4
votes
1answer
63 views

Complex Integration: theorem of Residues

I am trying to prove a theorem that is doing my head in a bit. I have tried to simplify the problem as much as possible and leave out the details, even though it might look a bit too big. The ...
3
votes
2answers
94 views

Evaluate by contour integration $\int_0^1\frac{dx}{(x^2-x^3)^{1/3}}$

Evaluate by contour integration [i am learning complex analysis - calculus of residues] $$\int_0^1\frac{dx}{(x^2-x^3)^{1/3}}$$ I tried by taking $x^3$ out from the denominator but that didnt work.
2
votes
1answer
40 views

Using calculus of residues, prove that $\int z^2log[(z+1)/(z-1)]dz$

Using calculus of residues, how can it be proven that $$ \int z^2\log\left[\frac{z+1}{z-1}\right]\;dz $$ taken round the circle $\left\vert z\right\vert=2$ has the value $\frac{4\pi i}{3}$?
1
vote
1answer
38 views

Use calculus of residues to evlauate

Use calculus of residues to evaluate the integral $$\int_0^{2\pi}\cos^{2n}\theta d\theta$$ My Ateempt : $$\int_0^{2\pi}\frac{(1+\cos2\theta)^n}{2^n}d\theta$$ $$=\frac{1}{2^n} \int_C ...
2
votes
0answers
69 views

Finding the limit of sum $\sum \frac{1}{n^4}$

I'm trying to use the reside theorem to find the limit of $\sum \frac{1}{n^4}$. So I am considering the function $f(z) = \frac{\pi \cos(\pi z)}{\sin (\pi z)z^4}$ on a square contour. Now I am ...
1
vote
3answers
59 views

Laurent series and residue of $f(x)=\frac{1}{1+e^z}$

I am having trouble trying to expand this function using Laurent series, and finding the residue$$f(x)=\frac{1}{1+e^z}$$ If I replace $e^z$ with its series I get ...
0
votes
1answer
19 views

Contour Integration of this path circling branch point

If we let the semi-cricle blow up to infinity and the radius of the tiny circle encircling the branch point at origin go to zero, by residue theorem we have: $$\int_\gamma + \int_{AB} + \int_{BC} ...
1
vote
1answer
18 views

Conformal mapping, Residue

I have solved the second question till $z=0$ is a simple pole... But it's residue is mentioned as $2$. How come the laurent series expansion helps in finding $a_{-1}$, is there any other way to ...
1
vote
1answer
65 views

Understanding Poles and Zeros of a complex function

I am learning complex analysis and need help understanding and relating the concept of poles and zeros of a complex function. I understand that if a complex function $f(z)$ has an isolated ...
0
votes
2answers
22 views

How to calculate the Residue?

I need help how this $I_r$ is calculated. It says residue but can any one help me to reach the final results.
4
votes
3answers
295 views

A difficult integral evaluation problem

How do I compute the integration for $a>0$, $$ \int_0^\pi \frac{x\sin x}{1-2a\cos x+a^2}dx? $$ I want to find a complex function and integrate by the residue theorem.
2
votes
3answers
74 views

Quick question on infinite complex integral

I'm supposed to find: $$ \int_0^{\infty} \frac{(\ln x)^2}{1+x^2} dx $$ I start of by finding: $$ I_1 = \oint \frac{(\ln z)^2}{1+z^2} dz $$ $$ = \oint \frac{(\ln z)^2}{(z+i)(z-i)} dz $$ Now I ...
0
votes
1answer
27 views

Singularities of a function

Given $f(z):=\operatorname{Log}(\frac{z-2}{z-3})$, $\operatorname{Log}:\mathbb{C}\setminus\mathbb{R_{\le 0}}\to \mathbb{C}$. Is in $z_0=3$ a essential singularity of f? I'm not sure what is correct... ...
2
votes
2answers
68 views

help with integral using residue theorem

I'm solving the following integral: $$I=\int_{-\infty}^{+\infty}\frac{x\cdot\cos x}{x^2-2x+10}\mathrm{d}x$$ I used the residue theorem as follows: ...
1
vote
1answer
28 views

