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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Problem with $\int_{0}^{\infty} \frac{\log^2(x)}{1+x^2}$ (by residues) [duplicate]

I, I am trying solve the following integral $$\int_{0}^{\infty} \frac{\log^2(x)}{1+x^2}$$ Teachers teached me that I can solve the integral $$\int_{0}^{\infty} ...
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1answer
36 views

Complex integration on upper-half plane

In order to prove the normalisation property of a Lorentzian function, $L = \dfrac{1}{\pi}\displaystyle \int_{-\infty}^\infty \dfrac{b}{(z-a)^2+b^2} dz = 1$ we take a closed contour on the ...
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2answers
29 views

how could calculate $ \int_{C} \frac{1}{\sin(z)} \, dz $ when $C=C(0,1)$

i am trying calculate $$ \int_{C} \frac{1}{\sin(z)} \, dz $$ when $C=C(0,1)$ by complex methods, its said, by residues, some one could help me?
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2answers
30 views

How could I calculate $\int_{C} ze^{\frac{1}{z-1}}$ when $C=C(1,\frac{1}{2})$

I have to solve if $C=C(1,\frac{1}{2})$ $$I=\int_{C} ze^{\frac{1}{z-1}}$$ I know that $I=2\pi i \operatorname{Res}(f(z), 1)$, but I do not know how could I calculate that residue. What I did: ...
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3answers
36 views

Compute the integrals using the residue theorem

Compute the following integrals: $I:=\int_{|z|=2}\frac{1}{(z-3)(z^{13}-1)}dz$ $J:=\int_{|z|=10}\frac{z^3}{z^4-1}dz$ I do not know where to begin. I know I am supposed to use the substitution ...
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3answers
41 views

Contour Integration: non-convergent integral

The question is $$I=\int_{-\infty}^{\infty} \frac{\sin^2{x}}{x^2} dx$$ My attempt: $$I=-\frac{1}{4}\int_{-\infty}^{\infty} \frac{e^{2ix}-2+e^{-2ix}}{x^2} dx$$ $$I=-\frac{1}{4} \Big[ ...
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1answer
31 views

Contour Integration with pole on contour

I have come across an example I don't understand.. So, here is the problematic part: Consider the integrals: $ I = \int_C \frac{e^{iz}}{z} dz $ $ J = \int_C \frac{e^{-iz}}{z} dz $ Where $C,C_-, ...
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0answers
14 views

A clarification on an answer on residues and Polya fields

In this very informative and interesting answer about the relation between residues and representation of complex functions as vector fields the author states that the function $$f(z) = \frac{1}{z}$$ ...
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0answers
17 views

Verification on classification of singularities

In an exercise, I'm asked to classify the singularities of these functions: $\qquad i) f(z)=\frac{1}{(z-1)^2} \qquad ii)f(z)=\frac{1-\cos z}{z^2} \qquad iii) f(z)=\frac{z^2-1}{z-1}$ I don't know why ...
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2answers
61 views

Evaluate $\int_{-\infty}^{\infty}\frac{x^2dx}{(x^2+1)^2}$

$$\int_{-\infty}^{\infty}\frac{x^2dx}{(x^2+1)^2}$$ My attempt: $z=i$ is a pole with order $2$ in the upper part on the plain. $$\text{Res} (f,i)=\lim\limits_{x\to i}\frac{x^2}{(x+i)^2}=\frac ...
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2answers
21 views

Complex roots in order to apply residue theorem

$$\int_{0}^{2\pi}\frac{d\theta}{(4 + 2\sin\theta)^2}$$ $$\sin\theta = \frac{z - z^{-1}}{2i}$$ $$d\theta = \frac{dz}{iz}$$ $$\oint_c\frac{dz}{iz\left(4 + \frac{z - z^{-1}}{i}\right) ^2}$$ ending up ...
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1answer
20 views

Applying the residue theorem on a real integral

$$\int\frac{d\theta}{a + b\cos\theta}$$ Given that $$\cos\theta = \frac{z + z^{-1}}{2}$$ $$d\theta = \frac{dz}{iz}$$ We have $$\oint_c \frac{dz}{iz\left(a + b\frac{z +z^{-1}}{2}\right)}$$ ...
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1answer
8 views

