3
votes
1answer
83 views

Solve $\mathscr{F}^{-1} [ \cot{a \omega} \times \mathscr{F} \{ U(t) \sin{\omega_0 t} \} ] $ using contour integration

I wish to evaluate $y(t) = \mathscr{F}^{-1} [ \cot{a \omega} \times \mathscr{F} \{ U(t) \sin{\omega_0 t} \} ] $, where $\mathscr{F}$ represents the Fourier transform, and U(t) represents the ...
0
votes
1answer
24 views

Application of Residues

So in applying the residue theorem to solve improper real integrals, we agree to take our semicircles to be as large or as small as necessary such that all the poles we wish to work with lie inside ...
0
votes
0answers
94 views

Convergence of improper real integrals using the residue theorem

Let $f:\mathbb{R} \to \mathbb{R}$ be a funktion that satisfies $$ |xf(x)| \to 0 \qquad (|x| \to \infty) $$ If $f$ can be extended to a holomorphic function with finitely many singularities $S$ and the ...
15
votes
3answers
271 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.
0
votes
0answers
47 views

Integration by Residue Theorem - Is This Integral equal to Zero?

$$ \int_{-\infty}^{+\infty}\frac{\sin(x)}{2x^{5}-3jx^{3}+2x}dx=0 $$ ...
1
vote
3answers
76 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 ?
4
votes
4answers
274 views

Calculating $\int_0^\infty \frac {\sin^2x}{x^2}dx$ using the Residue Theorem.

I am trying to compute the following integral using the Residue Theorem but am quite stuck: $$\int_0^\infty \frac{\sin^2x}{x^2}dx$$ I have tried applying Jordan's lemma, having written $\sin(x)$ as ...
0
votes
1answer
48 views

Integral $\int_{0}^{+\infty}\frac{t \sin(t)}{t^{2}+b^{2}}dt$

I want to solve the integral $$\int_{0}^{+\infty}\frac{t \sin(t)}{t^{2}+b^{2}}dt$$ Which function and contour should I consider ?
1
vote
1answer
135 views

$\int_{0}^{+\infty}\frac{\sin x}{x^{k}(1+x^{2})}dx \ $ via residue calculus

I want to evaluate with calculus of residues $$\int_{0}^{+\infty}\frac{\sin x}{x^{k}(1+x^{2})}dx \ $$ $ k \in \mathbb{N}, k \geq 1$ If $k = 1$ we have $$\int_{0}^{+\infty}\frac{\sin ...
9
votes
1answer
332 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 ...
4
votes
2answers
293 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 ...
0
votes
0answers
36 views

perturbative series expansion of integral via complex integration

Define for real $x>0$ and $\epsilon>0,$ the function $$ f(x,\epsilon):= \int_{\epsilon}^\infty \frac{\mathrm{d}s}{s} \frac{1}{\sinh^2 s/2} e^{-sx}. $$ Question: is it possible to compute ...
5
votes
3answers
164 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 ...
1
vote
2answers
176 views

Integrating $\int_0^{\infty} \frac{dx}{1+x^3}$ using residues.

I want to calculate the integral: $$I \equiv \int_0^{\infty} \frac{dx}{1+x^3}$$ using residue calculus. I'm having trouble coming up with a suitable contour. I tried to take a contour in the shape ...
1
vote
1answer
289 views

Evaluating a Real Improper Integral by Residues

I am having trouble evaluating this improper integral due to its integrand and the singularities that are present. The question reads as Show that ...
3
votes
1answer
116 views

Improper integrals are “not totally Improper”

Question is to evaluate $$\int _{-\infty}^{\infty} \frac{dx}{(x^2+a^2)^2}\text {for } a>0$$ Idea is to calculate this using complex analysis/residue theory/contour integration. Approach is ...
3
votes
2answers
189 views

Evaluating $\int_{-\infty}^\infty \frac{dx}{\cosh(x-a)\cos(2x)}$

I have been asked to evaluate $$\int_{-\infty}^\infty \frac{dx}{\cosh(x-a)\cos(2x)}$$. I'm deliberating on whether this indefinite integral exists or not. The integrand diverges when ...
7
votes
5answers
383 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 ...
5
votes
1answer
111 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 ...
0
votes
3answers
227 views

Cauchy principal value of $\int_{- \infty}^{\infty}e^{-ax^2}\cos(2abx) \,dx$

How do I find out the Cauchy Principal value of $\int_{-\infty}^{\infty}e^{-ax^2}\cos(2abx) \,dx\,\,\,\,\,\,\,\,a,b>0$ using complex integration? The answer is $\sqrt{\frac{\pi}{a}}e^{-ab^2}$, and ...
1
vote
0answers
173 views

Improper integral equal to -pi with square root and Cauchy principal value

I'd like to know if the following proof for the value of $I$ is correct, and if there is a simpler solution to it. Also, I will probably encounter more improper integrals like this in the future, and ...
4
votes
3answers
196 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 ...
4
votes
1answer
103 views

evaluate $\int_0^\infty \dfrac{dx}{1+x^4}$ using $\int_0^\infty \dfrac{u^{p-1}}{1+u} du$

evaluate $\int_0^\infty \dfrac{dx}{1+x^4}$using $\int_0^\infty \dfrac{u^{p-1}}{1+u} du = \dfrac{\pi}{\sin( \pi p)}$. I am having trouble finding what is $p$. I set $u = x^4$, I figure $du = 4x^3 dx$, ...
6
votes
3answers
352 views

A generalized integral need help

I was thinking this integral : $$I(\lambda)=\int_0^{\infty}\frac{\ln ^2x}{x^2+\lambda x+\lambda ^2}\text{d}x$$ What I do is use a Reciprocal subsitution, easy to show that: ...
4
votes
1answer
128 views

Integral using residue theorem (maybe)

I came across the following integral in a book (Kato's Perturbation Theory for Linear Operators, $\S$3.5): $\int_{-\infty}^\infty (a^2+x^2)^{-n/2}\,dx$ where $n$ is a non-negative integer and $a$ is ...
2
votes
1answer
99 views

Residue Calculus Integral computation

I ran into this problem when I was doing some residue computations. For real $a\neq0$, compute, $$I=\int_{-\infty}^{+\infty} \frac{e^{iax}}{(x+i)^3} $$ Be sure to treat both cases when $a<0, ...
1
vote
3answers
348 views

Evalulate $\int_{-\infty}^{\infty}\frac{1}{(1+x^{2n})^2}dx$ by using residue theorem

I know the answer of the integral $$\int_{-\infty}^{\infty}\frac{1}{1+x^{2n}}dx=\frac{\pi}{n\sin\left(\frac{\pi}{2n}\right)}$$where $n\in\mathbb{N}$. But how to evalulate ...
1
vote
0answers
77 views

What is suitable contour shape for $\int_0^\infty\dfrac{b^2+2ab+k}{b(b^2+ab+l)}e^{bx}~db$

$\int_0^\infty\dfrac{b^2+2ab+k}{b(b^2+ab+l)}e^{bx}~db$ . What kind of contour is suitable for this integral?
8
votes
1answer
1k 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 ...