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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0answers
21 views

Guidance or advice with the determination of the type of singularities

I need to determine the type of singularity in $f$ at $z_0=0$ and calculate the Residue at that point. $$ \begin{align} f(z)=(z^2+z) \cos\left(\frac{1}{z}\right) \end{align} $$ I know that ...
0
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0answers
55 views

Residue with half order pole?

I'm having issues evaluating the following integral using Cauchy's residue theorem. $$\int_{-\infty}^{\infty} \frac{e^{ix}}{\sqrt{x^2 - 1}} dx $$ Here's what I have tried. We have to make a ...
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2answers
38 views

Evaluation of the type of singularity and Residue at that point

$$ \begin{align} f(z)=z^2e^{\frac{1}{z^3}} \end{align} $$ I need to determine the type of singularity and evaluate the Residue at $z_0=0$ We know that $e^{\frac{1}{z}}$ has an essential singularity ...
2
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1answer
38 views

Can use residue theorem for this integral

I need to compute $$I=\int_C \dfrac{e^{\sqrt{1+u}}\cdot\sqrt[4]{1+u}}{\sqrt{u}} \,\mathrm {d}u$$ where $C$ is the unit circle. I am confused about whether I can use the residue theorem to compute it? ...
1
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1answer
32 views

How to calculate complex residues

How would one best calculate the residue of $$f(z)=\frac{z^2}{z^6+1}$$ At its various poles? My method is to use L'hopital to calculate $\lim_{z\to root}(z-a)f(z)$ but this is rather slow and ...
1
vote
2answers
103 views

Complex analysis integral (maybe using residue theory?)

Trying to compute the following integral: $$ \int_{-\pi}^{\pi} \frac{d\theta}{1+\sin^2(\theta)}. $$ This is for a complex analysis course, so I'm trying to find a way to use residue theory of ...
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2answers
46 views

Residue of $\frac{z}{Log{z}}$ at $z=1$?

What is the residue of $\frac{z}{Log{z}}$ at $z=1$? I tried expanding log into it's series, factoring out a $z-1$ and then applying the geometric series expansion, but I'm just getting a series that ...
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0answers
73 views

Residue Integral: $\int_0^\infty \frac{x^n - 2x + 1}{x^{2n} - 1} \mathrm{d}x$

Inspired by some of the greats on this site, I've been trying to improve my residue theorem skills. I've come across the integral $$\int_0^\infty \frac{x^n - 2x + 1}{x^{2n} - 1} \mathrm{d}x,$$ where ...
4
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0answers
25 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 ...
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1answer
39 views

Writing the floor function as a contour integral

The function $f(z)=\frac{\pi}{\sin \pi z}$ has simple poles of residue 1 at the integers. Hence, by the residue theorem, I consider the interesting idea of drawing a (perhaps rectangular, for example) ...
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1answer
41 views

Is there an analytic function defined on $\Bbb C$ except for Gaussian integers where it has poles of order 1 and residue 1?

I need a function defined for all complex variables $z$, except for at all the Gaussian integers, where it has poles of order 1 and residue 1. The function has to be complex-analytic. Can anyone ...
0
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0answers
49 views

Proper way to set up “Pac-Man” contour integral

I'm trying to evaluate $$ \int_0^\infty \frac{x^a}{1+x} \: dx, \: -1<a<0 $$ using contour integrals. Actually, I have found the correct answer using a "Pac-Man" contour and residues. My only ...
10
votes
4answers
255 views

Contour Integral: $\int^{1}_{0}\frac{1}{\sqrt[n]{1-x^n}}dx$

I want to compute: $\int^{1}_{0}\frac{1}{\sqrt[n]{1-x^n}}dx$ for natural $n>1$ using Residue Calculus. I am thinking of using some kind of a keyhole or bone contour that could go around the ...
1
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2answers
38 views

Determine the nature of singularities and calculate the residue of $f(z)=\frac{e^z-\mathrm{sin}z-1}{z^5+z^3}$

$$f(z)=\frac{e^z-\mathrm{sin}z-1}{z^5+z^3},\;\;\;\;\;\;\; \mathrm{Res}[f(z),0]$$ I am having trouble determining the nature of singularities. This is what I managed to do: ...
2
votes
1answer
56 views

Calculating $\int_0^\pi \sin^2t\;dt$ using the residue theorem

I want to use the residue theorem to calculate $$I:=\int_0^\pi \sin^2t\;dt$$ Since $\sin^2$ is an even function, we've got $$I=\frac{1}{2}\int_0^{2\pi}\sin^2t\;dt$$ The solution of this exercise ...
1
vote
2answers
58 views

complex variable integral using residue theorem [duplicate]

I am asked to calculate a complex integral. how to compute $\displaystyle \int \limits_{-\infty}^{\infty}\frac{x^4}{1+x^8}dx$ with residue theorem?
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1answer
36 views

holomorphic function over the disk that is real on a closed curve must be constant

Let $f$ be holomorphic on $\{z\in \mathbb{C}\mid |z|\leq 3\}$ and real on the boundary of the square $\{z\in\mathbb{C}\mid Re(z)\leq1 \text{ and } Im(z)\leq 1 \}$. Prove $f$ is constant. How to ...
2
votes
1answer
22 views

Application of Rouché: Equality of a power series and a finite series

Let $f(z) = \sum_0^\infty{a_n z_0^n}$ be a complex power series with radius of convergence $R>0$ and let $z_0 \epsilon \, \mathcal{U}_R(0)$ an arbitrary point. I need to show with $Rouché$ : For ...
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2answers
35 views

Contour intergals of rational fuction

Consider $$F=\frac {x}{x^3+y^3}dx+\frac{y}{x^3+y^3}dy$$ 1) Show that $\int_GF=0$, where $G$ is the arc of a circle or radius $r$ in the first quadrant. 2) Compute the integral of $F$ along the ...
4
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1answer
73 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 ...
0
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1answer
72 views

Solving contour integral

I have the integral $$ I = \int_{-\infty}^\infty dk \frac{k}{\sqrt{k^2+m^2}} e^{ikx} $$ I want to show that it decays exponentially with large $x$. I was able to solve the integral with a few tricks. ...
2
votes
1answer
108 views

using residue for integration

Hi how do u calculate the integral which have square root ? for example for this integral (because of branches points I always baffle) : $$\int_0^1 \frac{(1-x)^{1/4}\, x^{3/4}}{5-x}\, dx$$
3
votes
2answers
99 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
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1answer
42 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
41 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
55 views

inverse laplace transform of $$p^{-3/2}e^{-\sqrt{pa}}(\cos(\sqrt{ap})+\sin(\sqrt{ap}))$$

I used the Residue theorem to solve this problem. But, I could not obtain the solution given by $$\mathscr{L}^{-1}\left( { p^{-3/2}e^{-\sqrt{pa}}\over{2\sqrt{2}}} [\cos(\sqrt{ap})+\sin(\sqrt{ap})] ...
2
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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
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3answers
65 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
24 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
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1answer
20 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
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1answer
83 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
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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.
5
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3answers
344 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
75 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
32 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
70 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
30 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
49 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
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0answers
40 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
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0answers
56 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
24 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
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1answer
36 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
109 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
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0answers
46 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 = ...
0
votes
3answers
62 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
votes
3answers
74 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
103 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
162 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
53 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
36 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 ...