Number of isolated singularities

If $f:G\longrightarrow\mathbb{C}$ is analytic except for the isolated singularities and has infinitely many singularities, why then the singularities only can accumulate on boundary of $G$? Which is ...
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0answers
47 views

Residue calculus: $\int_{-\infty}^\infty e^{-x^2} \mathrm{d}x$ [duplicate]

I am pretty sure I have read the answer somewhere on this site, but sadly I am unable to find the question. How to evaluate $\int_{-\infty}^\infty e^{-x^2} \mathrm{d}x$ using the residue theorem?
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24 views

Evaluate $I=\int_{-\infty}^{\infty}{\frac{\cos{ax}}{b^2+x^2}dx}$ [duplicate]

I am trying to find the value of the integral $$I=\int_{-\infty}^{\infty}{\frac{\cos{ax}}{b^2+x^2}dx}$$ Here, both $a,b\in\mathbb{N}$. Since $\cos{ax}=\frac{e^{iax}+e^{-iax}}{2}$, I can consider ...
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1answer
36 views

express as contour integral $ f(x) = \int_0^\infty dt \; e^{-t/g} \; \frac{1}{\sqrt{1 - 2 t x}} $

Let $0 < x < 1$, I have to compute this Laplace transform: $$ f(x) = \int_0^\infty dt \; e^{-t/g} \; \frac{1}{\sqrt{1 - 2 t x}} $$ I am not 100% this interal is defined. If $t > ...
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0answers
44 views

Why doesn't this residue method work for calculating $\sum_{k=1}^{k=\infty} \frac{\cos(k x)}{k^2}$

I want to calculate $S = \sum_{k=1}^{k=\infty} \frac{\cos(k x)}{k^2}$ using the method of residues. By using the function $f(z) = z \sum_{k=1}^{k=\infty} \frac{1}{k(z-k)}$, which has simple poles at ...
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3answers
162 views

How to do $\sum_{k=1}^{\infty} \frac{\cos(kx)}{k^2}$?

I'm a physicist with no clue how to calculate $S = \sum_{k=1}^{\infty} \frac{\cos(kx)}{k^2}$. One handbook says the answer is $\frac{1}{12}(3x^2 - 6 \pi x + 2 \pi^2)$ on the interval $0 \leq x \leq ...
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2answers
64 views

Using contour integrals to evaluate sum - Problem calculating residues

"Compute $$\sum_1^{\infty} \frac{1}{{n^2}{(n+1)^2}}$$ using contour integration" I have used the function $F(z) = \frac {\pi cot\pi z}{z^2(z+1)^2}$ Which has double poles at $z=0$ and $z=-1$ ...
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1answer
101 views

More 'conceptual' reason of why $\int_{-\infty}^{+\infty}\text{sin}(x^2) = \int_{-\infty}^{+\infty}\text{cos}(x^2)$

In our complex analysis course, as an application of the residue theorem and some clever contour integration, we computed the following integrals: ...
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1answer
36 views

Finding residue of double pole

I'm trying to find the residue of the pole $+ib$ of $$f(z) = \frac{e^{iaz}}{(z^2+b^2)^2}$$ (the pole $-ib$ lies outside of the contour). I'm trying to do this by $$Residue = ...
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2answers
76 views

I need to solve $\phi (x,y) = \frac{2V}{\pi} \int_{0}^{\infty} \frac{\sin(kx)\cosh(ky) dk}{k\cosh(ka)}$

I start with a integral in complex plane $$\oint_c \frac{e^{izx} e^{zy} dz}{z\cosh(za)}$$ where $c$ is a countour starting in $z = -R$ along the real axis and jumping the pole at origin and continuing ...
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3answers
71 views

How to evaluate the integral $\int_{0}^{\infty}\frac{\cos {(ax)}-\cos{(b x)}}{x^2 }dx$?

I'm wondering how to integrate the so-called integral using Residue theorem,as it has a pole of second order on the real axis(not simple) so we cannot use $\pi i Res(@ z=0)$.Would you please give me a ...
3
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1answer
33 views

Find the value of $\frac{i}{4-\pi}\int_{|z|=4}\frac{1}{z\cos{z}}dz$

Find the value of $$\frac{i}{4-\pi}\int_{|z|=4}\frac{1}{z\cos{z}}dz$$. My attempt: The integrand has singularities at $z=0, \frac{\pi}{2}, \frac{-\pi}{2}$, so ...
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2answers
60 views

what is $\int_{|z|=3}\frac{1}{(z-2)^n(z-4)^m}dz$ for $m,n\in\mathbb{N}$?

