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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2
votes
2answers
45 views

How to compute the residue of $(z^2+2z+1)\sin\left(\frac{1}{1+z}\right)$

This was an example given in my notes but all it concluded was with something about an infinite principal part. How do we compute it? we have it equal to $ \left( z + 1 \right)^2 \cdot \sin \left( ...
0
votes
2answers
57 views

How to find the residues of $\frac{1}{(z^4+4)^2}$?

How to find the residues of this function? $$\frac{1}{(z^4+4)^2}$$ So far, I found the poles: $z_1=-1-i$, $z_2 = -1+i$, $z_3=1-i$, $z_4=1+i$. I know they are of the second order. But I have ...
1
vote
0answers
20 views

calculating the residue of a complex function in an integral

I wonder how to solve this integral: $$P(q, \omega) = \frac{-2i}{(2\pi)^4}\times\int\frac{\mathrm d\mathbf{k}~\mathrm d\omega' e^{i\omega'\eta/h}}{\left[\omega' - E(\mathbf k) + ...
2
votes
1answer
48 views

Complex Analysis Integrals

I'm unsure how to apply what I've learned in complex analysis to the following question types: $$ \int_{-\pi}^\pi \frac 1 {1 + \sin^2(\theta)}\,d\theta $$ and $$ \int_{-\pi}^\pi \frac ...
2
votes
0answers
26 views

Calculate a complex integral using residues

Let $f(z)= \frac{2(e^\frac{1}{z}-1)(\sin^2z)}{z^3}$. Calculate $\int\limits_{\partial B_+(O,1)} f(z)\operatorname{d}z$ Could someone confirm my solution? Solution? I try to calculate the ...
1
vote
2answers
26 views

Compute $\int_\Gamma \frac{e^\frac{1}{z}}{z-1}dz$, where $\Gamma$ is the circle $|z-1|\le\frac{3}{2}$, positively oriented.

Compute $\int_\Gamma \frac{e^\frac{1}{z}}{z-1}dz$, where $\Gamma$ is the circle $|z-1|\le\frac{3}{2}$, positively oriented. The numerator is not analytic in $\Gamma$ so we can't use Cauchy ...
0
votes
2answers
73 views

How do I calculate the residue of $\sin(z+1/z)$?

How do I do this about $\displaystyle z=0$ ?. I tried creating a Laurent expansion and extracting it from there but I wasn't sure how to isolate the $\displaystyle 1/z$ expression. $$ \mbox{I ...
3
votes
0answers
37 views

Contour integral $\int_{|z|=1}\frac{2z^2+z}{z^2-1}\, dz$ using residues

I am trying to evaluate the contour integral $$\int_{|z|=1}\frac{2z^2+z}{z^2-1}\, dz.$$ In this case the two singular points lie on the boundary (on the contour). So do i count the residues at this ...
1
vote
1answer
60 views

Evaluation of $\int_0^{2\pi} \frac{1}{1+8\cos^2(\theta)}d\theta$ with Cauchy's residue Theorem

I have to proof $$\int_0^{2\pi} \frac{1}{1+8\cos^2(\theta)}d\theta = \frac{2\pi}{3}$$ with Cauchy's residue Theorem. I have showed it, but in my solution, there comes $-\frac{2\pi}{3}$. I Show you ...
0
votes
1answer
98 views

integration, laurent series, residue therorem

Evaluate the integral $\int_\gamma f(z)dz,$ where $\gamma(t)=e^{it}$, and $0\leqslant t\leqslant2\pi$. For $f(z)$ equal to: $$\dfrac{e^z}{z^3},\quad\dfrac1{z^2\sin z},\quad\tanh ...
0
votes
1answer
38 views

Residue theorem with contour integrals

I want to evaluate the integral $$ \int_{\gamma} \frac{1}{z^{2}\sin(z)} dz$$ where $\gamma(t) = e^{it}$ and $ 0 \leq t \leq 2\pi$ using the Residue theorem. I've tried expanding sin(z) with Taylor ...
2
votes
3answers
119 views

Evaluate $\int_0^{2\pi} \frac{\sin^2\theta}{5+4\cos\theta}\,\mathrm d\theta$

Evaluate $$\int_0^{2\pi} \frac{\sin^2\theta}{5+4\cos(\theta)}\mathrm d\theta$$ This is the final question on my review for my final exam tomorrow, and I will be honest and say that I have no clue ...
0
votes
0answers
48 views

Evaluate the Cauchy Principal Value of $\int_{-\infty}^{\infty} \frac{\sin x}{x(x^2-2x+2)}dx$

Evaluate the Cauchy Principal Value of $\int_{-\infty}^\infty \frac{\sin x}{x(x^2-2x+2)}dx$ so far, i have deduced that there are poles at $z=0$ and $z=1+i$ if using the upper half plane. I am ...
1
vote
1answer
28 views

