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Questions tagged [residue-calculus]

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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8 views

Missing $i$ while evaluating $\int_{-\infty}^{\infty}\frac{e^{iz}}{(z^2+2z+2)^2}$ using residue theorem

Okay, first I'm a bit ashamed to ask because I already asked a question yesterday about a similar question (it's from far not the same integral though), but I'm missing an $i$ somewhere in the process,...
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2answers
14 views

Residue of a product of series

I need to find the residue of $f(z)=\frac{e^{\frac{1}{z}}}{z-1}$ in $z=0$. To do this, I proceeded to find the Laurent series of $f$ which is: $\sum_{n=0}^{\infty} \frac{z^{-n}}{n!}\sum_{k=0}^{\...
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0answers
8 views

How to apply the method of steepest descent to solve this integral?

I want to reproduce Eq. (11) of this paper. It is the result of solving the integral $$ \int_{0}^{\Lambda} \text{d}q \frac{q (e^{i q r} - e^{-iqr})}{q^2 - x} $$ (where $\Lambda = \pi$ and $x = \omega/\...
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0answers
13 views

Classify the singularities of $f'(z)$ and its residue

I want to know which singularity does $f'(z_o)$ have if $f(z)$ has a singularity in zo. I know that if $f(z)$ has an essential singularity in $z_o$, then its Laurent series has infinite negative ...
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0answers
20 views

Proving $Res(f'/f,z)$ is an integer for f analytic on a domain $\Omega$ and $z \in \Omega$

Here is what I have so far, since f is analytic at z we know it is infinitely differentiable at z, so $f'(z) \not = 0$ and so we know that $Res(f'/f,z) = \frac{f'(z)}{f'(z)} = 1$ and any pole of ...
2
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2answers
69 views

Two different results with residue calculus. What went wrong?

So I have to evaluate the following integral :$$\int_{0}^{2\pi}\frac{\cos(3x)}{5-4\cos(x)}dx$$ So I solve as usual with the residue theorem and by using $\cos(3x)= Re(e^{3ix})$, $\cos(x)=\frac{e^{ix}+...
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2answers
42 views

Compute complex integral inside an open curve

I need to compute this complex integral: $$ \int_\gamma \frac{1}{(z-i)(z-2i)} dz $$ $\gamma$ is defined as: $$ \gamma (t) = t + i(3e^t\cos^2(t)) $$ The parameter t belongs to the following ...
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0answers
32 views

Solution of diffusion equation on infinite region using residue theorem

The problem is as follows: Solve the partial diffusion equation $\frac{\partial }{{\partial t}}C\left( {x,t} \right) = \frac{{{\partial ^2}}}{{\partial {x^2}}}C\left( {x,t} \right)$ with initial ...
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1answer
15 views

Possible values of the integration $\frac{1}{2\pi i}\int_\gamma \frac{2i}{z^2 + 1}dz$

Possible values of $I := $ $\frac{1}{2\pi i}\int_\gamma \frac{2i}{z^2 + 1}dz$ where $\gamma $ is any closed curve in upper half plane not passing through $i$. My approach: There are two cases ...
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1answer
43 views

Find using Residue Theorem the following integral

Find using Residue Theorem $$\int_{-1}^1 \dfrac{dx}{(\sqrt {1-x^2})(1+x^2)}$$ My try: I took the contour $C=C_1+C_2 $ where $C_1$ is the upper half of the circle with center at $0$ and and radius $...
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1answer
28 views

Derivative of Fourier transform using residue theorem

If we define the fourier transform of f as $$\hat{f}(\omega) = \frac{1}{\sqrt{2}}\int_{-\infty}^\infty f(x) e^{-i\omega x} dx$$ then if f is differentiable, and the integrals for $\hat{f}$ and $\...
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2answers
34 views

Evaluating a definite integral using residue theorem

I am trying to show $$\int_0^{\infty} \frac{\log (x)}{(x^2+1)^2}$$. We can use the integrand $g(z) = \frac{\log(z)}{(x^2+1)^2}$ defining log as $\log(\rho e^{i\theta}) = \log(\rho) + i\theta$ and ...
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0answers
43 views

