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
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1answer
23 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 &= ...
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} ...
-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 $U$. ...
1
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1answer
36 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 ...
0
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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 ...
1
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0answers
66 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
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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 ...
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 $ ...
0
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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, ...
0
votes
1answer
25 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) = ...
0
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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 ...
0
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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 ...
1
vote
1answer
24 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 ...
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0answers
46 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
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3answers
58 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
54 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 ...
1
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1answer
24 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
51 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
47 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
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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
24 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
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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
85 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
35 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} ...
4
votes
2answers
111 views

Problem over a definite integral, which surely needs contour integration

During my Master Thesis work I came up with an integral which I am going to consider as a hard challenge. I have been trying for days to crack it, but still nothing. The integral is the following ...
1
vote
1answer
28 views

Complex: evaluating integral with residues

Having a bit of trouble here. Having this integral $$ \int_{0}^{\infty} \frac{dx}{(x^{2}+1)(x^{2}+4)^{2}} $$ I can tell it's even, so it has symmetry. Thus, $$ \frac{1}{2} \int_{-\infty}^{\infty} ...
1
vote
0answers
22 views

Residue for $\frac{\zeta(s)}{\zeta(2s)}$ at zeros of $\zeta(2s).$

I want to calculate residue at the poles for $\frac{\zeta(s)}{\zeta(2s)}.$ For pole of numerator $s=1$ I have calculated the residue. I am having trouble at the zeros of denominator. Basically I am ...
0
votes
1answer
16 views

Write $\mbox{Res}(g, 0) $ in terms of the $\mbox{Res}(f, z)$ where $g(z)=\frac{1}{z^{2}}f\left(\frac{1}{z}\right)$.

Let $M\subseteq \mathbb{C}$ finite set and $f:\: \mathbb{C} \setminus M \longrightarrow \mathbb{C}$ be a holomorphic function. Consider $g(z)=\frac{1}{z^{2}}f\left(\frac{1}{z}\right)$. The quiestion ...
0
votes
1answer
30 views

Show that the numbers $3$,$3^2$,$3^3$,$3^4$,$3^5$,$3^6$ for a reduced residue system modulo 7.

Show that the numbers $3$,$3^2$,$3^3$,$3^4$,$3^5$,$3^6$ for a reduced residue system modulo 7. -A bit lost with this question, we just started a section on reduced residue sets and only covered ...
3
votes
1answer
73 views

Cauchy Principal Value of $\int_0^\infty \frac{x}{(x^2 + a^2) \, \sin(\mu x)} dx$

The problem here is to evaluate $$ \int_0^\infty \frac{x}{(x^2 + a^2) \, \sin(\mu x)} dx $$ for $a,\mu >0.$ Clearly this integral doesn't converge in the usual sense, but we can calculate its ...
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2answers
72 views

Inverse Laplace Transform of $e^{\frac{1}{s}-s}$

doing some work on a PDE system I have stumbled across a Laplace transform which I'm not sure how to invert: $$ F(s) = e^{\frac{1}{s}-s} $$ I can't find it in any table and the strong singular growth ...
0
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2answers
131 views

How can $\sin x = e^{iz}$?

This is probably a trivial question but I just don't see it. I'm solving the integral $$\int_{0}^{\infty}\dfrac{x \sin x }{x^2 + 5x + 4}dx$$ using the residue theorem. The thing is that they're ...
1
vote
1answer
25 views

Show that the numbers $-13, -9, - 4, -1, 9, 18, 21$ form a complete residue system modulo 7

Show that the numbers $-13, -9, - 4, -1, 9, 18, 21$ form a complete residue system modulo 7 We have just started he section on modular arithmetic so I am new to a residue system, we did a similar ...
4
votes
1answer
61 views

Cauchy Residue Theorem Integral

I have been given the integral $$\int_0^ {2\pi} \frac{sin^2\theta} {2 - cos\theta} d\theta $$ I have use the substitutions $z=e^{i\theta}$ |$d\theta = \frac{1}{iz}dz$ and a lot of algebra to transform ...
2
votes
1answer
70 views

Integrate $I=\int_e^\infty\left(\frac{\log\log y}{y(y+1)}\right)^2dy$ using residue calculus?

