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
2answers
35 views

Residue of $\cot z $ at $z= 0$

I need to calculate the Residue of $\cot z$ at the pole $0$ . Now, I think that the order of the pole is $1$. But I do not know how to show it. So, assuming the order of pole to be $1$, I calculated ...
0
votes
1answer
69 views

Integrating $\int_{-\infty}^{\infty}\frac{e^{ax}}{1+e^x}dx$ for $0<a<1$.

I'm supposed to find the value of $\int_{-\infty}^{\infty}\frac{e^{ax}}{1+e^x}dx$ for $0<a<1$. I wanted to integrate over the upper semicircle of radius $R$, and take the limit as $R\to\infty$. ...
0
votes
0answers
37 views

Using the $\cot (\pi z)$ to find $\sum \frac{1}{n^2}$ [duplicate]

I'm trying to prove the result that $$\sum_1^\infty \frac{1}{n^2}=\pi^2/6$$ using cotangents and residue theory. I know that $\sum f(n)=-$Sum of residues of $\pi \cot (\pi z)f(z)$ at the poles of $f(...
6
votes
3answers
108 views

Calculate $\int_0^{\infty}\frac{1}{(x+1)(x-2)}dx$ using residues

I'm supposed to calculate $$\int_0^{\infty}\frac{1}{(x+1)(x-2)}dx$$ using residues. The typical procedure on a problem like this would be to integrate a contour going around an upper-half ...
-2
votes
1answer
36 views

Integrate $\int_0^{2 \pi } \frac{1}{(a+b \cos^2 (x))^2} \, \mathrm{d}x$ [closed]

I'm having a trouble with this integral expression: $$\int_0^{2 \pi } \frac{1}{(a+b \cos^2 (x))^2} \, \mathrm{d}x$$ I want to solve to using residue but it seems hard.
2
votes
0answers
72 views

Sum of Gamma Function Residues

I was exploring Cauchy's residue theorem with the gamma function and came across an interesting identity. Consider $$\int_{C_R} \Gamma(z) \, dz $$ Over the complex plane where $C_R$ is the curve ...
2
votes
0answers
39 views

$\frac{\omega}{2\pi}\int_0^{\frac{2\pi}{\omega}}\frac{\sin^2\theta \cos^2\omega t}{(1+\beta\cos\theta \sin\omega t)^5} dt$

I'm going to write out the whole problem as it is given to me (bad grammar and all) even though some of the info may be irrelevant to finding a solution. A charge $e$ moving along a straight line ...
0
votes
1answer
21 views

Using Residue Theorem/Laurent Series to evaluate $\int_{c}$ $\frac{z^m}{2-\frac{1}{Z}} dz$

Let $m$ be an integer and $C$ be the circle $C(0; 1)$ traversed in the counterclockwise direction. What is the value of $\int_{c}$ $\frac{z^m}{2-\frac{1}{Z}} dz$ a) when $m\ge-1$? b) when $m<-1$?...
-1
votes
1answer
27 views

Pole on the Real Axis of Complex Integral

Save me! SOS!. Please see image. I'm completely baffled about how to go about solving this. Please can you explain the how you get -pi*i from the 4th line? Can you please explain by this paragraph as ...
0
votes
2answers
68 views

Complex Numbers - Equation manipulation

Please see link, I'm wondering how you end with RHS of equation on second line of image. Please can explain the step and algebra manipulation used Thanks
0
votes
0answers
43 views

Residue of $f(z)=e^{-1/z^2}$ using it's pole order

Find the residue of $f(z)=e^{-1/z^2}$ at $z=0$ Can somebody check both methods? I wrote out the series of $$e^z=1+z+\frac{z^2}{2}!+...$$ and subbed in $-1/z^2$ instead of $z$ to get: $$1-1/z^2+\...
0
votes
1answer
31 views

How to find and classify the singularities of $\frac{e^{-z} \sin(2(z-1)^2)}{(z^2-4)(z-1)^2}$?

How to find and classify the singularities of $$\frac{e^{-z} \sin(2(z-1)^2)}{(z^2-4)(z-1)^2}$$ Here is what I have: I think the singularities are all isolated and are located at $z=1,\pm2$. I have ...
3
votes
2answers
51 views

Contour Integration of $\sin^2(x)/(1+x^2)$

How should I calculate this integral $$\int\limits_{-\infty}^\infty\frac{\sin^2x}{(1+x^2)}\,dx\quad?$$ I have tried forming an indented semicircle in the upper half complex plane using the residue ...
3
votes
1answer
46 views

Using residue theorem to integrate from $-\infty$ to $\infty$

I'm trying to integrate $$\int_{-\infty}^{\infty} {x^2 \over {(x^2 + 1)}^2(x^2 + 2x + 2)} $$ given that the function $$f(z) = {z^2 \over {(z^2 + 1)}^2(z^2+2z+2)} $$ has residues $${9i - 12 \over 100},...
0
votes
0answers
13 views

