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

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5
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3answers
866 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?
5
votes
3answers
664 views

Evaluate $\int_0^{\pi} \frac{\sin^2 \theta}{(1-2a\cos\theta+a^2)(1-2b\cos\theta+b^2)}\mathrm{d\theta}, \space 0<a<b<1$

Evaluate by complex methods $$\int_0^{\pi} \frac{\sin^2 \theta}{(1-2a\cos\theta+a^2)(1-2b\cos\theta+b^2)}\mathrm{d\theta}, \space 0<a<b<1$$ Sis.
5
votes
4answers
290 views

How to prove $\int^{\pi/2}_0 \log{\cos{x}} \, \mathrm{d}x = \pi/2 \log{1/2}$

ALREADY ANSWERED I was trying to prove the result that the OP of this question is given as a hint. That is to say: imagine that you are not given the hint and you need to evaluate: $$I = ...
5
votes
3answers
296 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 ...
5
votes
1answer
170 views

A little rusty on my contour integration..

I've been asked to compute $\displaystyle\int_{0}^{\infty}\frac{x^2}{(x^2+1)^2(x^2+2x+2)}dx$ via finding the poles of the integrand and then construncting a contour to take advantage of cauchy's ...
5
votes
2answers
230 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
101 views

What is the Fourier transform of $\exp(2 \pi i / x)$?

The Fourier transform of $e^{2 \pi i / x}$ makes sense as a distribution, I believe. Does it have a nice expression in terms of functions and well-known distributions (e.g. Dirac delta)?
5
votes
2answers
178 views

Evaluate $\int_0^\infty \frac{\sqrt{x}}{x^2+1}\log\left(\frac{x+1}{2\sqrt{x}}\right)\;dx$

Prove that $$\int_0^\infty \frac{\sqrt{x}}{x^2+1}\log\left(\frac{x+1}{2\sqrt{x}}\right)\;dx=\frac{\pi\sqrt{2}}{2}\log\left(1+\frac{\sqrt{2}}{2}\right).$$ I managed to prove this result with some ...
5
votes
2answers
121 views

Is this a Morera´s Theorem Application?

Let $G \subset \mathbb C$ be a domain and $f: G \to \mathbb C$ a continuous function such that for any closed and rectifiable path $\gamma \subset G$, $$ \left| \oint_\gamma f(z)dz \right|\leq \left( ...
5
votes
1answer
349 views

Evaluating trigonometric integral using residues

I am trying to evaluate for real positive $\alpha,\beta$ $$\int_{0}^{\infty}\arctan\left(\frac{\alpha}{x}\right)\sin(\beta x)dx$$ using a hint to consider $$\int \log\left(\frac{z+ia}{z}\right) ...
5
votes
4answers
357 views

Complex integral - winding number

i want to find $$\frac{1}{2\pi i}\int _\gamma \frac{1}{z}dz$$ well $0$'s winding number is $2$, so $\frac{1}{2\pi i}\int _\gamma \frac{1}{z}dz=2$ but when I explicity calculate the integral I get ...
5
votes
3answers
700 views

Complex analysis: contour integration

Evaluate by contour integral: $$\int_0^1{ dx\over (x^2-x^3)^\frac 13}$$ Should I go for some kind of substitution so that the limit changes to $0$ to $\pi/2$?
5
votes
2answers
266 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) $$\int_0^\infty {{\pi^2\over 4}\cos^2x\over\left({\pi^2\over4}-x^2 \right)^2} dx={\pi\over 4}$$?
5
votes
2answers
251 views

A (basic?) contour integration problem

I am trying to prove the following using complex analysis: $$\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}}{a^{2}+n^{2}}=\frac{\pi}{a\sinh(a\pi)}$$ I am told to use the following function: ...
5
votes
2answers
759 views

Evaluating a real definite integral using the residue at infinity

I'm interested in evaluating $\displaystyle \int_{0}^{1} \frac{1}{(x^2-x^3)^{1/3}} \ dx $ using contour integration and the residue at infinity. But I'm not sure how to define $\displaystyle f(z) = ...
5
votes
1answer
256 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?
5
votes
1answer
146 views