Generating function of the Laguerre Polynomials

The Laguerre Polynomials have the following integral representations $$L_{n}^{\alpha} (x) = x^{-\alpha} e^x \frac{1}{2\pi i } \oint_c \frac{e^{-z} z^{n+\alpha}}{(z-x)^{n+1}} dz$$ where $c$ is an ...
2
votes
1answer
51 views

Countour integral using residue theorem

Evaluate the integral $$ \int_{\gamma} \tanh(z) dz $$ where $\gamma(t) = e^{it}$ and $0 \leq t \leq 2\pi$. I want to do this using the residue theorem but I am unsure of how to work out the poles of ...
0
votes
1answer
33 views

Evaluating $\int^{\infty }_{-\infty}\frac {z^3\sin az}{z^4+4}dz$

I'd like to evaluate following integral with contour integration $$\int^{\infty }_{-\infty}\dfrac {z^3\sin az}{z^4+4}dz$$ and I think the best way to solve is to recognize it is equal to the ...
2
votes
4answers
101 views

Evaluate $\int_{-\infty}^\infty \frac{1}{(x^2+1)^3} dx$

Evaluate $\int_{-\infty}^\infty \frac{1}{(x^2+1)^3} dx$ I wasnt exactly sure how to approach this. I saw some similar examples that used Cauchy's theorem.
1
vote
2answers
35 views

Poles of $\frac{1}{1+x^4}$

The integral I'd like to solve with contour integration is $\int^{\infty }_{0}\dfrac {dx}{x^{4}+1}$ and I believe the simplest way to do it is using the residue theorem. I know the integrand has four ...
0
votes
0answers
8 views

Proving susceptibility in Lorentz Model satisy Kramers-Kronig relations

My instructor asked me to prove that the real and imaginary parts of the electric susceptibility derived from Lorentz Model satisfy the Kramers-Kronig relations using the residue theorem. The problem ...
0
votes
1answer
43 views

residue of this function at infinity

How do I calculate the residue of $\frac{\sin(z)}{z}$ at infinity ? I tried to use wikipedia definition for the case, $\lim_{|z|\rightarrow\infty}f(z)=0$ then ...
0
votes
1answer
40 views

If $f(z)$ is a polynomial function of degree $n \ge 2$, prove that the sum of the residues of $\frac{1}{f(z)}$ is zero

Let $f(z)=a_nz^n +a_{n-1} z^{n-1} +...+a_1z+a_0$ be a polynomial of degree $n \ge 2$. Prove that the sum of the residues of $\frac{1}{f(z)}$ is zero. Ok, so here is my thinking process so far: At ...
21
votes
2answers
446 views
+50

How to solve $\int_0^{\infty}\frac{\cos{ax}}{x^3+1}dx$?

QUESTION. It is looked for a closed solution for following real integrals $\displaystyle\int_0^{\infty}\displaystyle\frac{\cos{ax}}{x^3+1}dx$ and ...
0
votes
0answers
56 views

Calculate $\int_0^3 \frac{x^{\frac{3}{4}} (3-x)^{\frac{1}{4}}}{5-x}\,dx$ using principal branch

I would like to calculate the following integral $$ I = \int_0^3 \frac{x^{\frac{3}{4}} (3-x)^{\frac{1}{4}}}{5-x}\,dx $$ using contour integration but using principal branch of the function, i.e. ...
5
votes
1answer
82 views

Residue of $\frac{\cos(\frac{\pi}{z-1})}{z^2 \sin z}$ at $z=1$

Residue of $$\frac{1}{z^2 \sin z}\cos\left(\frac{\pi}{z-1}\right)$$ at $z=1$. More importantly, I don't even know whether it exists or not. The one who creates this question has made questions that ...
1
vote
1answer
32 views

How can I calculate this complex integral?

The integral is the following: $$\int_{|z|=r} \frac{z+1}{z(z^2+4)} dz , r>0, r \neq 2 $$ I'm a little bit lost, I know that its partial fraction expansion is $$ \frac{z+1}{z(z^2+4)} = ...
6
votes
5answers
152 views

Finding $\sum_{n=1}^{\infty }\frac{243}{16(n\pi )^5}\sin(2n\pi /3)$

The WolfarmAlpha couldn't give me the sum of $$\sum_{n=1}^{\infty }\frac{243}{16(n\pi )^5}\sin(2n\pi /3)$$ therefore I thought that this problem is difficult so I used my calculator to get $(1/24)$ ...
3
votes
2answers
73 views

Complex integral using cauchy residue formula

I want to compute $ \displaystyle \int_{0}^{+\infty} \frac{dx}{x^n-1} $ I've proved that $ \displaystyle \int_{0}^{+\infty} \frac{dx}{x^n+1} = \frac{\pi}{n\sin\left(\frac{\pi}{n}\right)}$ in a ...
6
votes
3answers
142 views

Evaluating an Integral by Residue Theorem

Its been awhile since I have taken complex analysis and I am wondering how to solve the following integral when $a>0, \ a=0,$ and $a<0$ for $$\int^{\infty}_0\frac{\cos ax+x\sin ax}{1+x^2}dx.$$ ...
2
votes
1answer
28 views

How to compute contour integral?