Cauchy integral with matrix and imposed contour

Let's say I have a matrix $H$ (that is non hermitian), and a real number $E$. The matrix $H$ is a function of $k$, and $H(k)$ varies smoothly. I want to compute the following integral: $$\int_{\gamma}...
1
vote
1answer
70 views

Solve $\int_{0}^{\infty} \frac{x^{1/3}}{1+x^2}dx$ using residue calculus [duplicate]

I need to find the value of the following integral using residue calculus - $$I = \int_{0}^{\infty} \frac{x^{1/3}}{1+x^2}dx$$ I have used substitution to start with. Let $x = t^3$. Then $I $ changes ...
0
votes
1answer
23 views

How to determine the residues of $\frac{z}{\sinh(kz)}$?

What methods can I use to determine the residues of $\frac{z}{\sinh(kz)}$? Singularities occur at $z=\frac{i n \pi}{k}$ for $k \neq 0$ and $n$ an element of the integers. I've attempted a series ...
3
votes
2answers
41 views

Answer not matching for calculating the value of an integral

Consider the following integral $$\int_ {-\infty}^{\infty} \frac{x^2 - x + 2 }{x^4 + 10x^2 + 9} dx \!$$ I need to find its value using residue calculus. So, I considered the complex function $$\int_ {...
1
vote
1answer
46 views

Calculating residue of $f(z)$ around $a=\infty$

Let $f(z) := \frac{3\sin 4z}{z^4}$. Find $$\mbox{res}\left [f(z),\infty\right ]$$ For $a=\infty$, the residue is given as $-\frac{1}{2i\pi}\int_{|z|=R} f(z)\mbox{d}z$. $$\int_{|z|=R} \frac{3\sin 4\xi}{...
3
votes
1answer
22 views

How to choose the point at which we want to calculate residue?

Suppose I have to calculate the following integral using residue calculus - $$\int_ {-\infty}^{\infty} \! \frac{e^{-ix}}{x^2 + 1 } \, \mathrm{d}x. $$ Now my approach is to construct a contour in the ...
1
vote
0answers
43 views

Show that $\int_{C} \frac{e^{az}}{\sin(\pi z)} \,dz \rightarrow 0 \quad as \quad R \rightarrow \infty$

Given the straight line in the complex plane: $b+iR$ to $b+1+iR$ where $0<b<1$ and $|Im(a)|<\pi$, show the following: $$\int_{C} \frac{e^{az}}{\sin(\pi z)} \,dz \rightarrow 0 \quad as \quad ...
0
votes
1answer
37 views

Computing using residue $\int_{0}^{\infty}e^{-x^{2}}\cos(x^{2})\mathrm{d}x$ (but not Gaussian way)

I am wondering if there is a residue-trick for computing $\int_{0}^{\infty}e^{-x^{2}}\cos(x^{2})\mathrm{d}x$ without having to go through computing the Gaussian residue integral. For practice here ...
2
votes
2answers
70 views

Show that $\int_{0}^{\infty} \frac{\cosh(ax)}{\cosh(\pi x)} dx=\frac{1}{2}\sec(\frac{a}{2})$ using Residue Calculus

Show that the following expression is true $$\int_{0}^{\infty} \frac{\cosh(ax)}{\cosh(\pi x)} dx=\frac{1}{2}\sec(\frac{a}{2})$$ Edit: I forgot to mention that $|a|<\pi$ Specifically, using ...
0
votes
0answers
26 views

Complex integration with varying degrees

So I'm studying for an exam and going over past exams and one problem is causing me a little difficulty. Any help would be appreciated. The problem is: Let $0 \leq p < n \in \mathbb{Z}$. Calculate ...
0
votes
1answer
29 views

Find the residue of $f(z)=\frac{1-\cos z}{2\sin z-\sqrt{3}}$ at $\pi/3$

Find the residue of $f(z)=\frac{1-\cos z}{2\sin z-\sqrt{3}}$ at $\pi/3$. Here is my attempt: The Taylor series for $\cos z$ and $\sin z$ about $z=\pi/3$ are \begin{align*} \cos z &= \frac{1}{...
0
votes
1answer
35 views

Using Residue Theory to evaluate $\int_{0}^{\infty} \frac{x^3sin(kx)}{x^4+a^4} dx$

I'm having difficulty evaluating the following integral using residue theory, and would love some advice on proceeding. Below I develop my approach to the problem: $$\int_{0}^{\infty} \frac{x^3sin(kx)...
-2
votes
1answer
41 views

Show that there is no nth roots in $U$.

