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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Determine and classify all singular points

Determine and find residues for all singular points $z\in \mathbb{C}$ for (i) $\frac{1}{z\sin(2z)}$ (ii) $\frac{1}{1-e^{-z}}$ Note: I have worked out (i), but (ii) seems still not easy.
17
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3answers
1k views

A sine integral $\int_0^{\infty} \left(\frac{\sin x }{x }\right)^n\,\mathrm{d}x$

The following question comes from Some integral with sine post $$\int_0^{\infty} \left(\frac{\sin x }{x }\right)^n\,\mathrm{d}x$$ but now I'd be curious to know how to deal with it by methods of ...
18
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1answer
2k views

Compute the inverse Laplace transform of $e^{-\sqrt{z}}$

I want to compute the inverse Laplace transform of a function $$ F(z) = e^{-\sqrt{z}}. $$ This problem seems very nontrivial to me. Here one can find the answer: the inverse Laplace transform of ...
6
votes
5answers
462 views

Evaluate $\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$ using complex analysis

How do I compute $$\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$$ What I am doing is take $$f(z)=\frac{(\log z)^2}{1+z^2}$$ and calculating $\text{Res}(f,z=i) = \dfrac{d}{dz} \dfrac{(\log ...
6
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2answers
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Computing $\int_{-\infty}^{\infty} \frac{\cos x}{x^{2} + a^{2}}dx$ using residue calculus

I need to find $\displaystyle\int_{-\infty}^{\infty} \frac{\cos x}{x^{2} + a^{2}}\ dx$ where $a > 0$. To do this, I set $f(z) = \displaystyle\frac{\cos z}{z^{2} + a^{2}}$ and integrate along the ...
9
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1answer
3k views

Show that $\int_0^ \infty \frac{1}{1+x^n} dx= \frac{ \pi /n}{\sin(\pi /n)}$ , where $n$ is a positive integer.

Using residues, try the contour below with $R \rightarrow \infty$ and $$\lim_{R \rightarrow \infty } \int_0^R \frac{1}{1+r^n} dr \rightarrow \int_0^\infty \frac{1}{1+x^n} dx$$ I've ...
8
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2answers
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Computing $\int_{-\infty}^\infty \frac{\sin x}{x} \mathrm{d}x$ with residue calculus

This refers back to the integral of $\frac{\sin(x)}x = \frac\pi2$ already posted. How do I arrive at $\frac\pi2$ using the residue theorem? I'm at the following point: $$\int \frac{e^{iz}}{z} - \int ...
33
votes
4answers
1k views

Evaluation of $\int_0^{\pi/3} \ln^2\left(\frac{\sin x }{\sin (x+\pi/3)}\right)\,\mathrm{d}x$

I plan to evaluate $$\int_0^{\pi/3} \ln^2\left(\frac{\sin x }{\sin (x+\pi/3)}\right)\, \mathrm{d}x$$ and I need a starting point for both real and complex methods. Thanks ! Sis.
18
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5answers
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Prove $\int_0^\infty \frac{\sin^4x}{x^4}dx = \frac{\pi}{3}$

I need to show that $$ \int_0^\infty \frac{\sin^4x}{x^4}dx = \frac{\pi}{3} $$ I have already derived the result $\int_0^\infty \frac{\sin^2x}{x^2} = \frac{\pi}{2}$ using complex analysis, a result ...
11
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1answer
2k views

Calculating the Fourier transform of $\frac{\sinh(kx)}{\sinh(x)}$

I'm trying to compute $$\int_{-\infty}^\infty \frac{\sinh(kx)}{\sinh(x)}e^{-i\omega x} \ dx$$ i.e. the Fourier transform of $x\mapsto \frac{\sinh(kx)}{\sinh(x)}$, where $0<k<1$ is fixed. But ...
3
votes
2answers
455 views

Finding the poles and residues of a complex function $\frac{\cos(z)-1}{(e^z - 1)^2}$

I'm trying to find the poles and residues of: $$f(z) = \frac{\cos(z)-1}{(e^z - 1)^2}$$ I can see that this has a removable singularity at $z=0$ and double poles at $z=2k \pi i$. I'm having trouble ...
14
votes
2answers
827 views

Sum of the squares of the reciprocals of the fixed points of the tangent function

The sum of the squares of the reciprocals of the positive fixed points of the tangent function is $1/10$. I've seen this proved by means of residues, but I don't remember the details. I've also ...
4
votes
0answers
478 views

How to solve an integral with a Gaussian Mixture denominator?

