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

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20
votes
5answers
3k 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 ...
261
votes
7answers
80k views

Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \ \mathrm dx$

I need help with this integral: $$I=\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right)\ \mathrm dx.$$ The integrand graph looks like this: $\hspace{1in}$ The ...
3
votes
2answers
9k views

Calculate the Fourier transform of $b(x) =\frac{1}{x^2 +a^2}$ [closed]

I need help to calculate the Fourier transform of this funcion $$b(x) =\frac{1}{x^2 +a^2}\,,\qquad a > 0$$ Thanks.
5
votes
0answers
265 views

An interesting identity involving powers of $\pi$ and values of $\eta(s)$

It happens that, for any $m\geq 1$, $$\sum_{n=0}^{+\infty}\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{E_{2m}}{2\cdot(2m)!}\left(\frac{\pi}{2}\right)^{2m+1}\tag{1}$$ where $E_{2m}$ is an integer number. My ...
6
votes
5answers
486 views

Evaluate $\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$ using complex analysis

How do I compute $$\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$$ What I am doing is take $$f(z)=\frac{(\log z)^2}{1+z^2}$$ and calculating $\text{Res}(f,z=i) = \dfrac{d}{dz} \dfrac{(\log z)^...
7
votes
1answer
391 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 ...
173
votes
4answers
14k 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 ...
22
votes
5answers
1k views

Evaluating $\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$

I've been trying to evaluate the following integral from the 2011 Harvard PhD Qualifying Exam. For all $n\in\mathbb{N}^+$ in general: $$\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$$ ...
6
votes
3answers
475 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.
6
votes
5answers
2k 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 (z)>0\...
9
votes
4answers
628 views

Differentiation wrt parameter $\int_0^\infty \sin^2(x)\cdot(x^2(x^2+1))^{-1}dx$

Use differentiation with respect to parameter obtaining a differential equation to solve $$ \int_0^\infty \frac{\sin^2(x)}{x^2(x^2+1)}dx $$ No complex variables, only this approach. Interesting ...
9
votes
2answers
279 views

Integral $ \int_{-\pi/2}^{\pi/2} \frac{1}{2007^x+1}\cdot \frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx $

I am trying to solve this integral $$ \int_{-\pi/2}^{\pi/2} \frac{1}{2007^x+1}\cdot \frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx $$ A closed form does exist despite the looks of the integrand. ...
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 I'...
18
votes
2answers
694 views

Integral $\int_0^\pi \theta^2 \ln^2\big(2\cos\frac{\theta}{2}\big)d \theta$.

I am trying to calculate $$ I=\frac{1}{\pi}\int_0^\pi \theta^2 \ln^2\big(2\cos\frac{\theta}{2}\big)d \theta=\frac{11\pi^4}{180}=\frac{11\zeta(4)}{2}. $$ Note, we can expand the log in the integral to ...
10
votes
6answers
819 views

show that $\int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8}$

show that $$\int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8}$$ using different ways thanks for all
6
votes
1answer
2k views

Fourier transform of $\text{sinc}^3 {\pi t}$

$$f(t)=\frac{\sin^3(\pi t)}{(\pi t)^3}$$ I want to calculate the Fourier transform. I can't calculate this integral: $$\int_0^\infty\frac{\sin^3(\pi t)}{(\pi t)^3}\cos(ut)\,\mathrm{d}t$$
4
votes
1answer
3k views

Laplace transform of the Bessel function of the first kind

I want to show that $$ \int_{0}^{\infty} J_{n}(bx) e^{-ax} \, dx = \frac{(\sqrt{a^{2}+b^{2}}-a)^{n}}{b^{n}\sqrt{a^{2}+b^{2}}}\ , \quad \ (n \in \mathbb{Z}_{\ge 0} \, , \text{Re}(a) >0 , \, b >0 ...
4
votes
3answers
711 views

Improper Integral of $x^2/\cosh(x)$

I need to compute the improper integral $$ \int_{-\infty}^{\infty}{\frac{x^{2}}{\cosh\left(x\right)}\,{\rm d}x} $$ using contour integration and possibly principal values. Trying to approach this as ...
2
votes
1answer
1k views

Contour integration using the residue at infinity

I posted a similar problem a few months ago but got no responses. So I'm going to try again with a different problem. I want to evaluate $ \displaystyle I =\int_{-1}^{1}\frac{\sqrt{1-x^{2}}}{1+x^{2}}...
9
votes
3answers
2k views

Help with integrating $\displaystyle \int_0^{\infty} \dfrac{(\log x)^2}{x^2 + 1} \operatorname d\!x$ - contour integration?

