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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30
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.
23
votes
2answers
530 views

How to solve $\int_0^{\infty}\frac{\cos{ax}}{x^3+1}dx$?

QUESTION. It is looked for a closed solution for following real integrals $\displaystyle\int_0^{\infty}\displaystyle\frac{\cos{ax}}{x^3+1}dx$ and ...
18
votes
5answers
925 views

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 ...
18
votes
2answers
498 views

Show that $\int_0^{\pi/3} \big((\sqrt{3}\cos x-\sin x)\sin x\big)^{1/2}\cos x \,dx =\frac{\pi\sqrt{3}}{8\sqrt{2}}. $

I have run a FORTRAN code and I have obtained strong evidence that $$\int_0^{\pi/3} \!\! \big((\sqrt{3}\cos\vartheta-\sin\vartheta)\sin\vartheta\big)^{\!1/2}\!\cos\vartheta \,d\vartheta ...
18
votes
1answer
565 views

Integrating $\int_0^\infty\frac{\log (1+z^2)}{e^z-1}dz$ using residue calculus.

I've been looking at how to integrate the following definite integral using the residue calculus, but can't seem to get my thoughts together. I know the $\log$ term is a multivalued function and the ...
15
votes
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 ...
15
votes
3answers
382 views

Need help with $\int_0^\infty\frac{\log(1+x)}{\left(1+x^2\right)\,\left(1+x^3\right)}dx$

I need you help with this integral: $$\int_0^\infty\frac{\log(1+x)}{\left(1+x^2\right)\,\left(1+x^3\right)}dx.$$ Mathematica says it does not converge, which is apparently false.
15
votes
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 ...
13
votes
2answers
749 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 ...
12
votes
3answers
647 views

Integrating $\int_0^\infty \frac{\log x}{(1+x)^3}\,\operatorname d\!x$ using residues

I am trying to use residues to compute $$\int_0^\infty\frac{\log x}{(1+x)^3}\,\operatorname d\!x.$$My first attempt involved trying to take a circular contour with the branch cut being the positive ...
12
votes
3answers
707 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 ...
12
votes
2answers
207 views

$\int_0^\infty \frac{\cos(tx)}{(x^2 - 2x + 2)}\,\mathrm{d}x$ for $t$ real

This was a question on an old prelim exam in complex analysis: compute $$\int_0^\infty \frac{\cos(tx)}{x^2 - 2x + 2}\,\mathrm{d}x$$ for $t$ real. I've tried… Residue calculus—it's ...
11
votes
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 ...
11
votes
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 ...
11
votes
0answers
549 views

Proof of residue theorem (residue formula) for differential forms on curves over an arbitrary closed field.

I have been reading the book Algebraic Geometric Codes: Basic Notions by Tsfasman, Vladut and Nogin. They give a residue formula like this: Let $\mathbb{k}$ be an algebraically closed field and $X$ ...
10
votes
7answers
528 views

How to calculate $ \int_{0}^{\infty} \frac{ x^2 \log(x) }{1 + x^4} $?

I would like to calculate $$\int_{0}^{\infty} \frac{ x^2 \log(x) }{1 + x^4}$$ by means of the Residue Theorem. This is what I tried so far: We can define a path $\alpha$ that consists of half a ...
10
votes
5answers
360 views

Contour Integral: $\int^{1}_{0}\frac{1}{\sqrt[n]{1-x^n}}dx$

I want to compute: $\int^{1}_{0}\frac{1}{\sqrt[n]{1-x^n}}dx$ for natural $n>1$ using Residue Calculus. I am thinking of using some kind of a keyhole or bone contour that could go around the ...
10
votes
3answers
841 views

Evaluation by methods of complex analysis $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm{dx}$

How would we evaluate the below integral by methods of complex analysis? $$\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm{dx}$$ I asked the question a while ago, but at that time I didn't specify this ...
10
votes
1answer
122 views

Integrate $\int_0^\infty \frac{dx}{(x^2+2x+12)^2}$ using residues

I want to find the integral $$I=\int_0^\infty \frac{dx}{(x^2+2x+12)^2}$$ using contour integration; I am familiar with the trigonometric substitution in real analysis. There are no branch cuts, ...
10
votes
1answer
144 views

