Questions on the evaluation of integrals along a locus in the complex plane.

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5
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1answer
328 views

$\mathcal{B}^{-1}_{s\to x}\{e^{as^2+bs}\}$ and $\mathcal{L}^{-1}_{s\to x}\{e^{as^2+bs}\}$ , where $a\neq0$

http://en.wikipedia.org/wiki/Integral_transform#Table_of_transforms claims than the integral form of inverse bilateral Laplace transform and inverse Laplace transform are both the same. But are they ...
5
votes
1answer
55 views

Are these the correct residues?

$$\int_C \frac{z+1}{z^2-2z} dz$$ for the circle of $\lvert z \rvert = 3 $. Poles are obviously at $ z = {0,2}$. Can I calculate the residues by viewing the fraction in the integral as either $$\int_C ...
5
votes
2answers
126 views

Convergence of a line integral along semi-circular arc

There is a line integral in a form, $$\int_\mathrm{arc} \frac{\exp(iz)}{z^2+1} \, dz$$ "arc" is a semi-circular line with radius $R$ on the upper half complex plane. and i know that the integral ...
5
votes
1answer
157 views

Help with identity in complex contour

I am dealing with an integral in the complex plane, in particular I want to transform $$ \int_{0}^{2\pi} \frac{d\phi}{\sqrt{1+b^2 -2b \cos \phi}} $$ into another integral near the branch cuts for the ...
5
votes
1answer
1k views

Use rectangular contour to integrate $\sin(ax)/(\exp(2\pi x)-1)$

I have been self-studying CA and find it very interesting. So, working through problems in a book I have, I ran across $$\int_{0}^{\infty}\frac{\sin(ax)}{e^{2\pi ...
5
votes
0answers
57 views

Integral with contours

I want to evaluate the integral $\displaystyle \int_0^\infty \dfrac{\ln x}{e^x+1}\,{\rm d} x$ using contour integration. At first I though using a rectangular. Problem is that I cannot establish the ...
5
votes
0answers
56 views

Solving an integral (using Cauchy contour integral?)

I need to solve this integral: \begin{equation} f(t)=\int_0^\infty x^2 \sqrt x \left( e^{a x} -1\right)^{-1/2} \frac{e^{i(b-x)t}-1}{b-x} dx \end{equation} where $a$ and $b$ are real, positive ...
5
votes
2answers
92 views

Complex Numbers - Finding Roots

Hi there I was wondering if someone could help me? I am struggling to find the roots of the polynomial $z^4+2z+3=0$ It is not a quadratic so can't use the quadratic formula so am not quite sure ...
5
votes
0answers
63 views

Clarification of Contour Integration [duplicate]

I apologise if this seems like an elementary and silly question, but I am confused about the integral $$I=\int^{\infty}_{-\infty}\frac{\cos{x}}{1+x^2}dx=\frac{\pi}{e}$$ If I consider a semicircular ...
5
votes
0answers
704 views

contour integration of a function with two branch points .

Many of us have seen the evaluation of the integral $$\int^{\infty}_0 \frac{dx}{x^p(1+x)}\, dx \,\,\, 0<\Re(p)<1$$ It can be solved using contour integration or beta function . I thought of ...
4
votes
7answers
176 views

show that $\int_{0}^{\pi/2}\ln(\tan x)dx=0$

show that $$\int_{0}^{\pi/2}\ln(\tan x)dx=0$$ using two ways The first with real analysis and the second with contour integration
4
votes
3answers
412 views

show that $\int_{0}^{\infty}\frac{x\cos ax}{\sinh x}dx=\frac{\pi^2}{4} \operatorname{sech}^2 \left(\frac{a\pi}{2}\right) $

show that $$\int_{0}^{\infty}\frac{x\cos ax}{\sinh x}dx=\frac{\pi^2}{4} \operatorname{sech}^2\left(\frac{a\pi}{2}\right) $$ also I think we can solve it by contour integration but how and its better ...
4
votes
3answers
133 views

Compute $\int_{0}^{\infty}\frac{x \log(x)}{(1+x^2)^2}dx$

Given $$\int_{0}^{\infty}\frac{x \log(x)}{(1+x^2)^2}dx$$ I couldn't evaluate this integral. My only idea here was evaluating this as integration by parts. \begin{align} \int\frac{x ...
4
votes
3answers
253 views

when integrating a Laurent series $f(z)=\sum\limits_{j=-\infty}^{\infty}a_j(z-z_0)^j$, why do all terms for $j\neq-1$ dissappear?

