Questions on the evaluation of residues, on the evaluation of integrals using the method of residues or in the method's theory.

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3answers
75 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 ...
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1answer
30 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} ...
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1answer
23 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 ...
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1answer
125 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 ...
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2answers
24 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.
5
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3answers
374 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
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3answers
77 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 ...
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1answer
36 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... ...
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2answers
72 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: ...
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1answer
36 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 ...
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0answers
62 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 ...
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0answers
46 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 ...
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0answers
58 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
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1answer
28 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 ?
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1answer
43 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
118 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 ...
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0answers
47 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 = ...
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3answers
63 views

Contour integrals using residues

The question I'm working on is the following: Let $C_R$ be a contour in the shape of a wedge starting at the origin, running along the real axis to $x=R$, then along the arc $0 \leq \theta \leq ...
4
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3answers
82 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} ...
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2answers
127 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 ...
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4answers
168 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
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1answer
58 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 ...
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1answer
40 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
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1answer
90 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 ...
0
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1answer
27 views

Residue of $\frac{\cot{ax}}{x^2-b^2}$?

I am interested to find the residue of $$\frac{\cos{ax}}{(x^2-b^2)\sin{ax}}$$ at $x=b$. How would I go about doing this? I can see that the pole is second order, and so the formula $$\text{res} = ...
0
votes
1answer
39 views

Calculating residue of $(z^3+1)^{-3}$ at $z=\exp(i\pi/3)$

I'm struggling to calculate the residue of $(z^3+1)^{-3}$ at $z=\exp(i\cdot\pi/3)$, which is the pole for the given contour. Using the formula for residue has resulted in a prohibitive double ...
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1answer
87 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: ...
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0answers
41 views

Inverse Fourier transform using Residues for a ratio of hyperbolic functions.

I'm new and glad to be here. I have a problem relating to an inverse Fourier transform. I have $$g(w)= \frac{\sinh{w(a-b)}}{w \cosh{wa}}$$ and want to find $$G(t)$$. I cannot find this in tables so I ...
0
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1answer
26 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
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1answer
37 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
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3answers
342 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 ...
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3answers
54 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
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1answer
59 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 ...
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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
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1answer
74 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
91 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
53 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 ...
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1answer
71 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 ...
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1answer
20 views

Look for the group members $\left ( (\mathbb Z/24\mathbb Z)^*,\underset{24}{\odot} \right )$ and calculate their orders.

Look for the group members $\left ( (\mathbb Z/24\mathbb Z)^*,\underset{24}{\odot} \right )$ and calculate their orders. The elements are $\overline1,\overline2,\ldots,\overline{23}$. Have the ...
3
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1answer
48 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
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1answer
32 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 ...
5
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3answers
139 views

Find the Fourier Transform of $\dfrac{x}{x^4+4}$

I have a problem here that becomes quite difficult to manage. I have to find the fourier transform of: $$f(x)=\frac{x}{x^4+4}$$ I'm sure there will be many ways to do this and I'll post my method ...
2
votes
2answers
96 views

Riemann zeta, why are the residues either zero or one?

One more question, probably equally simple to answer but I don't know how this is true either: Why is the residue of Riemann zeta zero - trivial or non-trivial: $$\text{residue}\left(\zeta ...
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0answers
89 views

obtaining inverse z-transform by different methods

how can i obtain the z-transform of $X(z) = \frac{z+1}{(z-1)(z+2)^2}$ by: 1) Partial fraction expansion, 2) residue theorem, and 3) direct division method any help is appreciated.
5
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1answer
38 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}, \; ...
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1answer
54 views

Deducing Laplace Formulas

I have to compute the followings integrals $\forall\; b\in \mathbb{C},\; \text{Re} \;b \gt0,p\gt 0$ $$ \int_{-\infty}^\infty \frac{e^{ipx}}{x-ib}$$ $$ \int_{-\infty}^\infty \frac{e^{ipx}}{x+ib}$$ ...
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1answer
339 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
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1answer
49 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
94 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
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1answer
74 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 ...