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

learn more… | top users | synonyms

4
votes
3answers
84 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
83 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
118 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
299 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
424 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
968 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
260 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
98 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
145 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
195 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
155 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
113 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
87 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
68 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
2answers
148 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
554 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
77 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
62 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
193 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
103 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
1answer
120 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
255 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
654 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
3answers
155 views

Calculate $\int_{-\infty}^{+\infty}e^{-\frac{(x-it)^2}{2}}dx$ using contour integration

When one wants to calculate the characteristic function of a random variable which is of normal distribution, things boil down to calculate: $$\int_{-\infty}^{+\infty}e^{-\frac{(x-it)^2}{2}}dx$$ There ...
4
votes
1answer
194 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
384 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
1answer
50 views

Calculating a contour integral

I want to evaluate the integral $$\int_{\gamma} \sin{(2z)} \ {\rm d}z$$ where $\gamma$ is the line segment joining the point $i+1$ to the point $-i$. Thus $\gamma(t) = -i+t(2i+1)$ for $0\le t\le1$. ...
4
votes
1answer
167 views

Integral Using Harmonic Functions

Evaluate the integral: $$\int^{2 \pi}_0 \dfrac{\cos^2 \theta}{|2e^{i\theta}-z|^2} \, d \theta \qquad \mbox {when} \, |z| \neq 2.$$ Now, I thought about trying to change this to look like a Poisson ...
4
votes
1answer
64 views

Show $\int_{\gamma}e^{iz}e^{-z^2}dz$ same value on every line parallel to $\mathbb{R}$

From an old qualifier: Show that $$\large\int_{\gamma}e^{iz}e^{-z^2}\mathrm dz$$ has the same value on every straight line path $\gamma$ parallel to the real axis. Justify the estimates involved. My ...
4
votes
1answer
61 views

Complex integral, correct?

I am supposed to do the integral $$ \int_{\gamma_2} \frac{\sin(z)}{z+\frac{i}{2}} dz$$ where $\gamma_2:[-\pi, 3\pi] \rightarrow \mathbb{C}$ , $\gamma_2(t)=\exp(it)$ for $ t\in [-\pi,\pi]$, ...
4
votes
1answer
152 views

How do you integrate $\int_0^\infty \exp(it^k)\,\mathrm dt$ for $k \in \Bbb N$?

My problem is with the integral $$\int^\infty_0 e^{it^k}\,\mathrm dt$$ with $k\in\mathbb{N}$. Somehow it can be evaluated by use of Cauchy's theorem. But I don't see how. The best thing I can ...
4
votes
1answer
177 views

Residue of $p.v.\int_{-\infty}^{\infty}\frac{e^{2x}}{\cosh(\pi x)}dx=\text{sec}1$

Show that $$p.v.\int_{-\infty}^{\infty}\frac{e^{2x}}{\cosh(\pi x)}dx=\text{sec}1$$ by integrating $\frac{e^{2z}}{\cosh(\pi z)}$ around rectangles with vertices at $z=\pm p,p+i,-p+i.$ I asked ...
4
votes
1answer
411 views

Contour integration of $\int_{-\infty}^{\infty}e^{iax^2}dx$

Consider the following integral: $$\int_{-\infty}^{\infty}e^{iax^2}dx$$ Here I believe we have to consider the two cases when $a<0$ and $a>0$, as they need different contours. For $a>0$ ...
4
votes
1answer
1k views

Laplace transform of the Bessel function of the first kind

I can't figure out why my evaluation of $\displaystyle \int_{0}^{\infty} J_{n}(bx) e^{-ax} \ dx \ (a,b >0, \ n=0,1,2, \ldots)$ is off by a factor of $ \displaystyle \frac{1}{b}$. $$ \begin{align} ...
4
votes
1answer
354 views

Branch Cut Issues

I'm trying to evaluate what seems to be a straightforward contour integral: $$I=\int_{\gamma} \frac{dz}{\alpha + \beta z} $$ where $\gamma (t) = e^{-it}$, $t \in \left[ 0,\pi\right]$, $\alpha, \beta ...
4
votes
1answer
23 views

