5
votes
1answer
32 views

Meromorphic and even

I would like to do the following exercise : Let $f$ be a meromorphic function and $\mathcal{P}$ the set of its poles. We also assume that $f$ is even ($\forall z \in \mathbb{C}, \; ...
1
vote
2answers
133 views

$\int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx$ with residue calculus

I'm trying to compute $\displaystyle \int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx$, $(0<a<1)$ Let $f$ denote the integrand. I'm using the rectangular contour given by the following curves: ...
2
votes
1answer
59 views

Integral using residue theorem

We have the following problem given: $$ \int_{-\infty}^\infty \frac{\cos(t)^2}{t^4 + 5 t^2 + 4} \, \mathrm dt. $$ I thought that I could solve it using the residue theorem and by arguing that the ...
2
votes
2answers
147 views

How to show the residue of an analytic function's derivative is equal to zero?

Let $r>0$ . for $f: \Bbb D_r(0)-{0}\mapsto \Bbb C$ analytic function show that $Res(f';0)=0$ we know by residue therom $∫_Cf'(z)dz=2iπRes(f',0)$ What property of analytic functions will we use? ...
4
votes
2answers
79 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} = ...
3
votes
5answers
102 views

Residues at singularities

I have the following question: Show that the integral $$\int_{-\infty}^{+\infty}\frac{\cos\pi x}{2x-1}dx = -\frac\pi2$$ Clearly there is a singularity at $z=1/2$ but I think this is a removable ...
0
votes
1answer
43 views

Verity of Residue theorem of [0,2pi]

After I turn $$ cos\theta=\frac12(z+\frac1{z})$$and $$ d\theta=\frac1{iz}dz$$ the denominator become a mess $$ \frac{dz}{(a^2+\frac{b^2}4(z^2+2+\frac1{z^2})+\frac{ab}2(z+\frac1z))(iz)}$$ How can a ...
2
votes
2answers
143 views

Inverse Laplace transform of $\frac{s}{\sqrt{(s+a)^3}}$

Trying to find the inverse Laplace transform of $\frac{s}{\sqrt{(s+a)^3}}$. So solving $\oint_B dz \: \frac{z}{\sqrt{(z+a)^3}} e^{z t}$ (Bromwich contour). I tried doing a u-substitution with $u=z+a$ ...
0
votes
1answer
35 views

Residue Theorem for Denominator with $e^z$

$$ f(z)=\frac{z^3}{e^z-1} $$ Is this a simple pole at $z=0$ or some other types of pole? If it is a simple pole, what is its residue? Is it using this formula or other else? $$ \lim_{z\to 0}=zf(z) ...
1
vote
1answer
50 views

Prove on residue theorem

I have try to use the equation $$ Res(f;z_0)=\lim_{z\to z_0}\frac1{(m-1)!}\frac{d^{m-1}}{dz^{m-1}}[(z-z_0)^mf(z)] $$ But very soon I stuck, is that a good way to solve it?
0
votes
1answer
47 views

Integrating Real Function in the Complex Plane

Question: Evaluate the integral $$\int_{-\infty}^{\infty}\frac{\sin(x)}{x(x^2+a^2)}=Im\left ( \frac{e^{ix}}{x(x^2+a^2)} \right)$$ ...
0
votes
1answer
70 views

Calculating $\int_0^{2\pi} \cos^{2n} x \ dx $, please check my work.

In order to calculate the integral $$I \equiv \int_0^{2\pi} \cos^{2n} x \ dx, $$ I first express it in the form $$\int_0^{2\pi} f(e^{it}) ie^{it}dt = \oint_{|z|=1}f(z)dz.$$ By substituting for the ...
2
votes
2answers
52 views

Residues of Complex Functions

I need to find the residues of $f$ at the isolated singular points, namely $z=1,z=0$. Where $f(z)=\dfrac{2z+1}{z(z+1)}$. I already have that the residue at $z=0$ is $1$, and I know I need to do ...
2
votes
2answers
98 views

Complex contour integral with residue theory

I need to calculate the following contour integral using residue theory. $z \in \mathbb{C}$ $f(z)=\exp(-1/z) \sin(1/z)$ $\oint_C f(z) dz$ $C: \left | z \right |=1$ The difficult points I ...
1
vote
1answer
99 views

Summing a series by using residues

For the same series $$\sum_{n=0}^{\infty}\binom{3n}{2n} x^n$$ I am trying to calculate te sum by using residue theory. At the last line, I need to find the roots of $z^2-(z+1)^3x=0$ and one of ...
2
votes
0answers
131 views

The solution of the contour integral for $\epsilon =+1$

I understand the solution for $\epsilon =-1$. And I am trying the solve this question for $\epsilon =+1$. This is important for me. I want really to learn perfectly because I am continuously seein' ...
5
votes
1answer
171 views

Calculation of sum by using residue theory

I am studying an example about the calculation of a summation by using residue theory. I understand how to calculate the sum in general -the frame of the solution way-, but I dont know how some parts ...
1
vote
0answers
25 views

possible results of integral along closed path after defining branch of sqrt

Prove that one can define a branch of the function $\sqrt{1-z^2}$ in every region $D\subset \mathbb{C}$ such that the points $-1$ and 1 belong to the same connected component of the complement of $D.$ ...
0
votes
1answer
102 views

handwaving substitution in integral involving branch cut and derivative of sqrt + generalization

Want to compute $$ I = \int_0^i \mathrm{d}z \frac{z}{\sqrt{z^2-1}}$$ on the complex plane using complex methods. QUESTION: is the result $i \left( \sqrt{2}-1 \right)$ which one gets imposing ...
1
vote
1answer
110 views

integral involving square root using complex methods: what choice of path?