Can this integral be done using the residue calculus? $$I=\int_e^\infty\left(\frac{\log\log y}{y(y+1)}\right)^2dy$$ ? My (empirical) investigative attempts have been to use a keyhole contour centred ...
0
votes
1answer
43 views

Complex integral of the function $f(z)=\dfrac{1}{z^4+1}$

I must calculate this integral $$\int_C\dfrac{dz}{z^4+1}$$ , where $C$ is the circle $x^2+y^2=2x$. My result is $\int_C\dfrac{dz}{z^4+1}=-\dfrac{\pi}{\sqrt{2}}$ , but my book "A collection of problems ...
2
votes
2answers
66 views

Improper integral using residue theorem

I am meant to use the residue theorem to show that $\int\limits_{-\infty}^\infty \frac{\cos t}{(t^2+1)^2}dt=\frac{\pi}{e}$. So far I have deduced that I should take a contour over $\alpha$ the path ...
3
votes
1answer
292 views

Compute the series $\sum_{n=1}^{+\infty} \frac{1}{n^3\sin(n\pi\sqrt{2})}.$

I need to compute $$\sum_{n=1}^{+\infty} \frac{1}{n^3\sin(n\pi\sqrt{2})}.$$ This an exercice of "Amar and Matheron, complex analysis". I proved the convergence and now to compute the sum, I follow the ...
2
votes
0answers
36 views

Cauchy's Residue Theorem and Cauchy's Theorem

Cauchy's theorem in short says for a holomorphic function $f$ which is holomorphic on and inside a path $\gamma$ the path integral is $0$ I have calculated a path integral around a path where there ...
5
votes
1answer
137 views

Integral involving power of trigonometric functions

I'm having a technical problem evaluating the following integral: $$\int_{r=0}^1\int_{\theta=0}^{\pi \over2} \cos^{2\epsilon -1}\theta \sin^{\epsilon-1}\theta e^{-ikr\sin^\epsilon\theta}d\theta dr$$ ...
1
vote
0answers
75 views

Complex integration with infinitely many poles on imaginary axis

I'm trying to integrate with a closed contour on the upper-half of the complex plane. $I = \displaystyle\int_{-\infty}^\infty \dfrac{z\,\mathrm{sech(z)}}{[(z-a)^2+b^2][(z+a)^2+b^2]} dz$ There are ...
1
vote
0answers
34 views

Problem with $\int_{0}^{\infty} \frac{\log^2(x)}{1+x^2}$ (by residues) [duplicate]

I, I am trying solve the following integral $$\int_{0}^{\infty} \frac{\log^2(x)}{1+x^2}$$ Teachers teached me that I can solve the integral $$\int_{0}^{\infty} ...
1
vote
1answer
48 views

Complex integration on upper-half plane

In order to prove the normalisation property of a Lorentzian function, $L = \dfrac{1}{\pi}\displaystyle \int_{-\infty}^\infty \dfrac{b}{(z-a)^2+b^2} dz = 1$ we take a closed contour on the ...
1
vote
2answers
33 views

how could calculate $ \int_{C} \frac{1}{\sin(z)} \, dz $ when $C=C(0,1)$

i am trying calculate $$ \int_{C} \frac{1}{\sin(z)} \, dz $$ when $C=C(0,1)$ by complex methods, its said, by residues, some one could help me?
2
votes
2answers
32 views

How could I calculate $\int_{C} ze^{\frac{1}{z-1}}$ when $C=C(1,\frac{1}{2})$

I have to solve if $C=C(1,\frac{1}{2})$ $$I=\int_{C} ze^{\frac{1}{z-1}}$$ I know that $I=2\pi i \operatorname{Res}(f(z), 1)$, but I do not know how could I calculate that residue. What I did: ...
2
votes
3answers
49 views

Compute the integrals using the residue theorem

Compute the following integrals: $I:=\int_{|z|=2}\frac{1}{(z-3)(z^{13}-1)}dz$ $J:=\int_{|z|=10}\frac{z^3}{z^4-1}dz$ I do not know where to begin. I know I am supposed to use the substitution ...
1
vote
3answers
46 views

Contour Integration: non-convergent integral

The question is $$I=\int_{-\infty}^{\infty} \frac{\sin^2{x}}{x^2} dx$$ My attempt: $$I=-\frac{1}{4}\int_{-\infty}^{\infty} \frac{e^{2ix}-2+e^{-2ix}}{x^2} dx$$ $$I=-\frac{1}{4} \Big[ ...
0
votes
1answer
43 views

Contour Integration with pole on contour

I have come across an example I don't understand.. So, here is the problematic part: Consider the integrals: $ I = \int_C \frac{e^{iz}}{z} dz $ $ J = \int_C \frac{e^{-iz}}{z} dz $ Where $C,C_-, ...