answers
93 views

Complex integral with Residues Theorem

I've been going crazy with this complex integral I have to estimate with the Residues Theorem. I'm obviously missing a sign or something else, but I fear I may be commiting a conceptual mistake. $\...
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1answer
34 views

Solving a contour integral with Feynman prescription

My question relates to this question, where the integral $\displaystyle\int_{-\infty}^{\infty}\dfrac{e^{iax}}{x^2-b^2}dx=-\dfrac{\pi}{b}\sin(ab)$, where $a,b\gt 0$ is solved. Now, in many physics ...
2
votes
4answers
84 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
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1answer
52 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}}$...
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2answers
56 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 $$\widetilde\Gamma(z)\widetilde\Gamma(1-z)-\frac{\pi}{\sin(\pi z)}$$ ...
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1answer
48 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 ...
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3answers
36 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. $$\...
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1answer
38 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?
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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)...
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votes
1answer
33 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'...
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2answers
103 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 $ ...
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1answer
40 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
75 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
51 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 ...
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votes
3answers
195 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}^{+\...
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1answer
46 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:...
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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
44 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 $- \...
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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 $\...
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1answer
27 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?
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1answer
63 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
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1answer
71 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\...
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3answers
41 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. ...
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3answers
83 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
225 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 ...
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2answers
73 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
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4answers
76 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
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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
250 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
59 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
59 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
27 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 ...