A difficult one-variable exponential integral

I am trying to work out a closed form for the integral \begin{equation} \int_{0}^{1} \frac{1}{\sqrt{s(1-s)}} \exp\left(-\left(\frac{a}{s} + \frac{b}{1-s}\right) \right) \,ds \end{equation} where ...
5
votes
4answers
166 views

Integrating $ \int \limits_{-\infty}^{\infty} \dfrac{\sin^2(x)}{x^2} \operatorname d\!x $

I'm trying to evaluate $\displaystyle \int \limits_{-\infty}^{\infty} \dfrac{\sin^2(x)}{x^2} \operatorname d\!x $. My first though was to use residue calculus, since we've got the pole of order 2 ...
5
votes
1answer
996 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 ...
5
votes
3answers
99 views

Show that $\int_{0}^{+\infty} \frac{\sin(x)}{x(x^2+1)} dx = \frac{\pi}{2}\left(1-\frac{1}{e}\right) $

I'm trying to show that $$\int_{0}^{+\infty} \frac{\sin(x)}{x(x^2+1)} dx = \frac{\pi}{2}\left(1-\frac{1}{e}\right) $$ using Jordan's lemma and contour integration. MY ATTEMPT: The function in ...
5
votes
2answers
285 views

Using complex analysis to evaluate $\int_0^\infty\frac{(\ln x)^3}{1+x^2}d x$

Here is my attempt: Let $R>1>r$ and $C$ be the closed curve in $\mathbb{C}$ consists of the following pieces: $$C_1=\{Re^{it}: t\in(0,\pi)\},\quad C_2=[r,R],\quad C_3=\{re^{it}: ...
5
votes
2answers
217 views

Why do we need a branch cut for $\int_0^{\infty} \frac{x^{\frac{1}{2}}}{{(1 + x)^2}}dx$?

What is the significance of the $x^{\frac{1}{2}}$ in the numerator of this integral. I have read this kind of integral requires taking a branch cut. Why do we need a branch cut, what does it enable us ...
5
votes
1answer
120 views

Integrating $e^{a/x^2-x^2}/(1-e^{b/x^2})$

I want to solve the following two integrals analytically \begin{aligned} I_1 = & \int\limits_0^{\infty}\frac{e^{a/x^2}}{1-e^{b/x^2}}e^{-x^2}dx \\ I_2 = & ...
5
votes
2answers
263 views

Riemann Zeta function Analytic continuation integral

Following Riemann paper about analytic continuation of Zeta Function: http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZeta.pdf I can't understand the contour integral step: "If one now ...
5
votes
1answer
364 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 ...
5
votes
2answers
682 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
99 views

How to perform this contour integration with $\log$ in the denominator?

Let $k > 0$ and $ a>1$ be constants. As far as I can tell, the integral $$ J = \int_{-\infty}^\infty dx\frac{e^{i k x}}{1+x^2}\frac{1}{\log(a - ix)} $$ converges, since the argument of the ...
5
votes
1answer
189 views

Evaluate: $\int_{W(-1/\gamma)}^{W(1/\gamma)}\frac{e^{-u} \,\text{d}u}{\sqrt{1-(\gamma u e^{u})^2}}$

Evaluate the integral $$ P(\gamma)=\int_{W(-1/\gamma)}^{W(1/\gamma)}\frac{e^{-u} \,\text{d}u}{\sqrt{1-(\gamma u e^{u})^2}} $$ where $\gamma$ is a real number not equal to $0$ and has whatever ...
5
votes
1answer
131 views

Real integral using complex methods

Evaluate $\displaystyle\int_0^\infty \frac {x^\frac{1}{2}}{1+x^4}dx$ using complex methods. I'm totally locked up on this one and have thrown in the towel. My strategy was to integrate around a ...
5
votes
1answer
825 views

what is the best book to study contour integration?

what is the best book or website to study contour integration ? I find in some question answer using contour integration but I can't understand how they do that so is there any help ?
5
votes
2answers
200 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
2answers
569 views