Let $U\subseteq\mathbb{C}\setminus\left\{0\right\}$ be an open set and suppose that there is a path $\gamma$ in $U$ such that $\mbox{Ind}_{\gamma}(0)=1$. Show that there is no nth roots in $U$. $(n\...
1
vote
1answer
38 views

Residue Theorem: compute the integral $\int_0^\infty \frac{x \sin x}{x^4+4a^4}$

Compute the integral $$\int_0^\infty \frac{x \sin x}{x^4+4a^4}$$ Since, it's an even function I can rewrite the expression as $$\frac{1}{2}\int_{-\infty}^\infty \frac{x \sin x}{x^4+4a^4}$$. In the ...
0
votes
0answers
33 views

Let $R$ rational function. Show that $\int_0^\infty R(x)dx=-\sum_{w\in \widetilde{\mathbb{C}}}\mbox{Res}_{w}(R(z)\ln(z)).$

Let $R=\frac{P}{Q}$ where $P$ and $Q$ are polynomials such that $Q(x)\neq 0$ for all $x\geq 0$. Suppose that the degree of $Q$ exceeds that of $P$ be at least 2. Show that $$\int_0^\infty R(x)dx=-\...
1
vote
0answers
67 views

Finding a closed form for $\sum_{n=-\infty}^{n=+\infty}\frac{1}{n^{2k}+a^{2k}}$

I am trying to find a closed form for $S=\sum_{n=-\infty}^{n=+\infty}\frac{1}{n^{2k}+a^{2k}}$, $k \in \mathbb{N^{*}}$, $a>0$ I don't even bother to look for a closed form with an odd exponent, ...
0
votes
1answer
59 views

What is wrong in this use of Cauchy residue theorem?

Consider $F(z)$ a function such that $\overline{F(z)}=F(\overline{z})$, with no pole, decreasing faster than any power $\frac{1}{z}$ when z is imaginary going to $_{-}^{+}i \infty$. I define the ...
0
votes
1answer
25 views

How do I show the following modification of the Counting formula of zeros and poles?

Let $U\subset \mathbb{C}$ be an open and connected set, $g: U\rightarrow \mathbb{C}$ holomorphic function, $f$ meromorphic function in $U$ with zeros in $z_{1},z_{2},\ldots,z_{n}$ and poles in $p_{1},...
2
votes
1answer
41 views

Show that the Cauchy principal value of $ \int_{-\infty}^{\infty}\frac{P(x)}{Q(x)}dx$ exists when $\mbox{deg}(Q)=\mbox{deg}(P)+1$.

Let $R=\frac{P}{Q}$ where $P$ and $Q$ are polynomials such that $Q$ has not zeros in $\mathbb{R}$ and $\mbox{deg}(Q)=\mbox{deg}(P)+1$. Show that the Cauchy principal value of $ \int_{-\infty}^{\...
0
votes
1answer
33 views

Compute $\mathrm{Res}(\frac{e^{iz}}{z(z^2+1)^2},i)$

I have to compute $\mathrm{Res}(\frac{e^{iz}}{z(z^2+1)^2},i)$. Do I have to use the result from $Res[f/g,z_0]=\frac{f(z_0)}{g'(z_0)}$ -Proof (I think I have a pole of order $2$)? Otherwise, how ...
0
votes
1answer
27 views

Prove that $\mathrm{Res}(\frac{f(z)}{g(z)}, z_0) = \frac{f(z_0)}{g'(z_0)}$ [duplicate]

Show that if $f$ and $g$ are analytic on a neighborhood of $z_0$ with $f(z_0)\not= 0$ and $z_0$ is a simple zero of $g$, then we have $\mathrm{Res}(\frac{f(z)}{g(z)}, z_0) = \frac{f(z_0)}{g'(z_0)}$...
0
votes
1answer
20 views

Compute $\mathrm{Res}(\frac{1+2z+3z^2}{1+z+z^2-3z^3},1)$

I have to compute $\mathrm{Res}(\frac{1+2z+3z^2}{1+z+z^2-3z^3},1)$. I know that $\mathrm{Res}(f,z_0)+a_{-1}= \int_{C_p} \frac{f(z)dz}{z-z_0}$, where $C_p$ is simply the circle at $z_0$ with radius ...
0
votes
0answers
28 views

How to solve the laplace transform of $f_m(t_m)$ = $f_1(t)$ $\int_{0}^{\alpha} f_2(\tau) d\tau$ + $f_2(t)$ $\int_{0}^{\alpha} f_1(\tau) d\tau$.