In my complex analysis book we are looking at a Laurent series expression for $f(z)$ around a singularity $z_0$ that converges to $f(z)$ for all $z\in C, z\neq z_0$. The Laurent series looks like ...
4
votes
2answers
162 views

Improper integrals with singularities on the REAL AXIS (Complex Variable)

I'm having some troubles when I try to solve improper integrals exercises that have singularities on the real axis. I have made a lot of exercises where singularities are inside a semicircle in the ...
4
votes
3answers
1k 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 ...
4
votes
2answers
350 views

Contour integration for functions with residues that form an infinite oscillating sequence

I would like to evaluate some complicated integrals involving the hyperbolic secant, but the extension of the usual contour integration evaluation using the residue theorem isn't clear to me. I've ...
4
votes
3answers
133 views

Evaluation of integral $\int_{0}^{\infty}\frac{\sin x}{x\left ( 1+x^2 \right )^2}\,{\rm d}x$

I'm trying to evaluate the following integral: $$\mathcal{J}=\int_{0}^{\infty}\frac{\sin x}{x\left ( 1+x^2 \right )^2}\,{\rm d}x$$ Well there are $3$ poles , one lying on the real line the other on ...
4
votes
2answers
155 views

Why is the pole generally outside the contour loop when its outside the contour loop in 2D?

The following contour integral is path dependent with the following results \begin{align} \oint_C\dfrac{dz}{z} = \begin{Bmatrix} 2\pi i && \text{when $z=0$ is inside C} \\ 0 && ...
4
votes
1answer
362 views

Integrate $ \int_0^{\pi/2} \frac{x^{2p}}{1+\cos^2x}dx $

Hi I am trying to come up with a closed form expression for $$ \int_0^{\pi/2} \frac{x^{2p}}{1+\cos^2x}dx,\quad p\geq 0. $$ I am interested in this general case in terms of p. For small p, we can ...
4
votes
3answers
110 views

$\int_0^\infty \frac{\log(1+x^2)}{x^2} dx $ using contour integration

I am trying to evaluate $$\int_0^\infty \frac{\log(1+x^2)}{x^2} dx $$ by using contour integration. It is possible to compute this integral using real techniques; integration by parts yields the ...
4
votes
2answers
89 views

On the value of $e^{ix}$ at $\pm \infty$

Consider the integral $$ \int_{-\infty}^{+\infty} e^{ix} \, dx.$$ Integrating, we have $$\left[-ie^{ix}\vphantom{\frac11}\right]_{-\infty}^{+\infty},$$ and we need to evaluate the limits of $e^{ix}$ ...
4
votes
2answers
151 views

$\int_{0}^{\infty} \frac{\cos(x)}{1+x^2} dx$ and $\int_{0}^{\infty} \frac {\ln(x)}{x^2+b^2} dx$

Prove that $$\int_{0}^{\infty} \frac{\cos(x)}{1+x^2} dx = \frac {\pi}{2e}$$ My approach would be $$\lim_{n \to \infty} \int_{0}^{n} \frac{\cos(x)}{1+x^2} dx$$ and evaluate the limits of the sine and ...
4
votes
5answers
320 views

Integral, definite integral

How can we prove $$ \int_0^1 \frac{\ln x \cdot \ln(1+x)}{1+x}dx=-\frac{\zeta(3)}{8}? $$ This has been one of the integrals that came out of an integral from another post on here, but no solution to ...
4
votes
2answers
465 views

Integrating $\int_0^\infty \sin(1/x^2) \, \operatorname{d}\!x$

How would one compute the following improper integral: $$\int_0^\infty \sin\left(\frac{1}{x^2}\right) \, \operatorname{d}\!x$$ without any knowledge of Fresnel equations? I was thinking of using ...
4
votes
2answers
1k views

Complex part of a contour integration not using contour integration

A propos of a user's comment on this question, quoting Feynman to the effect that some integrals are only possible using contour integration, I wonder what the simplest example of such an integral ...
4
votes
2answers
263 views

Evaluation of the integral $\int_0^\infty \left(\frac{\pi^2}{4}-x^2\right)^{-2}\cdot\frac{\pi^2}{4}\cos^2 x\,dx$

