Questions on the evaluation of integrals along a locus in the complex plane.
105
votes
3answers
5k views
The Integral that Stumped Feynman?
In "Surely You're Joking, Mr. Feynman!," Nobel-prize winning Physicist Richard Feynman said that he challenged his colleagues to give him an integral that they could evaluate with only complex methods ...
17
votes
3answers
369 views
Evaluating the integral $\int_{-\infty}^\infty \frac {dx}{\cos x + \cosh x}$
Many recent questions have been asked here similar to this integral
$$\int_{-\infty}^\infty \frac {dx}{\cos x + \cosh x} = 2.39587\dots$$
whose "closed form" I cannot seem to figure out. I have ...
13
votes
2answers
277 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.
8
votes
4answers
185 views
contour integration of logarithm
I must compute the following integral
$$\displaystyle\int_{0}^{+\infty}\frac{\log x}{1+x^3}dx$$
Can someone suggest me the right circuit in the complex plane over which to do the integration? I ...
8
votes
2answers
368 views
What contour should be used to evaluate $\int_0^\infty \frac{\sqrt{t}}{1+t^2} dt$
Could anyone help me decide what contour to use to evaluate this integral?
$$\int_0^\infty \frac{\sqrt{t}}{1+t^2} dt$$
So we have simple poles at $i$,$-i$. Why does using a quarter of a circle in ...
8
votes
1answer
870 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 ...
7
votes
2answers
287 views
Showing that $\int_0^1 \log(\sin \pi x)dx=-\log2$
I need help with a textbook exercise (Stein's Complex Analysis, Chapter 3, Exercises 9). This exercise requires me to show that
$$\int_0^1 \log(\sin \pi x)dx=-\log2$$
A hint is given as "Use the ...
7
votes
1answer
283 views
using contour integration, or other means, is there a way to find a general form for $\frac{\sin^{n}(x)}{x^{n}}$
While studying some CA, I have ran across methods of solving $$\int_0^\infty \frac{\sin x}{x} \, dx, \;\ \int_0^\infty \frac{\sin^2 x}{x^2} \, dx, \;\ \int_0^\infty \frac{\sin^3 x}{x^{3}} \, dx.$$
Is ...
7
votes
2answers
365 views
Help with integrating $\displaystyle \int_0^{\infty} \dfrac{(\log x)^2}{x^2 + 1} dx$ - contour integration?
George Arfken's book: Mathematical Methods for Physicists has the following problem in a chapter on contour integration:
$\displaystyle \int_0^{\infty} \dfrac{(\log x)^2}{x^2 + 1} dx$.
Their ...
6
votes
1answer
222 views
If $(f'_n)$ converges uniformly, does $(f_n)$ necessarily converge uniformly?
I've been studying complex analysis problems, and get stuck on the following:
Let $D \subseteq \mathbb{C}$ be a domain (open connected set) and $z_0 \in D$. Assume that $(f_n)$ is a sequence of ...
6
votes
2answers
258 views
Is it possible to evaluate $ \int_0^1 x^n \, dx$ by contour integration?
It's been quite a time since I had the complex analysis course. The thing is now I don't know the answer to the following simple question:
Is it possible to find
$$ \int_0^1 x^n \, dx$$
using the ...
6
votes
0answers
162 views
Integrate $\ln(x^2+1)/(x^2+1)$ [duplicate]
How to evaluate $$\int_0^\infty \frac{\ln(x^2+1)}{x^2+1} \mathrm{d}x$$ using complex analysis?
I've spent ages trying to think of some clever contour integral which will give it, but I can't seem to ...
6
votes
1answer
114 views
Is this a correct way to calculate $\int_{-\infty}^\infty\frac{x\sin(\pi x)}{x^2+2x+5}dx?$
I have this integral to calculate: $$I=\int_{-\infty}^\infty\frac{x\sin(\pi x)}{x^2+2x+5}dx.$$
I think I have done it, but I would like to make sure my solution is correct.
I take the function ...
5
votes
4answers
325 views
Evaluating $\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx$
How would I go about evaluating this integral?
$$\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx.$$
What I've tried so far: I tried a semicircular integral in the positive imaginary part of the complex ...
5
votes
3answers
80 views
Evaluating $ \int_{-\infty}^{\infty} \frac{\cos \left(x-\frac{1}{x} \right)}{1+x^{2}} \ dx$
I'm curious about the proper way to evaluate $\displaystyle\int_{-\infty}^{\infty} \frac{\cos \left(x-\frac{1}{x} \right)}{1+x^{2}} \ dx = \text{Re} \int_{\infty}^{\infty} \frac{e^{i(x- ...