I am trying to solve this integral: $$t(v)\equiv\sum_{k=1}^{n}\sum_{j=1}^{n}\int_{-\infty}^{\infty}\frac{w_{k}N(x-x_{k},B_{k})N(x-x_{j},v)}{\sum_{m=1}^{n}w_{m}N(x-x_{m},B_{m}+v)}dx $$ where ...
3
votes
3answers
330 views

Evaluate $\int_0^{2\pi} \frac{\sin^2\theta}{5+4\cos\theta}\,\mathrm d\theta$

Evaluate $$\int_0^{2\pi} \frac{\sin^2\theta}{5+4\cos(\theta)}\mathrm d\theta$$ This is the final question on my review for my final exam tomorrow, and I will be honest and say that I have no clue ...
1
vote
3answers
144 views

Integral $\int_{0}^{\infty}e^{-ax}\cos (bx)\operatorname d\!x$

I want to evaluate the following integral via complex analysis $$\int\limits_{x=0}^{x=\infty}e^{-ax}\cos (bx)\operatorname d\!x \ \ ,\ \ a >0$$ Which function/ contour should I consider ?
2
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1answer
1k views

Contour integration using the residue at infinity

I posted a similar problem a few months ago but got no responses. So I'm going to try again with a different problem. I want to evaluate $ \displaystyle I ...
1
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1answer
159 views

Gaussian Integral using contour integration with a parallelogram contour

I'm having trouble figuring out how to use contour integration to compute the Gaussian integral. The contour I'm using is a parallelogram with function, $f(z) = \Large \frac{ e^{i \pi z^2}}{sin(\pi ...
2
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1answer
105 views

A Tough Problem about Residue

I tried my best to solve this problem from what I learned in residues, but the solution seems very far from what I was doing!! Is there any way other than using Laurent series expansion? Here is the ...
5
votes
3answers
2k views

Complex Integration poles real axis

In class my professor said that $$ \int_{-\infty}^{\infty}\frac{e^{iax}}{x^2 - b^2}dx = -\frac{2\pi}{b}\sin(ab) $$ where $a,b > 0$. However, since the poles are on the real axis, isn't the integral ...
2
votes
3answers
600 views

Use residues to evaluate $\int_0^\infty \frac{\cosh(ax)}{\cosh(x)}\,\mathrm{d}x$, where $|a|<1$

Use residues to evaluate $$ \int_0^\infty \frac{\cosh(ax)}{\cosh(x)}\,\mathrm{d}x $$ where $|a|<1$. Try considering the integral of the form $$ \int_C \frac{\exp(az)}{\cosh(z)}\,\mathrm dz, $$ ...
5
votes
3answers
308 views

Applications of the Residue Theorem to the Evaluation of Integrals and Sums

Evaluate the integral $$\int_{-\infty}^{\infty} \frac{1}{(1 + x^2)^{n+1}} dx. $$ I know that it equals $2\pi i$(the sum of the residues; at $z_k$) where $z_k$ are the poles of the function. I ...
3
votes
1answer
53 views

Find $ \int_0^{2\pi} \frac{1}{a^2\cos^2 t+b^2 \sin^2 t} dt \;; a,b>0$.

Using Residue Theorem find $\displaystyle \int_0^{2\pi} \frac{1}{a^2\cos^2 t+b^2 \sin^2 t} dt \;; a,b>0$. My Try: So, I am going to use the ellipse $\Gamma = \{a\cos t+i b \sin t: 0\leq t\leq ...
12
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1answer
1k views

Integral Representation of Infinite series

Let's take a look at the following integrals : 1) $\displaystyle \int\limits_{0}^{1} \frac{\log{x}}{1+x} \ dx = -\frac{\pi^{2}}{12} = -\frac 1 2 \sum\limits_{n=1}^{\infty} \frac{1}{n^2}= -\frac 1 2 ...
8
votes
3answers
566 views

use residues to evaluate sum involving square of csch

I have been trying to evaluate the following sum using residues $\displaystyle \sum_{n=1}^{\infty}\frac{1}{\sinh^{2}(\pi n)}=\frac{1}{6}-\frac{1}{2\pi}$ I am mainly interested in using residues to ...
3
votes
3answers
145 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 ...
9
votes
1answer
402 views

How to rigorously justify “picking up half a residue”?