What are the steps to evaluate the following definite integral? (Answer provided) $$\int_0^\infty {{\pi^2\over 4}\cos^2x\over\left({\pi^2\over4}-x^2 \right)^2} dx={\pi\over 4}$$?
4
votes
2answers
107 views

Methods of evaluating $\int_0^{\infty}\frac{{\rm d}x}{x^2+1}$

Methods of evaluating $$\int_0^{\infty}\frac{{\rm d}x}{x^2+1}$$ Firstly i know that directly: $$\int_0^{\infty}\frac{{\rm d}x}{x^2+1}=\arctan x\Bigg|_{0}^{\infty}=\frac{\pi}2$$ Also we can use the ...
4
votes
1answer
150 views

Using Cauchy integral formula to calculate $\int_\gamma \frac{\cos{z}}{z^n}$

Let $\gamma(\vartheta)=\mathrm{e}^{i\vartheta},\,\vartheta\in[0,2\pi]$, and consider the integral $$I(n)=\int_\gamma \frac{\cos{z}}{z^n},$$ where $n\in \{0,2,4,6,...\}$. Is there any way to prove ...
4
votes
4answers
236 views

Intuitive reason for why many complex integrals vanish when the path is “blown-up”?

It is a standard trick for evaluating difficult integrals along the real line to consider a closed-contour and "blow-up" the complex part till it vanishes, leaving us with the residues picked up along ...
4
votes
4answers
159 views

Contour integration of $\int \frac{dx} {(1+x^2)^{n+1}}$

I want to compute $$ \int_{-\infty}^\infty \frac 1{ (1+x^2)^{n+1}} dx $$ for $n \in \mathbb N_{\geq 1}$. If I let $$ f(z) := \frac 1 {(z+i)^{n+1}(z-i)^{n+1}} $$ then I see that $f$ has poles of ...
4
votes
1answer
126 views

Is it true that a complex function has a global antiderivative if and only if it integrates to zero over every closed curve?

I am somehow thinking that these properties must be equivalent, unfortunately I do not know a theorem that says it: $f$ has a global antiderivative iff the line integral $ \int_{\gamma}f$ over every ...
4
votes
1answer
271 views

Integrate: $\int_0^1 \frac{1}{\sqrt[3]{x^2 - x^3}}dx$

How to integrate using Residue theorem. $$\int_0^1 \frac{1}{\sqrt[3]{x^2 - x^3}}dx$$ How do I choose my branch-cut particularly? I was reading this article on wikiepdia and I think it is related. ...
4
votes
1answer
105 views

Evaluating $\int_0^{\infty} \frac{\sqrt{x}}{x^2+2x+5} dx$ using complex analysis

how do I compute $$\int_0^{\infty} \frac{\sqrt{x}}{x^2+2x+5} dx$$ with complex analysis? I feel like im calculating the residue wrong and I cant get to the answer correctly. I tried to branch cut ...
4
votes
3answers
69 views

Can $\frac1{z^2}$ be integrated on $|z+i|=\frac32$ using Cauchy's theorem?

$$ \begin{align} \int_{|z+i|=\frac{3}{2}}\frac{1}{z^2}dz=0 \end{align} $$ Is it safe to say the Integral is $0$ due to cauchy's Theorem? Does this apply for any $z_0$ that lies inside the circle ...
4
votes
1answer
417 views

Understanding Dogbone contour example

I am trying to understand example VI on the wikipedia page http://en.wikipedia.org/wiki/Methods_of_contour_integration, but one particular point has mystified me for hours. After it is shown that ...
4
votes
2answers
151 views

Use a change of contour to show that $\int_0^\infty \frac{\cos{(\alpha x)}}{x+\beta}dx = \int_0^\infty \frac{te^{-\alpha \beta t}}{t^2 + 1}dt$

A problem from an old qualifying exam: Use a change of contour to show that $$\int_0^\infty \frac{\cos{(\alpha x)}}{x+\beta}dx = \int_0^\infty \frac{te^{-\alpha \beta t}}{t^2 + 1}dt,$$ provided ...
4
votes
3answers
597 views

Evaluate $\int_{0}^{\pi} \frac{d\theta}{(2+\cos\theta)^2}$

How can one evaluate $\displaystyle\int_{0}^{\pi} \frac{d\theta}{(2+\cos\theta)^2}$? My attempt: $$\int_{0}^{\pi} \frac{d\theta}{(2+\cos\theta)^2} = \frac{1}{2}\int_{0}^{2\pi} ...
4
votes
1answer
122 views

