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

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4
votes
2answers
146 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
539 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
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
119 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
645 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
192 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
379 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
166 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
63 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
150 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
404 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
353 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
58 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
85 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
125 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
105 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
315 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
143 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
41 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
206 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
166 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
90 views

What is the difference between integrals and contour integrals?

I understand integrals but what are contour integrals?
4
votes
1answer
136 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
48 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
642 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
1answer
336 views

$\int_0^{2\pi} \log|1-ae^{i\theta}|d\theta=0$ when $|a| = 1$

I'm taking a course on complex analysis, and I've been working on this problem forever, but can't seem to figure out. The question asks you to show the following: $\int_0^{2\pi} \log|1-ae^{i\theta}| ...
4
votes
1answer
72 views

Contour integral using residue

Assume that $f(z) \in \{\sqrt{2z^2 + 1}\}$ $,f(0) = 1$ We have a cut: $\gamma = \{|z| = \frac{1}{\sqrt2}, Re(z) \geqslant 0 \}$ $\oint\limits_{|z|=1} \frac{z dz}{(z+2)(f(z) + 3)}$ I found ...
4
votes
0answers
62 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
0answers
30 views

$ \int_\gamma e^{\frac{1}{z^2-1}}\sin{(\pi z)}dz $ on a closed curve of index $N$ with respect to the point $1$.

Let $\gamma$ be a closed curve in the right half plane that has index $N$ with respect to the point $1$. Find $$ \int_\gamma e^{\frac{1}{z^2-1}}\sin{(\pi z)}dz $$ This is a problem from an old ...
4
votes
0answers
58 views

Does anyone have a good reference on calculating contour integrals around the unit circle (numerically or otherwise)?

I am looking for a reference that will help me calculate contour integrals around the unit circle or other curve. I have a particularly ugly function which isn't likely to have a nice closed form so I ...
4
votes
0answers
90 views

Ugly-nice double series

I'm trying to evaluate the following ugly double sum (presented in raw notation as used in my calculations): $\sum _{m=1}^{\infty } \sum _{n=1}^{\infty } \frac{4 m \cos \left(\frac{2 \pi m ...
4
votes
0answers
172 views

Integral $=\int_0^\infty x^{\alpha -1}Li_n (-\sigma x) Li_m(-\omega x^r)dx$.

I am trying to calculate an integral that can be expressed in terms of infinite hypergeometric series by using transforms and Residue method, the integral is $$ ...
4
votes
0answers
57 views

where is the mistake with my fake proof? [duplicate]

I tried to show that $$\int_{-\infty}^{+\infty} \frac{dx}{(x^2+1)^{n+1}}=\frac {(2n)!\pi}{2^{2n}(n!)^2}$$ using contour integration so $$\int_{C} \frac{dz}{(z^2+1)^{n+1}}=\int_{-\infty}^{+\infty} ...
4
votes
0answers
619 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
0answers
209 views

Where's my mistake applying Perron's Formula?

I applied Perron's Formula to Riemann Zeta Function and got a weird result. First, I started with a simple definition of Riemann Zeta Function, $$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}$$ where ...
3
votes
4answers
201 views

Evaluating $I(n) = \int^{\infty}_{0} \frac{\ln(x)}{x^n(1+x)}\, dx$ for real $n$

I am not sure how to handle the additional parameter $n$. I first need to find out for which real values of $n$ will the integral converge. Based on intuition and checking with mathematica, I believe ...
3
votes
2answers
1k 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
149 views

Does $\lim_{R \to \infty} \int_{C_{R}} e^{iz} \ dz \to 0$?

Let $C_{R}$ be the upper half of the circle $|z|= R$. Does $ \displaystyle \lim_{R \to \infty} \int_{C_{R}} e^{iz} \ dz \to 0 $? Jordan's lemma is not applicable here. And I'm not sure how to get a ...
3
votes
2answers
218 views

Improper integral involving $e^x$

How do you show that $$\int_{-\infty }^{+\infty }{\frac{{{x}^{2}}\, \text{d}x}{\left( \beta +{{\text{e}}^{x}} \right)\left( 1-{{\text{e}}^{-x}} \right)}}=\frac{\left( {{\pi }^{2}}+{{\ln }^{2}}\beta ...
3
votes
4answers
159 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
2answers
171 views

Integral $\int_0^a \ln \left( \frac{b-\sqrt{a^2-x^2}}{b+\sqrt{a^2-x^2}} \right)dx$

Hi I am trying to calculate, $$ \int_0^a \ln \left( \frac{b-\sqrt{a^2-x^2}}{b+\sqrt{a^2-x^2}} \right)dx $$ where $a,b$ are positive real constants. I Know $\ln(xy)=\ln x +\ln y$, but I do not ...
3
votes
2answers
63 views

what is the proper contour for $\int_{-\infty}^{\infty}\frac{e^z}{1+e^{nz}}dz$

what is the proper contour for $$\int_{-\infty}^{\infty}\frac{e^z}{1+e^{nz}}dz:2\leq n$$ I tried with rectangle contour but the problem which I faced how to make the contour contain all branches point ...
3
votes
3answers
108 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
3answers
581 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
65 views

Complex integral of $\frac{\cos x}{x^2+4} $

I want to evaluate: $$ \int_{-\infty}^{\infty}\frac{\cos x}{x^2 +4} dx $$ Using wolfram alpha, it gave an answer of $\frac{\pi}{2e^2}$. Wolfram Alpha is never wrong. Attempt $$ ...
3
votes
1answer
106 views

$\frac{5\pi^3}{154}=\int_{\pi/6}^{\pi/2}\bigg[\Re\big(\text{Li}_2(4\sin^2\theta)\big) +\text{Li}_2\bigg(\frac{1}{4\sin^2\theta}\bigg) \bigg]d\theta$

I am trying to prove $$ \int_{\pi/6}^{\pi/2}\bigg[\Re\big(\text{Li}_2(4\sin^2\theta)\big) +\text{Li}_2\bigg(\frac{1}{4\sin^2\theta}\bigg) \bigg]d\theta=\frac{5\pi^3}{54}. $$ Clearly, this closed form ...
3
votes
2answers
215 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?