Residues of $\frac{e^{imz}}{1+z^4}$

I am trying to calculate the residues of $\frac{e^{imz}}{1+z^4}$. For context, I am calculating $\int\limits_0^\infty \frac{e^{imz}}{1+z^4}dx$ and taking the real part of the integral to calculate $...
1
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1answer
45 views

Limitations of Bromwich integral for inverting Laplace transform

Suppose: $$f(t)=e^{at}+e^{bt};\quad a>b>0$$ Its Laplace transform is: $$\mathbf{L}[e^{at}+e^{bt}]=\frac{1}{s-a}+\frac{1}{s-b}$$ for $Re(s)>a$ where $Re$ stands for the real part; for $Re(s)&...
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2answers
143 views

Solving integrals using complex analysis

There are many times when one uses methods from complex analysis to solve integrals that would otherwise be very difficult to solve. My question is: Can all integrals that are usually solved this way ...
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2answers
69 views

Evaluate $\int_0^{\infty}\frac{\ln x}{x^a(x+1)}dx$ where $0<a<1$

I'm trying to compute this integral, $$\int_{0}^{\infty}\frac{\ln x}{x^{a}(x+1)}dx \hbox{ where } 0<a<1$$ I drew a typical Pacman contour with branch cut at positive real axis. Then, we have $$\...
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1answer
28 views

Solving Trignometric integral with the aid of residues.

If $\alpha, \beta, \gamma$ are real numbers such that $\alpha^2> \beta^2+\gamma^2$ show that $$\int_0^{2\pi}\frac{d\theta}{\alpha+\beta \cos \theta +\gamma \sin \theta} = \frac{2 \pi}{\sqrt{\alpha^...
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1answer
42 views

When can a rational function be represented by a Taylor series?

I am trying to prove that $$ \int_C \frac{P(z)}{Q(z)} dz = 0 $$ If polynomial order of $Q$ is 2 or more than that of $P$, using the theorem stating that if a function has a finite number of singular ...
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1answer
53 views

Integral by Residue Theorem

I'm working through a Complex Analysis text and am working through the Residue chapter. I am not sure if I am approaching this question correctly. $$ \int_\gamma \frac{1}{(z-1)^2(z^2+1)}$$ Such that ...
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1answer
49 views

Calculating the residue of $\frac{1}{z^2 \sin z}$ at $z = 0$

I am having some difficulty calculating the residue of $\frac{1}{z^2 \sin z}$ at $z = 0$. From what I can tell, we are dealing with an essential singularity here and so the problem becomes that of ...
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1answer
17 views

Solution verification: finding the residues of a function

So in a given problem, I am seeking to compute the residues of the function $$f(z) = \frac{cos(z)}{z^2 (z-\pi)^3}$$ at its singularities. I feel pretty weak in this subject, so I want to just double-...
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2answers
56 views

Finding the residues of $\frac{\cos z -1}{(e^z-1)^2}$.

I've found the poles of $\frac{\cos z -1}{(e^z-1)^2}$ to be double poles at each $z_k = 2k\pi i$, where $k\in\mathbb{Z}$ and $k\neq 0$. (At $k=0$ this is a removable singularity instead.) I have no ...
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1answer
28 views

Find all natural $n$ for which the integral is non-zero

I have to find all natural $n$ such that $$I=\int_{|z|=2}\frac{z^n}{z^{10} - 1}dz \neq 0$$ Since all the $10^{\text{th}}$ roots of unity lie in $|z|=2$, by residue theorem we have $$\int_{|z|=2}\...
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2answers
57 views

Find residues of $f(z)=\frac{1}{(e^{z}-1)^{2}}$

How to find the residues of $f(z)=\dfrac{1}{(e^{z}-1)^{2}}$ I have found that the poles $z=2\pi i n$. But when I apply the formula $\dfrac{1}{(m-1)!}\lim\limits_{z\rightarrow z_{0}}\dfrac{d^{m-1}}{...
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1answer
62 views

How to find the Fourier transform of $\frac{(t^2+2)}{(t^4+4)}$ and use it to evaluate $\frac{(t^2+2)^2}{(t^4+4)^2}$?

I'm asked to evaluate the Fourier transform of $\dfrac{(t^2+2)}{(t^4+4)}$, and then use it to evaluate the integral from minus infinity to plus infinity of $\dfrac{(t^2+2)^2}{(t^4+4)^2}$. Part 1: ...
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1answer
59 views

Find the residues at the singularities of $\frac{z^2 - z}{1-\sin{z}}$.