Application of the residue theorem

I want to prove that $$\sum_{j=1}^n \frac{1}{\left[\text{cos} \left( \frac{j \pi}{2n+1} \right)\right]^4}=\frac{8n(n+1)(n^2+n+1)}{3}$$ for $n \in \mathbb{N}$ by using the residue theorem. Which ...
9
votes
6answers
252 views

Evaluate $\int_0^{\infty} \frac{\log x }{(x-1)\sqrt{x}}dx$ (solution verification)

I tried to find the integral $$I=\int_0^{\infty} \frac{\log x }{(x-1)\sqrt{x}}dx \tag1$$ I substituted $x=t^2, 2tdt=dx$ and chose $\log x$ and $\sqrt{x}$ to be principal values. We have ...
9
votes
3answers
2k views

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 ...
9
votes
3answers
330 views

How to show $\int^{\infty}_{-\infty}\frac{\sin(ax)}{x(x^2+1)}dx=\pi(1-e^{-a})$? ($a\ge0$)

$$\int^{\infty}_{-\infty}\frac{\sin(ax)}{x(x^2+1)}dx=\pi(1-e^{-a}), \ a\ge0$$ I tried to solve but came up with $\pi(2-e^{-a}) $. Could you tell me where did I do the mistake? if $x=z$ then ...
9
votes
1answer
292 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 $z_0$. ...
9
votes
2answers
204 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
2answers
203 views

Integrating around simple pole and semicircle

Let $f$ be a holomorphic function on $\mathbb{C}$ with simple pole at $z_0$. Then if $\Gamma$ is a circle around $z_0$ oriented counter-clockwise with radius $r$ and $r\rightarrow 0$, then ...
9
votes
1answer
316 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
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 ...
8
votes
2answers
228 views

Why is the (-1)-th coefficient of $f^n f'$ equal to 0, without dividing by $n+1$?

Let $R$ be a commutative ring, and $n$ be a nonnegative integer. Let $f\in R\left[t,t^{-1}\right]$ be a Laurent polynomial in one variable $t$ over $R$ (this means a formal $R$-linear combination of ...
8
votes
1answer
139 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 ...
7
votes
3answers
432 views

Evaluate $\int_{0}^{\infty}\dfrac{\mathrm dx}{(e^{\pi x}+e^{-\pi x})(16+x^2)}$

Find the integral $$I=\int_{0}^{\infty}\dfrac{1}{(e^{\pi x}+e^{-\pi x})(16+x^2)}dx$$ My try:let $x=-t$ $$I=\int_{-\infty}^{0}\dfrac{1}{(e^{\pi x}+e^{-\pi x})(16+x^2)}dx$$ so ...
7
votes
3answers
184 views

How prove this sum$\sum_{k=0}^{\infty}\frac{(-1)^{\frac{k(k+1)}{2}}}{(2k+1)^2}=\frac{\sqrt{2}\pi^2}{16}$

How show that $$\sum_{k=0}^{\infty}\dfrac{(-1)^{\frac{k(k+1)}{2}}}{(2k+1)^2}=\dfrac{\sqrt{2}\pi^2}{16}$$ My idea:I know how to prove the following sum ...
7
votes
4answers
227 views

Help with the contour for this integral using residues

$$ PV \int_0^\infty \frac{dx}{\sqrt{x}(x^2-1)} $$ A keyhole contour can't be used because we have a pole in the real positive axis, isn't it?
7
votes
3answers
508 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 ...
7
votes
4answers
279 views

Integral by residue - “dog bone”

Let $I=\int_{-1}^{1}\frac{x^2 dx}{\sqrt[3]{(1-x)(1+x)^2}}$. I used complex function $f(z)=\frac{z^2}{\sqrt[3]{(z-1)(z+1)^2}}$, which we can define such that it is holomorphic on ...
7
votes
2answers
1k views

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 ...
7
votes
3answers
140 views

Calculation of $\int^{\pi/2}_{0}\cos^{n}(x)\cos (nx)\,dx$, where $n\in \mathbb{N}$

Compute the definite integral $$ \int^{\pi/2}_{0}\cos^{n}(x)\cos (nx)\,dx $$ where $n\in \mathbb{N}$. My Attempt: Using $\cos (x) = \frac{e^{ix}+e^{-ix}}{2}$, we get $$ \begin{align} ...
7
votes
1answer
594 views

To calculate residue of the function $f(z) = \frac{z^2 + \sin z}{\cos z - 1}$.

