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
60 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
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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, how ...
0
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1answer
29 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
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1answer
21 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
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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
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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
52 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
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
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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
54 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
56 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
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
21 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
91 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_{...
6
votes
2answers
146 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
25 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
78 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 ...
1
vote
2answers
73 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
votes
2answers
133 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
28 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
66 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
71 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
44 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
67 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 ...
4
votes
1answer
303 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
39 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
149 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
89 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
35 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} \frac{\log^2(x)}{1+x^2}=\frac{d^2}{d\...
1
vote
1answer
56 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 upper-...
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: $$f(...
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 $z\...
1
vote
3answers
47 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[ \int_{-\infty}^{...
0
votes
1answer
48 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_-, ...
1
vote
0answers
17 views

A clarification on an answer on residues and Polya fields

In this very informative and interesting answer about the relation between residues and representation of complex functions as vector fields the author states that the function $$f(z) = \frac{1}{z}$$ ...
1
vote
0answers
27 views

Verification on classification of singularities

In an exercise, I'm asked to classify the singularities of these functions: $\qquad i) f(z)=\frac{1}{(z-1)^2} \qquad ii)f(z)=\frac{1-\cos z}{z^2} \qquad iii) f(z)=\frac{z^2-1}{z-1}$ I don't know why ...
0
votes
2answers
67 views

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

$$\int_{-\infty}^{\infty}\frac{x^2dx}{(x^2+1)^2}$$ My attempt: $z=i$ is a pole with order $2$ in the upper part on the plain. $$\text{Res} (f,i)=\lim\limits_{x\to i}\frac{x^2}{(x+i)^2}=\frac 1 4$...
1
vote
2answers
23 views

Complex roots in order to apply residue theorem

$$\int_{0}^{2\pi}\frac{d\theta}{(4 + 2\sin\theta)^2}$$ $$\sin\theta = \frac{z - z^{-1}}{2i}$$ $$d\theta = \frac{dz}{iz}$$ $$\oint_c\frac{dz}{iz\left(4 + \frac{z - z^{-1}}{i}\right) ^2}$$ ending up ...
1
vote
1answer
21 views

Applying the residue theorem on a real integral

$$\int\frac{d\theta}{a + b\cos\theta}$$ Given that $$\cos\theta = \frac{z + z^{-1}}{2}$$ $$d\theta = \frac{dz}{iz}$$ We have $$\oint_c \frac{dz}{iz\left(a + b\frac{z +z^{-1}}{2}\right)}$$ $$\...
0
votes
1answer
13 views

Number of isolated singularities

If $f:G\longrightarrow\mathbb{C}$ is analytic except for the isolated singularities and has infinitely many singularities, why then the singularities only can accumulate on boundary of $G$? Which is ...