I've been set a question where I'm asked to classify all of the singularities of $$f(z) = \frac{z^2 - z}{1-\sin{z}}$$ and then calculate the residue of each of its singularities. I've found the ...
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1answer
25 views

Prove this improper integral is finite

I tried to expand the term in the integral but it turns out to be of order 1/x and diverges... Any help is appreciated! Thanks in advance! $\int_{0}^{\infty}\sqrt{\log (1+1/x^2)}dx<\infty?$
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1answer
48 views

Evaluating an integral using residue theorem

I'm evaluating $\frac{1}{2\pi i}\oint_C \frac{e^{zt}}{z(z^2+1)}dz$, $t > 0$ where $C$ is defined by the vertices $2+2i$, $-2+2i$, $-2-2i$, and $2-2i$. I'm pretty sure this is done by finding the ...
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2answers
39 views

Residue of order 3 -

Find the Laurent Series for the function \begin{align} f(z) = \frac{1}{(z^2+4)^3} \end{align} about the isolated singular pole $z = 2i$. What is the pole order? What is the residue at the ...
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1answer
20 views

Show that $\text{Res}_c\left(\frac{f}{g}\right)=\dots$

Let $D\subseteq\mathbb C$ be open, $c\in D$ and $f,g:D\to\mathbb C$ holomorphic. Assume that $g(z)$ has a zero in $c$ of order 2. Show that $$\text{Res}_c\left(\frac{f}{g}\right)=\frac{6f'(c)g''(c)-2f(...
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1answer
22 views

Residual calculus [closed]

I wish to construct a function $f(z)$ with the properties as only singularities of $f(z)$ in the extended complex plane are poles of order of $1$ and $2$ at $z= 1$ and $z=-1$ respectively. Also it ...
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1answer
33 views

Using the Residue Theorem for Complex Integrals [closed]

I want to calculate the following integral: $\int_{|z-i| = 10} \left(z+\frac{1}{z}\right)^4$. I have been told to use the Residue Theorem, but I couldn't accomplish a correct calculation. Can anyone ...
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1answer
26 views

How to compute residues for contour integral of matrix?

I would like to compute $Y = \int^{\infty}_{-\infty} e^{-itx} (Ix-A)^{-1}dx$, where $A$ is a known square matrix. I am using the semi-circle contour from $-\infty$ to $\infty$. From the Cauchy ...
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4answers
84 views

Evaluating the integral $\int_0^{\infty}\frac{dx}{\sqrt[4]{x}(1+x^2)}$ using Residue Theorem

I need to evaluate the integral $$\int_0^{\infty}\frac{dx}{\sqrt[4]{x}(1+x^2)}$$ I've been given the hint to use the keyhole contour. So I would first choose the principal branch of $\sqrt[4]{\cdot}$,...
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2answers
7 views

Residues at a finite point

What is the residue of cotz at z=nπ ,where n is integer ? I have calculated the residue of cotz at z=0 and it is equal to 1 via expansion of cotz ....but how can I find the residue at nπ with the help ...
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0answers
32 views

To find $\text{Res}\left(\sin\left(\frac{z}{z-1}\right),1\right)$

I have to find $\text{Res}\left(\sin\left(\frac{z}{z-1}\right),1\right)$ We have $$\sin\left(\frac{z}{z-1}\right) = \sin\left(1+\frac{1}{z-1}\right)\\= \left(1+\frac{1}{z-1}\right)-\frac{1}{3!}\left(...
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0answers
22 views

Evaluating an integral of a real function using the residue theorem

Using the residue theorem, I have shown that $\oint_C f(z) dz = \frac{\pi i}{e^2}$ where $f(z) = \frac{ze^{iz}}{(z^2+4)^2}$ and $C$ is the closed curve consisting of the horizontal line $y = 0$ from $...
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1answer
35 views

Find poles of $f(z)=\frac{z}{(z-1)(z-2)^2}$ , calculate residues at the poles and then evaluate $\int_C f(z)dz$ where $C:|z+1-i|=2$

The poles are clearly 1 and 2. $C$ is the circle with equation $(x+1)^2+(y-1)^2=4$. Putting $y=0$, we get $x=\sqrt{3} -1, -\sqrt{3} - 1$. This means both poles lie outside the circle. So both residues ...
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0answers
19 views

An alternative to the (contour) integration of a sixth order simple (but nasty) pole?