I saw many examples how to calculate integrals with the residue theorem. But now I'm stuck with this integral: $$\int_{|z|=3}\frac{1}{(z-2)^n(z-4)^m}dz,$$where $m,n\in\mathbb{N}$ and $z=2$ and $z=4$ ...
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3answers
40 views

Laurent expansion of $1/(1+z^n)$ for $n \in \mathbb{N}$.

I've seen in many texts and answers on this website that the residue of $$1/(1+z^n)$$ can be computed easily since it has a simple pole at $z=e^{i \pi / n}$. That is all well and good but nothing ...
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1answer
71 views

How to calculate $\int_{-\infty}^{\infty}\frac{x^2}{\cosh(x)}\mathrm{d}x$ [duplicate]

I know the poles are $z=i\pi/2+i n\pi$ and therefor I got an rectangular contour for the integration which wasn't so useful. I also know with change of variables I can get to ...
3
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3answers
99 views

How to show that $\int_0^{\infty} dx \frac{\log{x}}{1+x^2}$ is zero using complex analysis

I want to show this using contour integration, the appropriate contour is a keyhole I think.
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2answers
170 views

how to calculate $\int_{0}^{\infty}\frac{x}{\sqrt{e^x-1}}\mathrm{d}x$

I was trying to solve another integral when then I reached this, I've no idea of how to select the contour for the integration.
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2answers
25 views

Improper integral computation using complex analysis

$$pv\int_{-\infty}^{\infty} \frac{x}{(x^2+4)(x^2+2x+2)}dx$$ I get an answer of $-\frac{\pi}{5}$ but wolframalpha disagrees by a factor of $2$ ($-\frac{\pi}{10}$): ...
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33 views

Residue theorem for an integral

Can I use the Residue Theorem to evaluate $g(x)=\int_{-\infty}^{\infty}\frac{sin(tx)sin(t)}{t}dt$? My guess is that $g(x)=0$ since that is the residue of the integrand. I'm not quite sure if I am ...
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1answer
40 views

What is the Residue of this function?

What is $$\operatorname*{Res}_{z=i \pi}{\frac{e^{(1-a)z}}{\left (1+e^{z} \right )^n}}$$ where $0<a<1$ and $n$ is an integer??
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1answer
28 views

Express ${1\over 2\pi i}\int_{\partial \Omega}{g(z){f'(z)\over f(z)}}dz$ with poles and zeros of $f$

The question states, in fact, show ${1\over 2\pi i}\int_{\partial \Omega}{g(z){f'(z)\over f(z)}}dz=\sum_{a_k}g(a_k)-\sum_{b_k}g(b_k)$, where $a_k$ are the zeros of $f$ in $\Omega$ and $b_k$ are the ...
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0answers
37 views

Evaluating the residue of $(1 - e^{-z})^n$ at $z = 0$ with $n \in \mathbb{Z}$

For $n \in \mathbb{Z}$, I want to find a way to determine the residue of the function $f(z) := (1 - e^{-z})^n$ at $z = 0$. I must admit that I haven't come too far yet. In case that $n ≥ 0$, $f$ ...
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2answers
159 views

What is the integral of 1/(z-i) over the unit circle?

At present there is a simple pole on the closed contour, so the Residue Theorem appears to be inapplicable. But I want to claim that we can enlarge this circle to make sure that it encloses the ...
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Integration (Residue theorem )

$$ \phi (x,y)=\int_{0}^{\infty }\left ( \frac{1-\cos zx}{z} \right )\sin zx \,e^{-zy}dz. $$ I have solved this integration this way: $$ \frac{\partial \phi (x,y) }{\partial y}=\int_{0}^{\infty }(\cos ...
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1answer
38 views

Integrating secans over the imaginary axis using the residue theorem

I am trying to integrate $\sec(z)$ over the whole imaginary axis using the residue theorem. i.e., I want to calculate the integral $$\int_{\Gamma} \frac{dz}{\cos{z}}$$ where $\Gamma$ is the (open) ...
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1answer
69 views

