Tagged Questions

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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0
votes
1answer
12 views

Why is the equation $\frac{(z-i)^2}{(z^2+1)^2}=\frac{1}{(z+i)^2} $ in the residue theorem accurate?

I don't understand the reasoning here: $\frac{(z-i)^2}{(z^2+1)^2}=\frac{1}{(z+i)^2} $
3
votes
1answer
46 views

A suitable integration path for $\cos z/(2 + \cos z)$

I was solving the exercises and got stuck when trying to solve this with tools of residual calculus $$ \int_{0}^{2 \pi} \frac{\cos (z)}{2 + \cos (z)} \, dz = \int_{0}^{2 \pi} f(z) \, dz. $$ Isolated ...
0
votes
1answer
18 views

Find the Residue of $\frac{e^z}{sin^2(z)}$ at each finite singularity

The problem states: Find the Residue of $f(z)=\frac{e^z}{sin^2(z)}$ at each finite singularity. The poles are clearly at $z=k\pi (k\in\mathbb{Z})$, and the order are all 2, since: $\lim_{z \to ...
0
votes
1answer
9 views

Apply residue formula 1

This seems to be a bit of a stupid question, but I can't get it right and need som help. I've a formula that says: If $$f(z) = (z-a)^{-N}g(z)$$ then $$Res_{z=a} f(z) = \frac{g^{N-1}(a)}{(N-1)!} $$ I ...
1
vote
1answer
33 views

determining a residue by taking a limit

To determine a residue, I need to take this limit: $$\lim_{z\to 2\pi ik} \frac{d}{dz}\frac{(z-2\pi ik)^2}{z(e^z-1)^2}$$ with $k$ any integer number (like -1, -1, 0, 3, 7) I have tried l'Hopital's rule ...
0
votes
0answers
27 views

A Contradiction of Riemann Zeta Residues

We can show (1+2+3...+n)^2 = 1^3 + 2^3 + ... +n^3, which holds for any finite n, shouldn't this imply Z(-1)^2 = Z(-3)? However, this does not hold if we look at the residues of the zeta function ...
1
vote
2answers
44 views

$\tan(z)$ with residue theorem

Calculate $$\oint_{|z|=2}\tan(z)\,dz$$ because $\tan(z)=\dfrac{\sin(z)}{\cos(z)}$ the poles are when $\cos(z)=0$ at $z=\pm\pi/2\pm n\pi, \;n\in\mathbb{Z}$ Poles inside $|z|=2$ are $\pm\pi/2$ and ...
0
votes
2answers
24 views

Integral with residues

Calculate integral $$\oint\limits_{\gamma}\frac{e^z}{z^4+5z^3}dz$$ Where $\gamma$ is parameterization of one rotation of circle $A(0,2)$ So if I write the integral like this ...
0
votes
1answer
35 views

Evaluating the integral of $1+z+1/\tan z$ over a circle

I am a beginner and I want to learn how to solve these kind of integrals: $$\int_{|z|= \pi/4}\left(1+z+\frac{1}{\tan z}\right)\,dz$$ So should I divide it in three integrals, calculate each integral ...
0
votes
1answer
14 views

Singularity and residue in z = 0

How can I classify the singularity in $z = 0$ and determine the respective residue in $z = 0$ for the following function ? f(z) = $ cos(1/z)(z+1)^2$ Do I have to use Taylor expansion of $cos(1/z)$ ...
2
votes
1answer
34 views

how evaluate $\int_0^{\pi}\frac{1}{(a+\cos{\theta})^2}, a>1 $, using residues theorem?

how evaluate $\int_0^{\pi}\frac{1}{(a+\cos{\theta})^2}, a>1 $, using residues theorem? This problem is an exercise book Complex Analysis of Conway.
0
votes
2answers
34 views

Evaluation of real integral by residues

I have to calculate the integral $$I=\int_{0}^{2\pi}\frac{\cos^2(3\theta)}{5-4\cos(2\theta)}d\theta$$ using residues, but I'm having trouble calculating the residue in $z=0$. If you rewrite the ...
2
votes
0answers
24 views

complex integral with non integer power

I want to calculate this integral ...
1
vote
3answers
61 views

Using Residue Theorem to calculate the integral

for $$I=\int_{|z|=1}{z^m \cos\left(\frac{1}{z}\right)}\,dz$$ where $m=0,1,2...$ Is the singularity $z=0$ or there are some other singularities? if it is $z=0$, what's order of pole?
1
vote
1answer
55 views