I was trying to find the residue of the function $$f(z) = \frac{z^2 + \sin z}{\cos z - 1}.$$ Here is the my attempt: The given function has a pole of order two at $z = 2n\pi$. So, we use the ...
7
votes
1answer
181 views

How to find closed form formula for a sum

I am a PhD student in electrical engineering. I need to find a closed form formula for the following series: $$\sum_{k=1}^{\infty}\frac{1}{2}A_k^2e^{-k^2\sigma_m^2}(e^{k^2\sigma_m^2}-1)$$where $A_k= ...
6
votes
3answers
185 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 $$\int^{\infty}_0\frac{\cos ax+x\sin ax}{1+x^2}dx.$$ ...
6
votes
1answer
119 views

A particular integral: $\int_{-\infty}^{+\infty}\frac{\sin(\pi x)}{\prod_{k=-n}^{n}(x-k)}\,dx$

I have to show summability, then compute the following integral: $$\int\limits_{-\infty}^{+\infty} \frac{\sin(\pi\,x)}{\prod_{k = - n}^n (x - k)}\,dx = \frac{(-4)^n}{(2\,n)!}\,\pi $$ for every $n\in ...
6
votes
3answers
506 views

Using residue theory to evaluate $ \int_0^\infty \frac{ \sin \pi x}{x(1-x^2)} \;\text{ dx}$

I'm on the last question of my homework and it's involving using the residue theory, which I dont really understand, so could somebody lend me a hand? I have to evaluate the real convergent improper ...
6
votes
1answer
105 views

Looking for guidance on a Fourier integral

Working with a Fourier transform problem, I've encountered the following integral: $$ \int_{-\infty}^{\infty}\frac{\exp\left(-a^2x^2+ibx\right)}{x^2+c^2}dx $$ where $a$, $b$, and $c$ are real ...
6
votes
1answer
215 views

Evaluating the (complex) integral $\int_\gamma \frac{e^{z+z^{-1}}}{z}\mathrm dz$ using residues.

I am trying to evaluate the following integral. $$\int_\gamma \frac{e^{z+z^{-1}}}{z}\mathrm dz$$ where $\gamma$ is the path $\cos(t)+2i\sin(t)$ for $0\leq t <4\pi$. So, $\gamma$ is an ...
6
votes
4answers
2k views

Using the Residue Theorem for a contour integral along the Riemann sphere

Given the integral $\int_{-\infty}^{\infty}\frac{x}{x^2+1}dx,$ we can clearly see this is the integral of an odd function with limits which are symmetric about the origin, and thus its integral is ...
6
votes
1answer
2k 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 ...
6
votes
1answer
73 views

Evaluating $\int_0 ^{\infty}\frac{dx}{x^{1/3}(1+x)}$ using Complex Analysis

I am trying to use the residue theorem to evaluate $$I=\int_0 ^{\infty}\frac{dx}{x^{1/3}(1+x)}$$ I'll explain my difficulty in finding a contour, then I explain my difficulty in finding a new contour ...
6
votes
2answers
144 views

Evaluating sums using residues $(-1)^n/n^2$ [duplicate]

I am an alien towards compelx analysis, with very little know I am posing a question, who someone may want to help with. Evaluate: $$\frac{1}{4}\cdot \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$$ In ...
6
votes
2answers
664 views

The law of large numbers: How does the convergence take place? How does the “remainder” look like?

I have the independent and identically distributed random variables $X_1,X_2,\ldots$ with a finite expectation $\mu$. I also have defined $S_n = X_1 + \cdots + X_n$. According to the law of large ...
6
votes
3answers
113 views

Residue Formula application

Using the Residue formula, I've been trying to prove $$\int_0^{2\pi}\frac{1}{a^2\cos^2\theta+b^2\sin^2\theta}\,d\theta=\frac{2\pi}{ab},\quad\quad a,b\in\Bbb R.$$First, it seems like the formula should ...