Use Residue theorem to compute contour integral $$\int_C \frac{4e^z}{\sin z} dz$$ I need help figuring out singularities that are within the circle $|z|= 4$. I am stuck at that part. Thanks in ...
1
vote
1answer
70 views

Evaluate $\int_0^{2\pi} \frac{d\theta}{\left(1+\beta \cos (\theta )\right)^2}$

I am trying to evaluate the integral $$\int_0^{2\pi} \frac{d\theta}{\left(1+\beta \cos \left(\theta\right)\right)^2}$$ via change of variables and applying Cauchy's Residue Theorem. Here is how I'm ...
2
votes
1answer
34 views

Complex integral and Laurent series

Could you help with solving this complex integral: $$\int_C z^3\exp{\left(\dfrac{-1}{z^2}\right)} dz$$ where $C$ is $|z|=5$. I am expecting that the Residue Theorem will be needed. The answer should ...
3
votes
2answers
77 views

Find the Fourier transform of $u(x) = \frac{x \cos(2x)}{(1+x^2)^2}$

Find the Fourier transform of $$u(x) = \frac{x \cos(2x)}{(1+x^2)^2}$$ My work Okay so we want $$\int_\mathbb R \frac{e^{-ixt}x\cos(2x)}{(1+x^2)^2}dx$$ Of course we want to apply the residue ...
2
votes
1answer
46 views

Using complex analysis to find the Inverse Laplace transform

I have been reviewing for my comprehensive graduation exam where I have been solving the Inverse Laplace transform via complex analysis. Consider $$ H(s) = \frac{s^2 - s + 1}{(s + 1)^2} $$ Then we ...
3
votes
1answer
85 views

Evaluating $\int_{-\infty}^{\infty}\frac{1}{(x^2+b^2)^2}dx$

Find $$\int_{-\infty}^{\infty}\frac{1}{(x^2+b^2)^2}dx$$ We see that the only poles are at $x=\pm bi$. Integrating over the semicircular contour implies that it is equal to $2\pi i*Res_{(+bi)}$ ...
3
votes
2answers
87 views

Help finding the residue of $1/(z^8+1)$

Help finding the residue of $1/(z^8+1)$ I'm integrating over $\{ Re^{it} | 0 \leq t \leq \pi \}$, and I found 4 simple poles at $z_0=e^{in\pi/8}$ where $n = 0,...,3$ and I'm trying to calculate ...
2
votes
0answers
25 views

Solution of gaussian integral with hyperbolic cotangent

I was wondering if the integral $$I=\int_{-\infty}^{\infty}d\omega \omega e^{-(\omega/a)^2}\coth(\frac{b\omega}{2})\cos(\omega c)$$ where $a,b,c>0$ can be solved using complex countour ...
5
votes
2answers
63 views

Prove that $\zeta(4)=\pi^4/90$

I am asked to "use the calculus of residues" to prove that $$\displaystyle\sum\limits_{n=1}^{\infty} \frac{1}{n^4}=\frac{\pi^4}{90}$$ I think I can do this given the Laurent series for $\cot z$ ...
1
vote
1answer
38 views

Counting number of roots inside a circle, using Rouche's theorem,

Using the Argument Principle and applying Rouche's Theorem, I know that there are 6 zeroes of the polynomial $$z^{10}-6z^6+3z^4-1$$ inside the unit circle $|z|=1$, no zeroes inside $|z|=1/2$, but I'm ...
3
votes
1answer
38 views

Determining the value of an integral using complex methods

I need to find the value of the following integral using complex analysis: $$\int_{-\infty}^{\infty}\frac{\sin(k_1\ x)+\sin(k_2\ x)}{x^2-a^2}\ dx$$ where $k_1, k_2, a$ are real coefficients. The ...
4
votes
1answer
72 views

Contour integral using residue

Assume that $f(z) \in \{\sqrt{2z^2 + 1}\}$ $,f(0) = 1$ We have a cut: $\gamma = \{|z| = \frac{1}{\sqrt2}, Re(z) \geqslant 0 \}$ $\oint\limits_{|z|=1} \frac{z dz}{(z+2)(f(z) + 3)}$ I found ...
0
votes
0answers
24 views

Problem computing the residue at a simple pole.

I know that the residue of $\frac{1}{(z + 2i)(2z + i)}$ at $z = -i/2$ is $-i/3$, but using the method I was taught; taking the limit as $z \rightarrow -i/2 $ of $\frac{2z+i}{(z + 2i)(2z + i)}$ gives ...
4
votes
0answers
28 views

Are these the correct residues?

$$\int_C \frac{z+1}{z^2-2z} dz$$ for the circle of $\lvert z \rvert = 3 $. Poles are obviously at $ z = {0,2}$. Can I calculate the residues by viewing the fraction in the integral as either $$\int_C ...
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
58 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
40 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
10 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
37 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
34 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
48 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
28 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 ...