5
votes
2answers
159 views
Where have I gone wrong? Contour integration $\int_{-a}^a {u\over 1+u+u^2} du$ as $a\to \infty$
I would like to integrate $\int_{-a}^a {u\over 1+u+u^2} du$ as $a\to \infty$.
So I thought I might use the residue theorem. In the complex plane, the singularities occur at $z=e^{\pm i2\pi\over 3}$. ...
5
votes
2answers
182 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 ...
5
votes
2answers
146 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 ...
5
votes
3answers
137 views
Evaluate the integral $\int_{0}^{\pi} \frac{d\theta}{(2+cos(\theta))^2}$
$\int_{0}^{\pi} \frac{d\theta}{(2+cos(\theta))^2}$
My attempt:
$\int_{0}^{\pi} \frac{d\theta}{(2+cos(\theta))^2} = \frac{1}{2}\int_{0}^{2\pi} \frac{d\theta}{(2+cos(\theta))^2}$
To find the ...
5
votes
2answers
224 views
Calculating the residues of $f(z)=\frac{e^{az}}{1+e^z}$
Let $$f(z)=\frac{e^{az}}{1+e^z}$$
where $0<a<1$
Can anyone help me find the residues of this function?
So $$e^z+1=0 \Rightarrow z=i\pi(1+2k)$$ where $k\in \mathbb{Z}$, so these are simple ...
5
votes
1answer
66 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 ...
5
votes
1answer
320 views
Evaluating $\int\limits_0^\infty \frac{\log x} {(1+x^2)^2} dx$ with residue theory
I need a little help with this question, please!
I have to evaluate the real convergent improper integrals using RESIDUE THEORY (vital that I use this), using the following contour:
...
5
votes
1answer
241 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
424 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
$$\int_{0}^{\infty}\frac{\sin(ax)}{e^{2\pi ...
4
votes
3answers
192 views
$\int_0^\infty\frac{\log x dx}{x^2-1}$ with a hint.
I have to calculate $$\int_0^\infty\frac{\log x dx}{x^2-1},$$
and the hint is to integrate $\frac{\log z}{z^2-1}$ over the boundary of the domain $$\{z\,:\,r<|z|<R,\,\Re (z)>0,\,\Im ...
4
votes
3answers
135 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
3answers
450 views
How to measure the volume of rock?
I have a object which is similar to the shape of irregular rock like this
I would like to find the volume of this. How to do it?
If I have to find the volume, what are the things I would need. eg., ...
4
votes
2answers
518 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
1answer
214 views
Integration Problem
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
155 views
Integrate: $\int_0^{\infty} \frac{\sin (ax)}{e^{\pi x} \sinh(\pi x)}dx$
How to evaluate the following
$$\int_0^{\infty} \frac{\sin (ax)}{e^{\pi x} \sinh(\pi x)} dx $$
Given hints says to construct a rectangle $0\to R\to R+i\to i \to 0$ and consider $\displaystyle ...
4
votes
1answer
111 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.
$$ \oint_C ...
4
votes
1answer
277 views
Branch Cut Issues
I'm trying to evaluate what seems to be a straightforward contour integral:
$$I=\int_{\gamma} \frac{dz}{\alpha + \beta z} $$
where $\gamma (t) = e^{-it}$, $t \in \left[ 0,\pi\right]$, $\alpha, \beta ...
4
votes
2answers
128 views
Contour integration using Cauchy's integral formula
I need to show that
$$\int_{-\infty}^\infty\frac{\sin^2(x)}{x^2+1}dx=\frac{\pi}{2}\left(1-\frac{1}{e^2}\right)$$
but I don't really know why I'm not getting the result using contour integration ...
4
votes
1answer
80 views
Period Homomorphisms and closed 1-forms
This is from Otto Forster's "lectures on Riemann Surfaces", on integration of forms.
Let $\Gamma = \alpha_1 \mathbb{Z} + \alpha_2 \mathbb{Z}$ be a lattice in $\mathbb{C}$ (i.e. $\alpha_i \in ...
4
votes
1answer
81 views
Evaluating $\int_{0}^{\infty} \frac{2 \cos x \ln x + \pi \sin x}{x^2+4} \ dx$
I want to show that $\displaystyle\int_{0}^{\infty} \frac{2 \cos x \ln x + \pi \sin x}{x^2+4} \ dx = \frac{\pi \ln 2}{2e^{2}}$.
The recommendation is to let $\displaystyle f(z) = \frac{e^{iz} ...
4
votes
1answer
47 views
What is the difference between integrals and contour integrals?