George Arfken's book: Mathematical Methods for Physicists has the following problem in a chapter on contour integration: $\displaystyle \int_0^{\infty} \dfrac{(\log x)^2}{x^2 + 1} \operatorname d\!x$....
19
votes
3answers
1k views

How do I evaluate this integral $\int_0^\pi{\frac{{{x^2}}}{{\sqrt 5-2\cos x}}}\operatorname d\!x$?

Show that $$\int\limits_0^\pi{\frac{{{x^2}}}{{\sqrt 5-2\cos x}}}\operatorname d\!x =\frac{{{\pi^3}}}{{15}}+2\pi \ln^2 \left({\frac{{1+\sqrt 5 }}{2}}\right).$$ I don't have any idea how to start, but ...
21
votes
3answers
1k 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 ...
15
votes
3answers
522 views

Evaluate $\int_0^{\frac{\pi}{2}}\frac{x^2}{1+\cos^2 x}dx$

Evaluate the following integral $$\int_0^{\frac{\pi}{2}}\frac{x^2}{1+\cos^2 x}dx$$ This function does not have an elementary anti-derivative. How can we solve this?
12
votes
6answers
1k views

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

Can this integral be solved with contour integral or by some application of residue theorem? $$\int_0^\infty \frac{\log (1+x)}{1+x^2}dx = \frac{\pi}{4}\log 2 + \text{Catalan constant}$$ It has two ...
11
votes
2answers
378 views

Sum $\sum^\infty_{n=1}\frac{(-1)^nH_n}{(2n+1)^2}$

I would like to seek your assistance in computing the sum $$\sum^\infty_{n=1}\frac{(-1)^nH_n}{(2n+1)^2}$$ I am stumped by this sum. I have tried summing the residues of $\displaystyle f(z)=\frac{\pi\...
12
votes
3answers
2k 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 ...
4
votes
3answers
619 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 $$\int_{0}^{\infty}\frac{x\cos ax}{\sinh x}dx=\frac{\pi^2}{4} \operatorname{sech}^2\left(\frac{a\pi}{2}\right) $$ I think we can solve it by contour integration, but how? and it would be ...
4
votes
1answer
230 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 2}\...
3
votes
1answer
445 views

Modification of Schwarz-Christoffel integral

I found two different formulations of the Schwarz-Christoffel formula (e.g. Link1, p.20 and Link2, p. 9). The first is \begin{align*} z=w(\zeta)=&A+C\int\limits^{z}\prod\limits_{k=1}^n\left(\zeta-...
1
vote
1answer
190 views

Gaussian Integral using contour integration with a parallelogram contour

I'm having trouble figuring out how to use contour integration to compute the Gaussian integral. The contour I'm using is a parallelogram with function, $f(z) = \Large \frac{ e^{i \pi z^2}}{sin(\pi z)}...
12
votes
3answers
720 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 ...
5
votes
3answers
2k 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 ...
1
vote
3answers
425 views

Integral of $\log(\sin(x))$ using contour integrals

I know the integral is possible with a simple fourier series expansion of $-\log(\sin(x))$ But I am interested in complex analysis, so I want to try this. $$I = \int_{0}^{\pi} \log(\sin(x)) dx$$ ...
7
votes
1answer
212 views

Evaluate Integral with $e^{ut}\ \Gamma (u)^{2}$

I am trying to integrate this integral: $$f(x)=\frac{1}{2\pi j}\int_{c-j\infty}^{c+j\infty}x^{-s}\sigma ^{ms-m}\left [ \frac{\Gamma \left ( \frac{s}{\beta} \right )}{\Gamma \left ( \frac{1}{\beta} \...
6
votes
3answers
1k views

show that $\int_{0}^{\infty } \frac {\cos (ax) -\cos (bx)} {x^2}dx=\pi \frac {b-a} {2}$

show that $$\int_{0}^{\infty } \frac {\cos (ax) -\cos (bx)} {x^2}dx=\pi \frac {b-a} {2}$$ for $a,b\geq 0$ I would like someone solve it using contour integrals, also I would like to see different ...
5
votes
3answers
1k views

erf(a+ib) error function separate into real and imaginary part

Is there an easy way to separate erf(a+ib) into real and imaginary part?
4
votes
2answers
140 views