Question about Residues at Poles

I've been asked to find the residues of the following function; $$f(z) = \frac{1}{(z^2+1)(z-1)^2}$$ So, I've got that the singularities are $\pm i$ and $1$, with order 1 and 2 respectively. I was ...
1
vote
0answers
48 views

Classification of Singularities and Residues

I'm kind of stumped on a question here. I've been asked to determine and classify the singularities of; $$f(z) = \frac{z^3}{(1+z)^3}$$ To me, it's pretty obvious that a singularity will occur when ...
0
votes
0answers
31 views

Laurent-Series Expansion and Residues

I was asked to find a Laurent-series expansion for $f(z) = z^{-1} \sinh(z^{-1})$ about the point 0, and to classify the singularity at 0. Now, that wasn't too bad - with some manipulation of the ...
0
votes
0answers
51 views

Computing the contour integral of $\frac{\log(z)}{z^2 +a^2}$.

I'm still a bit insecure when it comes to complex analysis and I wondered if you guys could take a look at my solution to this problem. Let $a > 0 $ and define $$f(z) = \frac{\log(z)}{z^2 +a^2}$$ ...
1
vote
1answer
22 views

How to compute residue of $f(z)=z^3e^{\frac{1}{z}}$?

How to compute residue of $$f(z)=z^3e^{\frac{1}{z}}$$ I find there is an essential singularity in $z=0$, and I have $$ f(z) = \sum_0^\infty \frac{z^{3-k}}{k!}$$ but how to compute residue in 0 ?
0
votes
1answer
34 views

what order is the pole at 0?

What order is the pole at $z=0$? $$\int\frac{\sin(3z)-3\sin(z)}{\sin(z)(\sin(z)-3)} dz$$ And do I calculate this this way: $$\frac{1}{(n-1)!} \lim_{z\to 0} \frac{d^{n-1}}{dz^{n-1}} f(z)$$ P.S. I ...
10
votes
1answer
106 views

Application of the residue theorem

I want to prove that $$\sum_{j=1}^n \frac{1}{\left[\text{cos} \left( \frac{j \pi}{2n+1} \right)\right]^4}=\frac{8n(n+1)(n^2+n+1)}{3}$$ for $n \in \mathbb{N}$ by using the residue theorem. Which ...
1
vote
0answers
45 views

Calculate integral when I have a pole on the unit circle

I'm having trouble integrating the following integral $$\int_0^{2\pi} \frac{d\theta}{3 -2\cos\theta + \sin\theta}$$ I make the substitutions $z= e^{i\theta}$ , $\sin\theta = ...
4
votes
3answers
63 views

Integral $I(a,b)= P\int_{0}^{\pi}\frac{d\theta}{a-b\cos\theta}$

Hi I am trying to calculate this integral $$ I(a,b)= P\int_{0}^{\pi}\frac{d\theta}{a-b\cos\theta},\quad 0 <a<b,\quad a,b\in \mathbb{R}. $$ We can first write $$ I(a,b)=\frac{1}{2} ...
3
votes
2answers
92 views

Finding the poles and residues of a complex function $\frac{\cos(z)-1}{(e^z - 1)^2}$

I'm trying to find the poles and residues of: $$f(z) = \frac{\cos(z)-1}{(e^z - 1)^2}$$ I can see that this has a removable singularity at $z=0$ and double poles at $z=2k \pi i$. I'm having trouble ...
4
votes
4answers
151 views

Integral $P\int_0^\infty \frac{x^{\lambda-1}}{1-x} dx$

I am trying to calculate the following principle value integral \begin{equation} P\int_0^\infty \frac{x^{\lambda-1}}{1-x} dx \end{equation} for $\lambda \in [0,1].$ I tried to turn this into a ...
3
votes
1answer
52 views

Prove that for an odd function, res(f, z0) = res(f, -z0)