Inverse Laplace transform of $\frac{\log(s)}{1 + s}$

Is it possible to find the inverse laplace transform $$\mathcal{L}^{-1}\frac{\log(s)}{1 + s}$$ using the Bromwich integral formula $$\mathcal{L}^{-1} \{F(s)\}(t) = f(t) = \frac{1}{2\pi ...
5
votes
1answer
43 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 ...
5
votes
1answer
119 views

A Cosine Integral

What is the value of the Cosine integral \begin{align} \int_{0}^{\infty} \cos\left( \frac{x (x^{2}-a^{2})}{x^{2}-b^{2}} \right) \, \frac{dx}{x^{2} + p^{2}} \, \, \, ? \end{align}
5
votes
2answers
95 views

How to compute $\int_C {e^{3z}-z\over (z+1)^2z^2}$?

I am asked to compute the integral $$ \int_C {e^{3z}-z\over (z+1)^2z^2} $$ where $C$ is a circle with the center at the origin and radius ${1 \over 2}$. My approach was to separate the integral as a ...
5
votes
1answer
509 views

Residue theorem with exponential and trig functions

The following integral should be doable using the residue theorum: $$\frac1{2\pi}\int_{0}^{2\pi}e^{\cos\theta}\cos(n\theta) \,d\theta$$
5
votes
1answer
142 views

Integrate: $\int_{a - i\infty}^{a + i\infty} \frac{e^{tz}}{z^2 + p^2}dz$

Q. Show that : $$\int_{a - i\infty}^{a + i\infty} \frac{e^{tz}}{z^2 + p^2}dz = \frac{\sin pt}{p}$$ I considered the following contour $$\int_\Gamma \frac{e^{tz}}{z^2 + p^2}dz + \int_{a - ...
5
votes
1answer
255 views

Evaluating $\int_{0}^{\infty} \frac{2 \cos (x) \ln x + \pi \sin x}{x^2+4} \ dx$

I want to show that $$\int_{0}^{\infty} \frac{2 \cos (x) \log x + \pi \sin x}{x^2+4} \ dx = \frac{\pi \log 2}{2e^{2}}.$$ The recommendation in the textbook is to let $$f(z) = \frac{e^{iz} ...
5
votes
2answers
184 views

Evaluating the integral $\int_{-1}^1 \frac{\ln|z-x|}{\sqrt{1-x^2}}\mathrm dx$

I don't know how to deal with this integral: $$\int_{-1}^{1}{\ln\left(\,\left\vert\,z - x\,\right\vert\,\right)\over \,\sqrt{\vphantom{\large A}\,1 - x^{2}\,}\,}\,{\rm d}x\,,$$ where ...
5
votes
2answers
110 views

How do I get $ \int_0^1 \frac{dz}{\sqrt{z(z - 1\,)(z+1\,)}} = \frac{\sqrt{\pi}}{2} \frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{9}{4})}$?

While reading physics papers I found a very interesting integral so I decided to write it down. Let $p(z) = z^ 3 - 3\Lambda^ 2 z$ where $\Lambda$ could be any number. If you want $\Lambda = 1$ and ...
5
votes
1answer
72 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..
5
votes
1answer
56 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
1answer
731 views

How do you integrate Gaussian integral with contour integration method?

How do you integrate $$\int^{\infty}_{-\infty} e^{-x^2} dx$$ with contour integration method? I do not even know how to setup the problem.
5
votes
2answers
142 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
164 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
81 views

Evaluate $\int_{-\infty}^{\infty}\frac{\sqrt{a+ix}}{a^2+x^2}\,dx$ using residue calculus

I'm asked to evaluate $$\int_{-\infty}^{\infty}\frac{\sqrt{a+ix}}{a^2+x^2}\,dx$$ $\mathbb{R}\ni a>0$, using residue calculus (where $\sqrt{\cdot}$ is the PV $\sqrt{}$). My approach is as follows: ...
5
votes
0answers
72 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
61 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
106 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 ...