Limit of a difference of integrals that both look almost identical,

Let $\gamma (t) = t+i(e^t-1)$ for $-1\le t \le 1$. find $$\lim_{\epsilon \to 0^+} \left[\int_{\gamma} \frac{\sin(z)}{(z-i\epsilon)^2} dz - \int_{\gamma} \frac{\sin(z)}{(z+i\epsilon)^2} dz\right]$$ ...
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1answer
38 views

Multivariate/multidimensional residues

My very specific question: Given $(z_1,z_2) \in \mathbb{C}^2 $ and $$ I = \frac{1}{z_1 z_2 +1} $$ a) Where are the poles of $I$? b) What are the residues of $I$? Note: $z_1$ and $z_2$ are not in ...
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3answers
678 views

Need a hint for this integral

I'm trying to evaluate the following integral $$\int_0^{\infty} \frac{1}{x^{\frac{3}{2}}+1}\,dx.$$ This is an old complex analysis exam question, so I plan to use the residue theorem. How can I ...
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Evaluating real trigonometric integral using contour, with pole order n

Use the residue theorem to compute the real integral: $$I = \int_{0}^{2\pi} \sin^{2n}\theta d\theta$$ I have considered a contour around a unit circle C, and used the substitutions: $sin\theta = ...
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2answers
47 views

Calculating residues of function with branch cut

Show that $$I= \int_{0}^{\infty} \frac{\ln x}{x^\frac{3}{4} (1+x)} dx = -\sqrt{2} \pi^2$$ I used a keyhole contour, with branch point at $z=0.$ Around $\Gamma$, $|zf(z)|$ tends to $0$ as $z$ ...
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1answer
33 views

Let $f(z) = \frac{z^{-2}}{\sin( \pi z )}$. What is the residue for $z \neq 0$?

Let $f(z) = \frac{z^{-2}}{\sin( \pi z )}$. What is the residue for $z \neq 0$? In a pdf online, it states we may calculate the residue using the "derivative trick" to get: $$ \mathrm{Res}(f,n) = ...
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0answers
35 views

On the dog-bone contour around [-1,1], what are the arguments of these two lines approaching the real axis from above and below?

I am using a dog-bone contour to integrate around the interval [-1,1]. (-1 and +1 are branch points of the integrand.) I am using the principal branch of log, so I am restricting its argument to ...
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0answers
24 views

Calculating the residues of trigonometric functions of functions

For a generic function f(x), with simple poles at $a_1, a_2,..., a_n,$ I want to know if there is any way you can calculate the residues at those poles for the function sin(f(x)). I assume that if ...
1
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1answer
20 views

Quick question on the roots and poles of a meromorphic function,

Does the degree of the polynomial in the numerator always equal the degree of the polynomial in the denominator? In other words, the number of zeros, counting multiplicity, equals the number of ...
6
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2answers
113 views

Computing $\sum\limits_{n=1}^\infty\frac{\sin n}{n}$ with residues

I'm running into some error in computing the sum. Since $\dfrac{\sin n}{n}$ is even, I'm considering the function $f(z)=\dfrac{\pi\sin z\cot\pi z}{z}$ and the contour integral $$\oint_\gamma ...
3
votes
1answer
34 views

Residue theorem with pole on integration path

I have to calculate the inverse Laplace transform of $\dfrac{1}{s^2+1}$ (which I know is sin(x)) by residue theorem: $\int^{i \infty}_{-i \infty}exp(t\cdot s)\cdot \dfrac{1}{s^2+1}\mathrm{d}s$. ...
3
votes
3answers
52 views

Contour integral with a logarithm squared

The integral I'd like to evaluate is $\int_0^\infty \frac{\log^2 x \, dx}{(1+x)^2}$. I consider the function $f(z) = \frac{\text{Log}^2 z}{(1+z)^2}$, which has a pole of order 2 at $z=-1$ and has a ...
3
votes
2answers
93 views

How can a positive integrand integrate to 0? [duplicate]

I integrated $\dfrac{\log x}{1+x^2}$ from $0$ to infinity with residue calculus and got... $0$. This also agrees with Wolfram Alpha. How can this be? Is it due to the behavior of $\log(x)$ near ...