Often in contour integrals, we integrate around a singularity by putting a small semicircular indent $\theta \rightarrow z_0 + re^{i\theta}$, $0 \leq \theta \leq \pi$ around the singularity at $z_0$. ...
5
votes
2answers
515 views

Evaluating real integral using residue calculus: why different results?

I have to evaluate the real integral $$ I = \int_0^{\infty} \frac{\log^2 x}{x^2+1}. $$ using residue calculus. Its value is $\frac{\pi^3}{8}$, as you can verify (for example) introducing the function ...
4
votes
3answers
569 views

Integral with branch cut ( Problem while calculating residue)

While calculating this integral $\int_{-1}^{1}\frac{dx}{\sqrt{1-x^2}(1+x^2)}$ , I am really struggling to calculate the residue at (-i), I am getting the value of residue as $\frac{-1}{2\sqrt{2}i}$, ...
4
votes
3answers
284 views

Integral $\int_0^\infty \exp(ia/x^2+ibx^2)dx$

Compute the integral: \begin{equation} \int_0^\infty \exp\left(\frac{ia}{x^2}+ibx^2\right)\,dx \end{equation} for $a$, $b$ real and positive. I tried complex variables, but don't really know how to ...
5
votes
3answers
228 views

Calculating $\int_{0}^{\infty} x^{a-1} \cos(x) \ \mathrm dx = \Gamma(a) \cos (\pi a/2)$

My goal is to calculate the integral $\int_{0}^{\infty} x^{a-1} \cos(x) dx = \Gamma(a) \cos (\pi a/2)$, where $0<a<1$, and my textbook provides the hint: integrate $z^{a-1} e^{iz}$ around the ...
6
votes
3answers
224 views

Evaluating an Integral by Residue Theorem

Its been awhile since I have taken complex analysis and I am wondering how to solve the following integral when $a>0, \ a=0,$ and $a<0$ for $$\int^{\infty}_0\frac{\cos ax+x\sin ax}{1+x^2}dx.$$ ...
4
votes
3answers
267 views

Residue Theorem for Gamma Function

I am kinda stuck and not sure what to do at this point of the calculation where: $$\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left(\,x^{-1}\sigma\,\sqrt{\, 2\,}\,\,\right)^{s}\Gamma\left(\,{s \over ...
4
votes
1answer
479 views

Why is the sum of residues of $\frac{1}{1+z^n}$ in the upper half plane $1/[in\sin(\pi/n)]$?

Suppose $F_n=1/(1+z^n)$ for $n$ even. I'm curious, why is the sum of residues of $F_n$ in the upper half plane a geometric series whose sum is $1/[in\sin(\pi/n)]$? I know that if ...
3
votes
3answers
123 views

Please more help me to find the convergence interval and the sum -by using residue theory- of the series.

The sum is that $$\sum_{n=0}^\infty \binom{3n}{2n} x^n$$ First of all, I need to check whether the sum converges or not and if it is convergent, which points? I am using ratio test. $$ ...
2
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1answer
313 views

Residue theorem:When a singularity on the circle (not inside the circle)

This is the integration I am trying to solve $$\int_{0}^{\pi} \sin^{2}(\theta)\sec^{3}(\theta)d\theta$$ putting $$z=e^{i\theta}$$ $$\int_{\gamma} \frac{-2{(z^{2}-1)}^2}{i(z-i)^{3}(z+i)^{3}}d\theta$$ ...
2
votes
2answers
288 views

How the calculate $\int_0^{+\infty} \frac{\sin^2 x}{x^2} \,\mathrm{d} x$?

Just as the title say, consider the integral: $$I=\int_0^{+\infty} \frac{\sin^2 x}{x^2} \,\mathrm{d} x=\frac{1}{2}\int_{-\infty}^{+\infty} \frac{\sin^2 x}{x^2} \,\mathrm{d} x,$$ how to apply the ...
2
votes
1answer
2k views

Finding the Fourier transform of $f(x) = \frac{a}{\pi} \frac{1}{a^2 + x^2}$ with the residue theorem

I keep getting the wrong answer for this problem! Find the Fourier transform of $f(x) = \frac{a}{\pi} \frac{1}{a^2 + x^2}$ using the residue theorem. Well, by definition: $$\hat f(k) = ...
1
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2answers
68 views

Residue Formula in complex analysis

I understand the residue formula but I just can't understand the cancelling down of $$ \operatorname{res}_{z=z_1} (f)= \lim \limits_{z \to z_1}(z-z_1) \frac {z^2}{z^4+1} = \frac {z_1^2}{4z_1^3}.$$ ...
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0answers
191 views

Fibonacci Numbers - Complex Analysis [duplicate]

Possible Duplicate: Complex Analysis - Integral over a circle of radius R Hey guys~ Does anyone know where to find the solutions to this problem set on page 106 involving the fibonacci ...
9
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3answers
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Applications of Residue Theorem in complex analysis?