My title said simple (linear), but the poles themselves are quite nasty, and hence my post on the forum. I have the following integral that I wish to integrate with respect to $\omega$ $$ I =\int_{-\...
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1answer
37 views

Residue of $(1-e^{-z})^{-n}$ at 0

Similar to Evaluating the residue of $(1 - e^{-z})^n$ at $z = 0$ with $n \in \mathbb{Z}$, but this question is unanswered. What is the residue at $0$ of $(1-e^{-z})^{-n}$? For $n=1$, we can just ...
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2answers
77 views

Evaluating the integral $\int_{-\infty}^{\infty}\ln|x|e^{-x^2}dx$

Here's the problem. Evaluate the definite integral $$I:=\int_{-\infty}^{\infty}\ln|x|e^{-x^2}dx.$$ I have made some good progress, but I'm unable to complete the solution. Here's what I've done: ...
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0answers
16 views

Residues of the Gamma function

I am trying to make sense of a proof that the poles of $\Gamma(z)$ are at $z=-n$ and have residue $\frac{(-1)^n}{n}$. The proof reduces $\Gamma(z)$ to the sum of an (entire) incomplete gamma function ...
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1answer
24 views

Finding residue of composite function.

Let $f$ be analytic at $z_0\in\Bbb{C}$ and $g$ have a simple pole at $z_0$. Find $\operatorname{Res}(f(g(z_0)),z_0)$. scratchwork (Would like big hints. I don't usually like asking for the full ...
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2answers
80 views

Using the Residue Theorem to Prove that $\int^{2\pi}_{0} \frac{1}{(a+\cos\theta)^{2}} d \theta=\frac{2\pi a}{(a^{2}-1)^{3/{2}}}.$

How do you evaluate the following integral? Here we take $a>1$. $$\int^{2\pi}_{0} \frac{1}{(a+\cos\theta)^{2}} d \theta=\frac{2\pi a}{(a^{2}-1)^{\frac{3}{2}}}.$$ I know I have to use the Residue ...
3
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2answers
43 views

Calculating the residue of $\frac{10z^4-10\sin(z)}{z^3}, z(0) = 0$

$$\frac{10z^4-10\sin(z)}{z^3}, \quad z(0) = 0.$$ I've gotten that $$\operatorname{Res} = 0$$ but I'm not quite sure if that is correct, or if I have even used a correct pathway towards it. How ...
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votes
0answers
12 views

A function of Laurent and the prime counting function $\pi(n)$.

The following result is apparently due to Laurent. If $$F_n(z)=\prod_{m=1}^{n-1}\prod_{l=1}^{n-1}(1-z^{ml}),$$ we can show that the series $$f(z)=-\sum_{n=2}^{\infty}\frac{F_n(z/n)}{\{(z/n)^n-1\}n^{n-...
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2answers
38 views

Order of the pole of $f(z)=\frac{e^{\sin{z}}-1}{z^3}.$

Let $$f(z)=\frac{e^{\sin{z}}-1}{z^3}.$$ a) Determine if $f$ has a pole at $0$ and determine its order. b) If $f(z)=\sum_{n=-\infty}^{\infty}a_n$ denote a Laurent serie of $f$ valid in ...
2
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1answer
40 views

Finding the residue with a Laurent series expansion.

I have a question about the following problem: "Detect the error in the following argument. The function $f(z)=\frac{1}{z(z-1)^2}$ has an isolated singularity at $z=0$. The Laurent series is $f(z)=\...
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2answers
24 views

Compute $i\oint_{|z|=1}\frac{(1-z^2)^2}{z^2(1+6z+z^2)} \ dz$

I was inspired to try and solve the integral from this post. When I get to computing the residues for $z_2=\sqrt{8}-3$ I run into troubles. We see that $z_1=0$ is a pole of order $2$ and $z_2=\sqrt{...
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votes
1answer
33 views

Evaluate $\int_0^{\infty}\frac{\cos(3x)}{x^2+12} \ dx$ using resiude theory.

I'm presenting my solution and I'd like to have it undergo some constructive scrutiny here. Is my solution mathematically correct and easy to follow? Any room for improvement and making it more ...