Could you please help me to solve the following : if $t_m$ = min($t_1$,$t_2$) The probability density function $t_1$ is $f_1(t_1)$ and $t_2$ is$f_2(t_2)$ then $f_m(t_m)$ is the probability density ...
1
vote
2answers
38 views

Residue Theorem for function quotients.

let $G$ be an open disc centered around $z_0$ of radius $r$. Let $f(z),g(z)$ be holomorphic functions on $G$. such that $f(z)$ has a simple zero at $z_0$. Find an expression for the residue of $\frac{...
1
vote
1answer
26 views

Use a rectangular contour to evaluate the integral

$$\int_{-\infty}^{\infty} \frac{\cos(mx) dx}{e^{-x}+e^x} = \frac{\pi}{e^{m\pi /2}+e^{-m\pi /2}}$$ I need to evaluate the above integral specifically using a rectangluar contour and at some point ...
1
vote
0answers
51 views

Laurent series for $\cot(\pi z)/z^2$

I'm looking for the series for $\frac{cot(\pi z)}{z^2}$ using the residue theory, where the function denotes a circle about the origin with a radius of $k+\frac{1}{2}$ I found that the residues of $\...
1
vote
3answers
63 views

Residue of $f(z) = \frac{1}{z-\sin z}$ at $z=0$

My attempt: $$ f(z) = \frac{1}{z-\sin z}$$ $$\frac{1}{z-(z-\frac{z^3}{6}+\frac{z^5}{120}-...)}$$ $$\frac{1}{z(1-(1-\frac{z^2}{6}+\frac{z^4}{120}-...))}$$ $$Res(f(z),0) = \lim_{z \to 0} z \cdot \...
0
votes
1answer
57 views

Using contour integration to solve this integral

We need to use contour integration to solve $$\int_{-\infty}^\infty {e^{ax}\over e^x+1}dx$$ given that $0<a<1$. My question is about what contour to use, knowing that the singularities are at $z=...
1
vote
1answer
33 views

Detailed proof of why integral over the upper semi-circle in $C$ of $\frac{e^{ix}}{x^2 + a^2}$ goes to $0$ as the radius goes to $\infty$?

This is a follow up question to this question: Computing $\int_{-\infty}^{\infty} \frac{\cos x}{x^{2} + a^{2}}dx$ using residue calculus. For clarity, I'll reproduce the question here: calculate $\...
2
votes
1answer
52 views

Pde using laplace transform

Could you help me to find a solution for this partial differntial equation by using laplace transform $$u_{t} - u_{xx} = xt$$ where $$u(0,t)=t, \quad u(1,t)=t^2, \quad u(x,0)= \sin \pi x$$ I tried ...
0
votes
2answers
48 views

Evaluate using complex integration: $\int_{-\infty}^\infty \frac{dx}{(x^2+1)(x^2+9)}$

Evaluate $$\int_{-\infty}^\infty \frac{dx}{(x^2+1)(x^2+9)}$$ Firsly I found the residues of this function: $$Res(i)=-i/16$$ $$Res(-i)=i/16$$ $$Res(3i)=i/48$$ $$Res(-3i)=-i/48$$ I then closed ...
1
vote
0answers
32 views

Use the theory of residue to evaluate $\int_0^{2\pi} \frac{\cos(3\theta)d\theta}{5-4\cos(\theta)}$ [duplicate]

My attempt: $$\int_0^{2\pi} \frac{\cos(3\theta)d\theta}{5-4\cos(\theta)}$$ I know I can substitute $\cos (\theta)$=$\frac{e^{i\theta}+e^{-i\theta}}{2}=\frac{1}{2}(z+\frac{1}{z})$, but I'm stuck on ...
1
vote
1answer
25 views

How to find the Residues of $f(z)=\frac{1}{(z^2+1)^2}$?

How to find the Residues of $$f(z)=\frac{1}{(z^2+1)^2}$$ So far I've wrote $$f(z)=\frac{1}{(z^2+1)^2}=\frac{1}{(z+i)^2(z-i)^2}$$ so $f$ has isolated singularities at $z=\pm i$. But I don't know ...
0
votes
1answer
20 views

Residual plot in the logarithmic model.

We've got some data containing two variables, where $x$ is the predictor and $y$ is the response variable. We make a model of the form of: $$y=\alpha+\beta \cdot x + \epsilon$$ Then we see that in the ...
3
votes
1answer
88 views

Use of residues to find I=$\int_0^\infty \frac{\sin^2(x)}{1+x^4} dx$

I'm working on the problem $$I=\int_0^\infty \frac{\sin^2(x)}{1+x^4} dx$$ I found 4 singularities and i would like to use the singularities in the 1st and 2nd quadrants to solve this integral; i.e. $...
1
vote
1answer
38 views

Evaluating contour in the form $\int_{o}^{\infty} \frac{x^{-a}}{1+x}dx $

Having this improper integral $$ \int_{0}^{\infty} \frac{x^{-a}}{1+x}dx$$ I apply the form as such $x^{a}R(x)$ as such $$ \oint_{\Gamma} z^{a}R(z)dz = \frac{2\pi i}{1-e^{2\pi ia}}\sum_{poles} Res_{...