Does anyone know the applications of Residue Theorem in complex analysis? I would like to do a quick paper on the matter, but am not sure where to start. The residue theorem The residue ...
4
votes
2answers
176 views

Evaluating $I_{\alpha}=\int_{0}^{\infty} \frac{\ln(1+x^2)}{x^\alpha}dx$ using complex analysis

Again, improper integrals involving $\ln(1+x^2)$ I am trying to get a result for the integral $I_{\alpha}=\int_{0}^{\infty} \frac{\ln(1+x^2)}{x^\alpha}dx$ - asked above link- using some complex ...
10
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1answer
2k views

Residue integral: $\int_{- \infty}^{+ \infty} \frac{e^{ax}}{1+e^x} dx$ with $0 \lt a \lt 1$.

I'm self studying complex analysis. I've encountered the following integral: $$\int_{- \infty}^{+ \infty} \frac{e^{ax}}{1+e^x} dx \text{ with } a \in \mathbb{R},\ 0 \lt a \lt 1. $$ I've done the ...
12
votes
3answers
970 views

Calculate $\displaystyle \int_0^\infty \frac{\ln x}{1 + x^4} \mathrm{d}x$ using residue calculus

I need to evaluate this integral using calculus of residues: $$\int_0^\infty\frac{\ln(x)}{1+x^4}\mathrm{d}x$$ I know I need to consider $\displaystyle ...
10
votes
2answers
396 views

Dog Bone Contour Integral

Would someone please help me understand how to integrate $$ \ \int_0^1 (x^2-1)^{-1/2}dx\, ? $$ This is a homework problem from Marsden Basic Complex Analysis. The text book suggested using a "dog ...
9
votes
1answer
341 views

Help in calculating the following integral $\int_0^{2\pi}\! \frac{(1+2\cos x)^n \cos (nx)}{3+2\cos x} \, \mathrm{d}x. $

I was asked to calculate this: $$\int_0^{2\pi}\! \frac{(1+2\cos x)^n \cos (nx)}{3+2\cos x} \, \mathrm{d}x. $$ My idea was to change the integration limits to $|z|=1$ in the complex plane and to ...
8
votes
1answer
179 views

Partial fraction expansion for non-rational functions

With regard to this answer to an inverse Laplace transform question, I derived the following result: $$\frac1{i 2 \pi} \int_{c-i \infty}^{c+i \infty} ds \, e^{s t} \Gamma(s)^2 = 2 K_0 \left ( 2 ...
5
votes
4answers
305 views

How to prove $\int^{\pi/2}_0 \log{\cos{x}} \, \mathrm{d}x = \pi/2 \log{1/2}$

ALREADY ANSWERED I was trying to prove the result that the OP of this question is given as a hint. That is to say: imagine that you are not given the hint and you need to evaluate: $$I = ...
5
votes
3answers
602 views

A difficult integral evaluation problem

How do I compute the integration for $a>0$, $$ \int_0^\pi \frac{x\sin x}{1-2a\cos x+a^2}dx? $$ I want to find a complex function and integrate by the residue theorem.
5
votes
2answers
205 views

Finding a generalization for $\int_{0}^{\infty}e^{- 3\pi x^{2} }\frac{\sinh(\pi x)}{\sinh(3\pi x)}dx$

$\;\;\;\;$I was reading the introduction of Paul J. Nain's book "Dr. Euler's fabulous formula" where he talks about the sense of beauty in mathematics and quotes the G.N.Watson as saying that a ...
3
votes
2answers
134 views

How find this sum $\sum_{n=1}^{\infty}\frac{1}{n^2-n+a}$

Today Question if $a>\dfrac{1}{4}$, show that $$\sum_{n=1}^{\infty}\dfrac{1}{n^2-n+a}=\dfrac{\pi}{\sqrt{4a-1}}\cdot\dfrac{e^{\pi\sqrt{4a-1}}-1}{e^{\pi\sqrt{4a-1}}+1}\tag{1}$$ I have konw that ...