Inverse Laplace Transform of e$^{-c \sqrt{s}}/(\sqrt{s}(a - s))$

I am trying to find the Inverse Laplace of the following function: $$ F(s) = \frac{\mathrm{e}^{-x b \sqrt{s}}}{ b (a - s)\sqrt{s}} $$ I really don't know where to start on this one as I have only ...
5
votes
3answers
319 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 ...
2
votes
2answers
1k views

Contour Integral: $\int^{\infty}_{0}(1+z^n)^{-1}dz$

I'm working through Priestley's Complex Analysis (really good book by the way) and this Ex 20.2: Evaluate $\int^{\infty}_{0}(1+z^n)^{-1}dz$ round a suitable sector of angle $\frac{2\pi}{n}$ for $n=1,...
2
votes
1answer
632 views

Integration of $\ln $ around a keyhole contour

I want to evaluate the following integral: $$\int_{0}^{\infty}\frac{\ln^2 x}{x^2-x+1}{\rm d}x$$ I use the following contour in order to integrate. I considered the function $\displaystyle f(z)=\...
2
votes
2answers
204 views

Calculating ${\int_{-\infty}^{\infty} \frac{\cos(\omega x)}{x^{2} + 25}\,{\rm d}x}$ using contour integration

I want to calculate $\displaystyle{ \int_{-\infty}^{\infty}\frac{\cos\left(\omega x\right)}{x^{2} + 25}\,{\rm d}x\,, \quad}$ for $\omega \in \mathbb{R}$ I thought of integrating along the line ...
1
vote
2answers
830 views

How to prove error function $\mbox{erf}$ is entire (i.e., analytic everywhere)?

How do I prove the error function $$ \mbox{erf}(z) = \frac{2}{\sqrt{\pi}} \int_{0}^{z} e^{-t^{2}} dt. $$ is entire? Could you give me some scratch proof?
0
votes
0answers
115 views

About evaluating $\mathcal{L}^{-1}_{s\to x}\bigl\{\frac{F(s)}{s}\bigr\}$ by considering contour integration with different entire functions $F(s)$

Detailedly compare the difficulties of different entire functions $F(s)$ where $F(0)\neq0$ when evaluating $\mathcal{L}^{-1}_{s\to x}\bigl\{\frac{F(s)}{s}\bigr\}$ by considering contour integration, e....
1
vote
1answer
201 views

Circular contour integration.

solving one of the 5 options would be much appreciated as this will give me an idea on how to solve the rest. Let $\gamma(w,R)$ denote the circular contour $t\mapsto w+Re^{it}$ where $0\lt t\lt2\...
1
vote
0answers
433 views

About the inverse laplace transform of sinc function

How to calculate $\mathcal{L}^{-1}_{s\to x}\{\text{sinc}(s)\}$ ? Note: $\text{sinc}(s)=\dfrac{\sin s}{s}$ when $s\neq0$ . Also note that $\lim\limits_{s\to\pm\infty}\dfrac{\sin s}{s}=0$ .
35
votes
4answers
966 views

Proving $\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}$

Wikipedia informs me that $$S = \vartheta(0;i)=\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}$$ I tried considering $f(x,n) = e^{-x n^2}$ so that its ...
11
votes
2answers
656 views

closed form of $\int_{0}^{2\pi}\frac{dx}{(a^2\cos^2x+b^2\sin^2x)^n}$

closed form of $$\int_{0}^{2\pi}\frac{dx}{(a^2\cos^2x+b^2\sin^2x)^n}$$ for $a,b>0$ n=1 we get $$\int_{0}^{2\pi}\frac{dx}{(a^2\cos^2x+b^2\sin^2x)^1}=\frac{2\pi}{ab}$$ n=2 we get $$\int_{0}^{2\...
3
votes
1answer
428 views

Calculating Inverse Laplace Transform of stretched exponential

I am trying to solve a Laplace transform problem that has gotten way over my head in terms of complex analysis knowledge. I would like to solve the Inverse Laplace Transform $(s\rightarrow t)$ of $$\...
11
votes
3answers
294 views

Prove that $\int_0^\infty \frac{\ln x}{x^n-1}\,dx = \Bigl(\frac{\pi}{n\sin(\frac{\pi}{n})}\Bigr)^2$

This question inspired me to ask the following. Prove that $$I_n = \int_0^\infty \frac{\ln x}{x^n-1}\,dx = \left(\frac{\pi}{n\sin\left(\frac{\pi}{n}\right)}\right)^2,$$ for $\Re(n)>1$. For some ...
8
votes
3answers
576 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 ...