I understand integrals but what are contour integrals?
4
votes
1answer
99 views
Integral using residue theorem (maybe)
I came across the following integral in a book (Kato's Perturbation Theory for Linear Operators, $\S$3.5):
$\int_{-\infty}^\infty (a^2+x^2)^{-n/2}\,dx$
where $n$ is a non-negative integer and $a$ is ...
4
votes
1answer
43 views
Evaluating $\lim_{n \to \infty} \oint_{ |z| = 1/4} \frac{1}{(4 z(1-z))^n} \frac{\mathrm{d}z}{z (1-2 z)} = \frac{1}{2}$
While working on an earlier question involving $\sum_{j=0}^n \binom{n+j-1}{j} \frac{1}{2^{n+j}}$ I rewrote the sum as a contour integral, using generating functions:
$$
\sum_{j=0}^n ...
4
votes
2answers
520 views
Help with an irregular integral
I am looking for help with doing the following integral :
$$\frac{1}{2\pi i}\int_{1}^{\infty}\ln\left(\frac{1-e^{-2\pi i x}}{1-e^{2\pi i x}} \right )\frac{dx}{x\left(\ln x+z\right)}\;\;\;\;z\in ...
4
votes
0answers
74 views
Contour integration of $\int_{-\infty}^{\infty}e^{iax^2}dx$
Consider the following integral:
$$\int_{-\infty}^{\infty}e^{iax^2}dx$$
Here I believe we have to consider the two cases when $a<0$ and $a>0$, as they need different contours. For $a>0$ ...
4
votes
0answers
185 views
contour integral with singularity on the contour
I want to compute the following integral
$$\oint_{|z|=1}\frac{\exp \left (\frac{1}{z} \right)}{z^2-1}\,dz$$
The integrand has essential singularity at the origin, and $2$-poles at $\pm 1$,which lie ...
3
votes
3answers
162 views
Improper integration involving complex analytic arguments
I am trying to evaluate the following:
$\displaystyle \int_{0}^{\infty} \frac{1}{1+x^a}dx$, where $a>1$ and $a \in \mathbb{R}$/
Any help will be much appreciated.
3
votes
4answers
135 views
Evaluating a complex contour
I need to show the following result:
$$
\int_{-\infty}^\infty \frac{1}{(1+x^2)^{n+1}}dx\, = \frac{1\cdot 3\cdot\ldots\cdot(2n-1)}{2\cdot 4\cdot\ldots\cdot(2n)}\pi
$$
With n=1,2,3,...
This function ...
3
votes
3answers
76 views
Find $I:=\lim\limits_{R\to \infty}\int\limits_{-R}^R \frac{x \sin(3x)}{x ^2+4}dx$ using residues
Find $I:=\lim\limits_{R\to \infty}\int\limits_{-R}^R \frac{x \sin(3x)}{x ^2+4}dx$ using residues. Let $f(z)= \frac{z \sin(3z)}{z ^2+4}$. First define two contours:
$$\Gamma_1: z=t \text{ where } ...
3
votes
2answers
216 views
using contour integration
I am trying to understand using contour integration to evaluate definite integrals. I still don't understand how it works for rational functions in $x$. So can anyone please elaborate this method ...
3
votes
2answers
60 views
Complex integration help
The integral given is
$$\int_{-\infty}^{\infty} \frac{\cos(x)-1}{x^2}\,dx $$
Ok, so, I've used the upper semi circular contour with the function
$$f(z) = \frac{e^{iz}-1}{z^2}$$
Now the residue I ...
3
votes
3answers
104 views
ln and rational function using contour integration
I was playing around with $\displaystyle \int_{0}^{1}\frac{\ln^{2}(x)}{x^{2}-x+1}dx=\frac{1}{2}\int_{0}^{\infty}\frac{\ln^{2}(x)}{x^{2}-x+1}dx$ and managed to solve it using real methods via digamma ...
3
votes
3answers
298 views
contour integral computations
Let $C$ be the boundary of the square whose vertices are $1+i$, $1-i$,
$-1 + i$ and $-1 -i$. Suppose that $C$ is oriented counterclockwise. How to compute
a) $$\int_C \frac{e^z}{z-1/2} \, dz$$
b) ...
3
votes
1answer
68 views
contour integration of $f(z) = z^{a-1}(z+z^{-1})^{b-1}$
I want to evaluate $\displaystyle f(z)= z^{a-1} \left(z + z^{-1}\right)^{b} \ (a >b >-1)$ counterclockwise around the right half of the circle $|z|=1$.
So I close the contour with the vertical ...
3
votes
1answer
54 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 ...
