1
vote
2answers
32 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
20 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
13 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
33 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.
2
votes
0answers
21 views

complex integral with non integer power

I want to calculate this integral ...
1
vote
3answers
55 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
47 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
0answers
76 views

Another theorem of Principal value

Let $f$ holomorphic function with isolated singularities in neighborhood of $\bar{\mathbb{H}}^+\lbrace z\in \mathbb{C}\vert 0\leq\vert z\vert\rbrace$ and suppose that f only have one sigularity on ...
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 $\partial C$), ...
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
63 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: ...
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
20 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
75 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
46 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
35 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 ...
3
votes
0answers
81 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
0answers
59 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
302 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
59 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
60 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
25 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 ...
4
votes
1answer
83 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
101 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 ...
2
votes
0answers
96 views

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

I'm trying to use the residue theorem to find the limit of $$\sum_{n=1}^{\infty} \frac{1}{n^4}.$$ So I am considering the function $$f(z) = \frac{\pi \cos(\pi z)}{\sin (\pi z)z^4}$$ on a square ...
1
vote
3answers
74 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
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} ...
1
vote
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 ...
1
vote
1answer
122 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
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
votes
3answers
372 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
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 ...
0
votes
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... ...
2
votes
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: ...
1
vote
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 ...
1
vote
0answers
60 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
45 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
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
vote
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 ?
0
votes
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 ...