I am having difficulty proving that, for an odd function, the residue function is symmetric. i.e $res(f, z_0) = res(f, -z_0)$ I am using the Laurent series expansion of a function ...
1
vote
1answer
34 views

integral using residual theorem

I have the following problem: $$\int_C\cot z\ \mathrm{d}z,\ \ \ C(t)=2e^{it},\ t\in[0,2\pi]$$ my solution is: $$\mathrm{res_0}\cot z=\frac{\cos(0)}{\cos(0)}=1$$ The circle only surrounds one ...
3
votes
1answer
83 views

Solve $\mathscr{F}^{-1} [ \cot{a \omega} \times \mathscr{F} \{ U(t) \sin{\omega_0 t} \} ] $ using contour integration

I wish to evaluate $y(t) = \mathscr{F}^{-1} [ \cot{a \omega} \times \mathscr{F} \{ U(t) \sin{\omega_0 t} \} ] $, where $\mathscr{F}$ represents the Fourier transform, and U(t) represents the ...
1
vote
1answer
70 views

Calculate the residue of $\cot\pi z$ at poles $z=n$

I'm having trouble calculating the residue of $f(z) =\cot\pi z$. The function has a simple pole for every integer n, and i'm, trying to find the residue at n. I know that by the residue theorem: ...
0
votes
1answer
23 views

Pole and residue of $f(z) = \frac{1}{1+z^n}$

Let $f(z) = \frac{1}{1+z^n}$ for $n \ge 3$. How can we compute poles and residue in first point over real axis? Firstly, we have to solve $1+z^n=0$. I think, that solution of this are $$z_k = e^{2 ...
2
votes
1answer
35 views

Calculating the residues of $\frac{\zeta^{\prime}{(s) x^{s}}}{\zeta(s)\cdot s}$

Calculating the poles of $\frac{\zeta^{\prime}{(s) x^{s}}}{\zeta(s)\cdot s}$, where x is a fixed real number. I am trying to calculate the poles of this function at the trivial zeros of $\zeta$, ...
2
votes
3answers
305 views

A nasty simple pole… A residue calculation.

We are asked to compute: Res$[\frac{z^n+1}{z^n-1},e^{2\pi ki/n}]$, where I am assuming $k\in\mathbb Z$. The only tools I am aware of to compute residues comes from relating the function to power ...
1
vote
3answers
42 views

Residue Calculation

I am stuck on what should be a trivial residue calculation. Any suggestions? Compute the residue of $\frac{e^{2iz}-1}{z^2}$ at $z=0$. Isn't it a double pole, i.e., shouldn't I be able to take the ...
1
vote
1answer
47 views

Integrating around pie-slice domain

We are asked to show $$\int_0^{\infty}\frac{\log(x)}{x^3+1}dx=-\frac{2\pi^2}{27}$$, and $$\int_0^{\infty}\frac{1}{x^3+1}dx=\frac{2\pi}{3\sqrt{3}}$$ By integrating around a pie slice with angle ...
0
votes
0answers
23 views

What if a line integral is independent of the function?

This concerns the proof that the $n$ times punctured plane has $\mathbb Z ^n $ as his second homotopy group. We choose for each puncture $a_i$ a loop $\sigma_i$ which circles it once counterclockwise, ...
1
vote
1answer
66 views

Calculate integral using the residue theorem

I want to calculate the integral $$\int_C {{z^2-2z}\over{(z+1)^2(z^2+4)}}dz$$, where $C=\{z:|z|=4\}$ I want to use the Residue theorem to tackle this integral. Now, $f(z)$ has a pole of degree 2 at ...
2
votes
1answer
64 views

Calculating residues of multiple poles

I have $f(z)=\dfrac{z^2e^{iz}}{(z^2+1)^2}$, and I need its residue at $z=i$, which is a double pole. I tried expanding $f(w+i)$, where I got to $f(w+i)=w^{-2}\dfrac{(w^2+2iw-1)e^{iw-1}}{(w+2i)^2}$ and ...
3
votes
1answer
46 views

