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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1answer
32 views

Evaluating contour in the form $\int_{o}^{\infty} \frac{x^{-a}}{1+x}dx $

Having this improper integral $$ \int_{0}^{\infty} \frac{x^{-a}}{1+x}dx$$ I apply the form as such $x^{a}R(x)$ as such $$ \oint_{\Gamma} z^{a}R(z)dz = \frac{2\pi i}{1-e^{2\pi ia}}\sum_{poles} ...
4
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2answers
109 views

Problem over a definite integral, which surely needs contour integration

During my Master Thesis work I came up with an integral which I am going to consider as a hard challenge. I have been trying for days to crack it, but still nothing. The integral is the following ...
1
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1answer
28 views

Complex: evaluating integral with residues

Having a bit of trouble here. Having this integral $$ \int_{0}^{\infty} \frac{dx}{(x^{2}+1)(x^{2}+4)^{2}} $$ I can tell it's even, so it has symmetry. Thus, $$ \frac{1}{2} \int_{-\infty}^{\infty} ...
1
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0answers
22 views

Residue for $\frac{\zeta(s)}{\zeta(2s)}$ at zeros of $\zeta(2s).$

I want to calculate residue at the poles for $\frac{\zeta(s)}{\zeta(2s)}.$ For pole of numerator $s=1$ I have calculated the residue. I am having trouble at the zeros of denominator. Basically I am ...
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1answer
15 views

Write $\mbox{Res}(g, 0) $ in terms of the $\mbox{Res}(f, z)$ where $g(z)=\frac{1}{z^{2}}f\left(\frac{1}{z}\right)$.

Let $M\subseteq \mathbb{C}$ finite set and $f:\: \mathbb{C} \setminus M \longrightarrow \mathbb{C}$ be a holomorphic function. Consider $g(z)=\frac{1}{z^{2}}f\left(\frac{1}{z}\right)$. The quiestion ...
0
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1answer
29 views

Show that the numbers $3$,$3^2$,$3^3$,$3^4$,$3^5$,$3^6$ for a reduced residue system modulo 7.

Show that the numbers $3$,$3^2$,$3^3$,$3^4$,$3^5$,$3^6$ for a reduced residue system modulo 7. -A bit lost with this question, we just started a section on reduced residue sets and only covered ...
3
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1answer
70 views

Cauchy Principal Value of $\int_0^\infty \frac{x}{(x^2 + a^2) \, \sin(\mu x)} dx$

The problem here is to evaluate $$ \int_0^\infty \frac{x}{(x^2 + a^2) \, \sin(\mu x)} dx $$ for $a,\mu >0.$ Clearly this integral doesn't converge in the usual sense, but we can calculate its ...
1
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2answers
72 views

Inverse Laplace Transform of $e^{\frac{1}{s}-s}$

doing some work on a PDE system I have stumbled across a Laplace transform which I'm not sure how to invert: $$ F(s) = e^{\frac{1}{s}-s} $$ I can't find it in any table and the strong singular growth ...
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2answers
131 views

How can $\sin x = e^{iz}$?

This is probably a trivial question but I just don't see it. I'm solving the integral $$\int_{0}^{\infty}\dfrac{x \sin x }{x^2 + 5x + 4}dx$$ using the residue theorem. The thing is that they're ...
1
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1answer
23 views

Show that the numbers $-13, -9, - 4, -1, 9, 18, 21$ form a complete residue system modulo 7

Show that the numbers $-13, -9, - 4, -1, 9, 18, 21$ form a complete residue system modulo 7 We have just started he section on modular arithmetic so I am new to a residue system, we did a similar ...
4
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1answer
55 views

Cauchy Residue Theorem Integral

I have been given the integral $$\int_0^ {2\pi} \frac{sin^2\theta} {2 - cos\theta} d\theta $$ I have use the substitutions $z=e^{i\theta}$ |$d\theta = \frac{1}{iz}dz$ and a lot of algebra to transform ...
2
votes
1answer
69 views

Integrate $I=\int_e^\infty\left(\frac{\log\log y}{y(y+1)}\right)^2dy$ using residue calculus?

Can this integral be done using the residue calculus? $$I=\int_e^\infty\left(\frac{\log\log y}{y(y+1)}\right)^2dy$$ ? My (empirical) investigative attempts have been to use a keyhole contour centred ...
0
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1answer
43 views

Complex integral of the function $f(z)=\dfrac{1}{z^4+1}$

I must calculate this integral $$\int_C\dfrac{dz}{z^4+1}$$ , where $C$ is the circle $x^2+y^2=2x$. My result is $\int_C\dfrac{dz}{z^4+1}=-\dfrac{\pi}{\sqrt{2}}$ , but my book "A collection of problems ...
2
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2answers
59 views

Improper integral using residue theorem

I am meant to use the residue theorem to show that $\int\limits_{-\infty}^\infty \frac{\cos t}{(t^2+1)^2}dt=\frac{\pi}{e}$. So far I have deduced that I should take a contour over $\alpha$ the path ...
3
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1answer
278 views

Compute the series $\sum_{n=1}^{+\infty} \frac{1}{n^3\sin(n\pi\sqrt{2})}.$

I need to compute $$\sum_{n=1}^{+\infty} \frac{1}{n^3\sin(n\pi\sqrt{2})}.$$ This an exercice of "Amar and Matheron, complex analysis". I proved the convergence and now to compute the sum, I follow the ...
2
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0answers
32 views

Cauchy's Residue Theorem and Cauchy's Theorem

Cauchy's theorem in short says for a holomorphic function $f$ which is holomorphic on and inside a path $\gamma$ the path integral is $0$ I have calculated a path integral around a path where there ...
5
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1answer
132 views

Integral involving power of trigonometric functions

I'm having a technical problem evaluating the following integral: $$\int_{r=0}^1\int_{\theta=0}^{\pi \over2} \cos^{2\epsilon -1}\theta \sin^{\epsilon-1}\theta e^{-ikr\sin^\epsilon\theta}d\theta dr$$ ...
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0answers
71 views

Complex integration with infinitely many poles on imaginary axis

I'm trying to integrate with a closed contour on the upper-half of the complex plane. $I = \displaystyle\int_{-\infty}^\infty \dfrac{z\,\mathrm{sech(z)}}{[(z-a)^2+b^2][(z+a)^2+b^2]} dz$ There are ...
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0answers
34 views

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} ...
1
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1answer
46 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
33 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?
2
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2answers
32 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: ...
2
votes
3answers
44 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 ...
1
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3answers
46 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[ ...
0
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1answer
42 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
17 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
24 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 ...
0
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2answers
63 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 ...
1
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2answers
23 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 ...
1
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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
10 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 ...
1
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0answers
48 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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0answers
27 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 ...
2
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1answer
44 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 > ...
0
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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
170 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 ...
2
votes
2answers
79 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 $z=-1$ ...
7
votes
1answer
112 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: ...
1
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1answer
39 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 = ...
3
votes
3answers
146 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
75 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
votes
1answer
34 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 ...
1
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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 $z=4$ ...
1
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3answers
45 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 ...
3
votes
1answer
86 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
votes
3answers
109 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.
2
votes
2answers
174 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.
1
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2answers
26 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 $2$ ($-\frac{\pi}{10}$): ...
0
votes
0answers
37 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 ...
0
votes
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??