The Residues of an even function or an odd function on $U$ subset open symmetric

I have to proof that for $f$ even function holomorphic with singularities isolated then $$res_{z}f=-res_{-z}f$$ an simmetric for $f$ odd, $i.e.$ $$res_{z}f=res_{-z}f$$ My hint is proof that laurent ...
3
votes
1answer
143 views

Another theorem of Principal value

Let $f$ holomorphic function with isolated singularities in neighborhood of $ \overline{\mathbb{H}}^+ = \{ z\in \mathbb{C} : \operatorname{Im} z \geqslant 0\}$ and suppose that f only have one ...
1
vote
2answers
60 views

Evaluate $\int_{\partial C} \frac{dz}{(z-a)(z-b)}$ where $\partial C$ is the boundary of a rectangle ($a$ and $b$ are not on $\partial C$)

In discussing the possible outcomes of the integral $$\int_{\partial C} \frac{dz}{(z-a)(z-b)}$$ where $\partial C$ is the boundary of a rectangle ($a$ and $b$ are complex and not on $\partial C$), ...
1
vote
1answer
36 views

Inverse Laplace of $\frac{\sinh{x\sqrt{s}}}{s^2\sinh{\sqrt{s}}}$

What is the inverse Laplace of $\frac{\sinh{x\sqrt{s}}}{s^2\sinh{\sqrt{s}}}$? Using the residues, I can calculate the residues at $s_n=2n\pi i$, but I have problem in calculating residue at $s=0$. ...
0
votes
0answers
15 views

How do I compute the following residue

Not really sure how to tackle this problem. Compute the following residue: 3^3 + 2 mod 5 Any hints in the right direction would be appreciated.
0
votes
1answer
25 views

Residue Calculus (Computing an Improper Integral)

Use residue calculus to compute the integral $\int_{-\infty}^{\infty}\frac{1}{(z^{2}+25)(z^{2}+16)}dz$ My solution If we add to the interval $I_{R}=[-R,R]$ add the semicircle $\gamma_{R}$ in the ...
0
votes
0answers
19 views

Complex analysis, poles and singularities and boundedness

So I am on the following problem: Prove that an isolated singularity of $f(z)$ is removable as soon as either $\text{Re}f(z)$ or $\text{Im}f(z)$ is bounded above or below. The hint is to use a ...
1
vote
1answer
64 views

Can I use the residue calculus here?

I know from an alternative to the residue calculus that $$I=\int_{-\infty}^{\infty}\frac{e^x}{e^{e^x}+1}dx = \log 2.$$ However, I see no reason I cannot apply the residue calculus here. My attempt: ...
0
votes
2answers
27 views

How to orient the integration contour when applying the residue theorem?

For fixed $s$ and $k$ real positive numbers, I consider the $2\pi$-periodic function $f:\mathbf R\to\mathbf C$ defined by $$f(x)=\frac1{s+\mathrm i k\cos x}$$ and want to compute its Fourier series ...
1
vote
1answer
28 views

Why is the residue of $\dfrac{1}{z-w}R(z)$ at $w$ not continuous as a function of $w$?

Let $R(z)$ be some fixed rational function, and define $Res_R(w)$ to be the residue of $f(z)=\dfrac{1}{z-w}R(z)$ for any $w\in\mathbb{C}$. I would have thought that $Res_R$ would be continuous in ...
1
vote
0answers
22 views

Validity of Laurent series's principal part calculation

I need to calculate the principal part of the Laurent expansion of $f$ around a given $z_0$ in an annulus of the form $\{z\in \mathbb{C}:0<|z-z_0|<r$} and then use this to find $Res(f,z_0)$ ...
0
votes
0answers
27 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
votes
0answers
78 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 ...
0
votes
2answers
48 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
votes
1answer
42 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
vote
1answer
36 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
111 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 ...
0
votes
2answers
48 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 ...
3
votes
0answers
84 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
votes
0answers
28 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 ...
0
votes
1answer
42 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) ...
0
votes
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
votes
0answers
63 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
5answers
304 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
vote
2answers
41 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
61 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
61 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?
0
votes
1answer
39 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
26 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 ...
0
votes
2answers
37 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
votes
1answer
84 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
votes
1answer
74 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
110 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
102 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
43 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
43 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 ...