I'm asked to compute using complex methods the following integral: $$ I(a)= \int_0^1 \mathrm{d}x \frac{\sqrt{1-x^2}}{x^2-a^2},$$ where $a>1.$ What I know is the following: for $|z|<1,$ the ...
2
votes
2answers
59 views

How do I find the residue of a function with a huge exponent?

How would I find the remainder of a function that has a huge exponent that would take ages to work out? Say I have something like this: $\frac{5x^{110} + x^4 - 7x^2 - 6}{x-1}$ I honestly don't know ...
5
votes
1answer
167 views

Evaluating the (complex) integral $\int_\gamma \frac{e^{z+z^{-1}}}{z}dz$ using residues.

I am trying to evaluate the following integral. $$\int_\gamma \frac{e^{z+z^{-1}}}{z}dz$$ where $\gamma$ is the path $\cos(t)+2i\sin(t)$ for $0\leq t <4\pi$. So, $\gamma$ is an ellipse ...
1
vote
1answer
153 views

Locate poles and calculate residue

Let $a = \sqrt{\pi}e^{\pi i/4} = (1+i)\sqrt{\pi/2} $ Consider the function $$ f(z) = \frac{e^{-z^2}}{1+e^{-2az}} $$ I have already shown that $f(z) - f(z+a) = e^{-z^2}$ if that helps at all. The ...
4
votes
2answers
146 views

How does $\int_{z=-R+0i}^{R+0i} \frac{e^{2iz}-1-2iz}{z^2}\ dx$ become $\int_{-R}^R \frac{\sin^2x}{x^2}\ dx$?

While trying to compute $\int_0^\infty \frac{\sin^2 x}{x^2}\ dx$, the author of this book suggests computing $\int_{C_R} \frac{e^{2iz}-1-2iz}{z^2}\ dz$ on a semi-circular contour in the upper ...
0
votes
1answer
71 views

Residue Theorem, Find the sum of residues when z is an integer

We have the function $$ f(z)=\frac{\pi \cot(\pi z)}{(u + z)^2} $$ I already found the residue at the pole when $z = -u$. However there are more poles when z is an integer. How do I go about finding ...
0
votes
2answers
78 views

$\textrm{Res}\left(\frac{\log z}{z^3+8}; z_k\right) = \frac{-z_k \log z_k}{24}$ when $z_k$ solves $z^3+8=0$

The problem in the book is Compute $\int_0^\infty \frac{dx}{x^3+8}$. I set up the keyhole contour, apply the residue theorem, and go through the tedious algebra. I get stuck in doing so, but ...
12
votes
5answers
666 views

Prove $\int_0^\infty \frac{\sin^4x}{x^4}dx = \frac{\pi}{3}$

I need to show that $$ \int_0^\infty \frac{\sin^4x}{x^4}dx = \frac{\pi}{3} $$ I have already derived the result $\int_0^\infty \frac{\sin^2x}{x^2} = \frac{\pi}{2}$ using complex analysis, a result ...
1
vote
2answers
66 views

Explain why the residue is equal to the limit?

I'm studying for a midterm and my teacher warned this would be a good question to understand for the test. The problem is, I do not know how to go about explaining it. Suppose g(x) has a pole of ...
4
votes
1answer
103 views

evaluate $\int_0^\infty \dfrac{dx}{1+x^4}$ using $\int_0^\infty \dfrac{u^{p-1}}{1+u} du$

evaluate $\int_0^\infty \dfrac{dx}{1+x^4}$using $\int_0^\infty \dfrac{u^{p-1}}{1+u} du = \dfrac{\pi}{\sin( \pi p)}$. I am having trouble finding what is $p$. I set $u = x^4$, I figure $du = 4x^3 dx$, ...
4
votes
2answers
388 views

Evaluating real integral using residue calculus: why different results?

I have to evaluate the real integral $$ I = \int_0^{\infty} \frac{\log^2 x}{x^2+1}. $$ using residue calculus. Its value is $\frac{\pi^3}{8}$, as you can verify (for example) introducing the function ...
1
vote
2answers
224 views

How the calculate $\int_0^{+\infty} \frac{\sin^2 x}{x^2} \,\mathrm{d} x$?

Just as the title say, consider the integral: $$I=\int_0^{+\infty} \frac{\sin^2 x}{x^2} \,\mathrm{d} x=\frac{1}{2}\int_{-\infty}^{+\infty} \frac{\sin^2 x}{x^2} \,\mathrm{d} x,$$ how to apply the ...
1
vote
1answer
312 views

Cauchy principle value of $\int_{-\infty}^{\infty}\sin(x)/(x-a)dx$

I need to find the cauchy principle value of $\int_{-\infty}^{\infty}\sin(x)/(x-a)dx$ ? I'm think of rewriting in terms of $e^{i\theta}$ and try to rewrite as contour integral? Need some aid on how ...
3
votes
2answers
498 views

Another residue theory integral

this is the last from me I need to evaluate the following real convergent improper integral using residue theory (vital that i use residue theory so other methods are not needed here) I also need to ...
5
votes
1answer
518 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: ...
6
votes
3answers
465 views

Using residue theory to evaluate $ \int_0^\infty \frac{ \sin \pi x}{x(1-x^2)} \;\text{ dx}$

I'm on the last question of my homework and it's involving using the residue theory, which I dont really understand, so could somebody lend me a hand? I have to evaluate the real convergent improper ...