Keyhole Domain Residue problem with logarithm

Show $$\int^{\infty}_{0}\frac{\log(x)}{x^a(x+1)}dx = \frac{\pi^2\cos(\pi a)}{\sin^2(\pi a)}, \ 0<a<1$$ I have tried tackling this by using a keyhole domain, where $\gamma_{\epsilon}$, where ...
0
votes
1answer
68 views

Application of Residue Theorem and limits

I am trying the following problem from Fisher's Complex Variables book: If $f$ is analytic on a plane except at poles $\gamma_1, \cdots \gamma_N$ and none of them are integers and ...
3
votes
1answer
45 views

Residue of $\frac{1}{(1-z)^3}$ at $z=1$

I know there is a singularity of $z=1$ but I am a bit confused on how to find the residue at that point since if we have that $f(z)=\frac{g(z)}{h(z)}$ with $g(z)=1$ and $h(z)=(1-z)^3$ then $g(z)$ has ...
0
votes
1answer
24 views

Application of Residues

So in applying the residue theorem to solve improper real integrals, we agree to take our semicircles to be as large or as small as necessary such that all the poles we wish to work with lie inside ...
4
votes
1answer
29 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
1answer
51 views

Deducing Laplace Formulas

I have to compute the followings integrals $\forall\; b\in \mathbb{C},\; \text{Re} \;b \gt0,p\gt 0$ $$ \int_{-\infty}^\infty \frac{e^{ipx}}{x-ib}$$ $$ \int_{-\infty}^\infty \frac{e^{ipx}}{x+ib}$$ ...
0
votes
1answer
133 views

Is f(z)=1/z truly an analytic function

For an analytic function $f(z)$, we have $$\frac{\partial f}{\partial \bar{z}}=0.$$ Consider the function $f(z)=\frac{1}{z}$, which, at first sight, is a bona fide analytic function. However, we can ...
2
votes
1answer
40 views

I want to compute $\int_0^\infty \frac{x^t}{1+x^2}dx \; \forall t \in (-1,1)$ using residue theroem.

I want to compute $$\int_0^\infty \frac{x^t}{1+x^2}dx \qquad \forall t \in (-1,1)$$ using residue theroem. I consider $$f(z) = \frac{z^t}{1+z^2}$$ I find two pole of order 1 in $z=i$ and $z=-i$ with ...
3
votes
0answers
91 views

Is there a simple and fast way of computing the residue at an essential singularity?

Is there a simple and fast way of computing the residue at an essential singularity ? I mean if we have a pole of order $n$ at $c$ we can use the formula : $$\mathrm{Res}(f,c) = \frac{1}{(n-1)!} ...
1
vote
1answer
62 views

Complex contour integral with sign function:$-i \int \limits_{-\infty}^\infty \frac{{\rm sgn}(x)^2 ~x~ e^{i x}}{1+ax^2} dp$

I am trying to evaluate the integral: $-i \int \limits_{-\infty}^\infty \frac{{\rm sgn}(x)^2 ~x~ e^{i x}}{1+ax^2} dx$ with sgn$(x)$ the sign function and $a$ positive real. Naively applying the ...
2
votes
2answers
62 views

compute the integral using residue theory

I am trying to compute an integral in an example in my complex analysis textbook: $$\int_{-\infty}^\infty {xsinx\over x^4+1}dx$$ The book gives some startup hints, but I don't quite follow, I set ...
0
votes
0answers
20 views

computing integral using residue theory [duplicate]

I want to compute the integral $\int_{-\infty}^\infty {x^4\over {1+x^8}}dx$ by using residue theory. I find the zero of $Q(x)$ is $i^{1/4}$. Do I have to factor the denominator into 8 different ...
2
votes
1answer
59 views

Find the Laurent series of the function and give the residue

I have $f(z)={z^2\over {z^2-1}}$. I want to find the Laurent series of $f$ and the residue at the point $z_0=1$. Can I say that $f$ has a pole of order 2 at $z=1?$ Or is that only used when the ...