Integral principal value with $\cos$ and $x^2$

Could you tell me how to solve this integral? $$\int_0^{\infty} \frac{\cos x -1}{x^2}dx$$ I think I should focus on this integral $$\int_{\Gamma} \frac{e^{iz}-1}{z^2+ \varepsilon^2}$$ where ...
4
votes
1answer
62 views

inverse laplace transform by using complex integral

given function $$f(s)=\frac{1}{s}\frac{\sqrt{s}-1}{\sqrt{s}+1}$$ and $$\int_{0}^{\infty}{\frac{e^{-xt}}{\sqrt{x}(x+1)}dx=\pi e^t {erfc}(\sqrt{t})}$$ my steps: ...
4
votes
2answers
89 views

Establish $\int_0^{\infty} \frac{x^a}{x^2 + b^2}dx = \frac{\pi b^{a-1}}{2 \cos(\pi a /2)}$ when $-1 < a < 1$

My attempt at a solution: (this is homework, btw) Let $f(z) = \frac{z^a}{z^2 + b^2}dz$ then the singularities of $f$ occur at $\pm ib$. $$ Res(f; ib) = \frac{z^a}{z + ib} \biggr |_{ib} = ...
4
votes
2answers
127 views

show that $\int_{0}^{\pi/2}\tan^ax \, dx=\frac {\pi}{2\cos(\frac{\pi a}{2})}$

show that $$\int_{0}^{\pi/2}\tan^ax \, dx=\frac {\pi}{2\cos(\frac{\pi a}{2})}$$ I think we can solve it by contour integration but I dont know how. If someone can solve it by two way using complex ...
4
votes
1answer
106 views

Real integral $ \int_{-\infty}^{\infty} \frac{dx}{1+x^2} $ with the help of complex friends

I have to solve the integral $$ \int_{-\infty}^{\infty} \frac{dx}{1+x^2} $$ by doing this: Given a rectangle that is defined by the points $ r+i, -r+i,-r-i,r-i$, $r>0$ and $\gamma_r$ is a closed ...
4
votes
2answers
318 views

Contour integration using Cauchy's integral formula

I need to show that $$\int_{-\infty}^\infty\frac{\sin^2(x)}{x^2+1}dx=\frac{\pi}{2}\left(1-\frac{1}{e^2}\right)$$ but I don't really know why I'm not getting the result using contour integration ...
4
votes
1answer
145 views

Period Homomorphisms and closed 1-forms

This is from Otto Forster's "lectures on Riemann Surfaces", on integration of forms. Let $\Gamma = \alpha_1 \mathbb{Z} + \alpha_2 \mathbb{Z}$ be a lattice in $\mathbb{C}$ (i.e. $\alpha_i \in ...
4
votes
1answer
49 views

Contour integral $\int_{|z|=1}\frac{2z^2+z}{z^2-1}\, dz$ using residues

I am trying to evaluate the contour integral $$\int_{|z|=1}\frac{2z^2+z}{z^2-1}\, dz.$$ In this case the two singular points lie on the boundary (on the contour). So do I count the residues at this ...
4
votes
1answer
47 views

How can I find the Cauchy Principal Value of this integral using complex analysis?

I'm supposed to solve the real integral using a contour integral (The Cauchy Principal Value). Can someone give me a hand? I cannot seem to be able to do it... This is what I've tried so far: I ...
4
votes
1answer
47 views

Complex analysis $\int^{2\pi}_0 \frac{d \theta}{(2-\sin \theta)^2}$

how do I compute $$\int^{2\pi}_0 \frac{d \theta}{(2-\sin \theta)^2}$$ I tried substituting $z=e^{i\theta}$ but it just got very messy..
4
votes
1answer
99 views

Contour integration of $\int_{-\infty}^{\infty}\frac {\sin^3 x}{x^3} dx$: where are the singularities?

I have just begun to study complex analysis and I'm trying to calculate $$ \int_{- \infty}^{\infty} \frac {\sin^3 x}{x^3} dx $$ with the "help" of an exercisebook. I have followed these steps: ...
4
votes
1answer
211 views

Laplace transform of and impulse sampled function using “frequency” convolution

This is a long question, but assume we have this: The book uses the frequency convolution theorem to solve this problem. To solve the integral, it uses a contour + residue theorem to solve it. The ...
4
votes
2answers
176 views

Contour Integration on a finite domain

Evaluate by contour integration $$\int_{0}^{1}\frac{dx}{(x^2-x^3)^\frac{1}{3}}$$ I am having trouble selecting the right contour for the problem as the usual problems involve selecting real axis ...
4
votes
1answer
96 views

What is the difference between integrals and contour integrals?

I understand integrals but what are contour integrals?
4
votes
1answer
138 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 ...