A bessel function integral

$$\int_{-\infty}^{\infty} dy \frac{J_1 \left ( \pi\sqrt{x^2+y^2} \right )}{\sqrt{x^2+y^2}} = \frac{2 \sin{\pi x}}{\pi x} $$ How do I show this?
4
votes
1answer
102 views

Complex integration $\int_{-\pi}^{\pi} \frac{\sin^2 t}{3+\cos t}dt$

I'm trying to evaluate the integral $$\int_{-\pi}^{\pi} \frac{\sin^2 t}{3+\cos t}dt$$ using complex numbers. Meaning, instead of calculating $$\int_{-\pi}^{\pi} \frac{\sin^2 t}{3+\cos t}dt,$$ I want ...
4
votes
1answer
73 views

Integrate using residue theorem

This was a question on my complex analysis take home final. Since the semester is over and grades have been posted I believe I can post it now. Let $a > 0$ and $b > 0$. Verify that ...
4
votes
1answer
207 views

Evaluate Complex Integral with $\frac{\Gamma(\frac{s}{2})} {\Gamma\big({\beta +1\over 2} - {s\over 2}\big)}$

I am proving this integral: $$ \int_{c\ -\ j\infty}^{c\ +\ j\infty} \left(\,x^{-1}\sigma\beta^{1 \over 2}\,\right)^{s}\ \Gamma\left(\,s \over 2\,\right) \Gamma\left(\,{\beta +1 \over 2} - {s \over ...
4
votes
1answer
160 views

Solving this complicated integral using the Residue Theorem

The following is an integral I am trying to evaluate $$I= \int_{-\infty}^\infty f(s) \, ds = \int_{-\infty}^\infty \frac{\frac{1}{(1- \ \ 2 \pi j s )^{m}}-1}{2\pi j s }\ e^{-2\pi j s \ \theta}\ ds ...
4
votes
2answers
95 views

How to integrate $\int_C{\frac{\sin\pi z}{(z^2-1)^2}}dz$, where $C: |z-1|=1$ using Cauchy's formula?

How can evaluate $$\int_C{\frac{\sin\pi z}{(z^2-1)^2}}dz$$, where $$C: |z-1|=1$$ by using Cauchy's formula. I have to use Cauchy's formula. Cauchy's formula $$f(z_0)=\frac{1}{2\pi ...
4
votes
1answer
122 views

1 dimensional integral, definite integral.

Trying to show $$ \int_0^1 \frac{\ln x \cdot \ln(1+x)}{1-x}dx=-\frac{1}{4}\pi^2 \ln(2)+\zeta(3). $$ I am unsure how to approach this integral as I do not know how to use a power series representation ...
4
votes
4answers
260 views

Using residue theory, show that $\oint_C\frac{e^{z/2}}{1+e^z}dz = 2\pi$

Using residue theory, show that $$\oint_C\frac{e^{z/2}}{1+e^z}dz = 2\pi$$ I've been attempted this problem using residue theory and the Cauchy integral formula that over a closed contour ...
4
votes
1answer
750 views

Contour integral with branch cut

This is a question based on the method here: http://en.wikipedia.org/wiki/Methods_of_contour_integration#Example_.28V.29_.E2.80.93_the_square_of_the_logarithm The author chose a contour which ...
4
votes
1answer
210 views

Evaluating the contour integral: $\oint_C \frac{\sin 2z}{(6z-\pi)^3}dz$

I am trying to evaluate the following integral, but don't know how to take the coefficient of $z$ out of the parenthesis to get it into the Cauchy integral form. Any help is appreciated. $$ \oint_C ...
4
votes
2answers
425 views

Contour Integral Question

I'm working through a contour integral question, which is rounded off by finding the integral: $$\int^{\infty}_{0} \frac{x-\sin(x)}{x^3} dx$$ I have already shown that the residue at $0$ of the ...
4
votes
2answers
84 views

Computing $\int_{0}^{+\infty}\frac{\log(x)}{\sqrt x(1+{x^2})}dx$.

I would like to compute the following integral : $$\int_{0}^{+\infty}\frac{\log(x)}{\sqrt x(1+{x^2})}dx$$ using Residue theorem. I took the contour corresponding to half of the "donuts" ...