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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17 views

How well does $L_{n,f}$ approximate $f$?

My Try: I did part a,b and c. Having trouble with d. Followed the hint and got $\displaystyle f(z)-RHS=\frac{1}{2\pi i}\Bigg[\int_\Gamma \frac{f(\zeta)}{(\zeta - ...
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1answer
86 views

How to compute this contour integral with a modulus sign in the integrand,

Evaluate the integral $$∫_{∣z∣=ρ} \frac {1}{|z−a|^{2}}|dz|$$ where ρ is a positive number, a is a complex number, and |a|<ρ. I welcome any hints on how to get started on this problem. The ...
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1answer
33 views

Taking inverse laplace transform of $\dfrac{1}{(s^2+1)^2}$ using residues, something's wrong

I am trying to compute the inverse laplace transform of $\dfrac{1}{(s^2+1)^2}$ using residues So $$\mathcal{L}^{-1}\dfrac{1}{(s^2+1)^2} = res(\dfrac{e^{st}}{(s^2+1)^2}, i) + ...
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2answers
106 views

Residue theorem for Multi-valued functions

Im stuck with this problem show that: $$\int_0^\infty{\frac{x^a}{(x^2+1)^2}dx} = \frac{\pi (1-a)}{4cos(a \pi /2)}, \, -1<a<3, \, a \neq 1$$ I have the solution for it and everything but i ...
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1answer
73 views

Continuous extension of $\int_\mathbb{R} dt\, e^{-t^2}/(t-z)$ from $\operatorname{Im} z < 0$ onto $\mathbb R$

I am asked to show that the continuous extension of $$ F(z) = \int_{-\infty}^{\infty} dt\, \frac{e^{-t^2}}{t-z}, \quad \operatorname{Im} z < 0 $$ onto $\mathbb R$ is given by $$ ...
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1answer
64 views

how to check if a singularity is isolated?

I have a function $1/(\sin(1/z))$ and I must show if the singularities are isolated or not. Is taking the limit of the number a little to the right and a little to the left enough? If not, how can ...
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2answers
47 views

Compute integral with residue theorem

I have the following integral $$\int_0^{\infty} \frac{\sin^2x}{x^2}\mathrm{d}x$$ And a hint: integrate $\frac{e^{2iz}-1-2iz}{z^2}$ within a semi circle. But this function residue zero (what I ...
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0answers
11 views

How to calculate residues of 1/(GaussianMixture)?

If I numerically calculate the complex roots of a Gaussian Mixture... $$g(z)\equiv\sum_{k=1}^{n}\frac{1}{\sqrt{2\pi B_{k}}}e^{-\frac{\left(z-x_{k}\right)^{2}}{2B_{k}}} $$ how do I calculate the ...
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33 views

When can we solve integrals by residue method without explicitly finding the poles?

An issue that prompted a Question earlier got me thinking about a more general class of problem. $$\int_{-\infty}^{\infty}\frac{f(z)}{g(z)}dz$$ I know I've seen some integral problems where it was ...
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478 views

How to solve an integral with a Gaussian Mixture denominator?

I am trying to solve this integral: $$t(v)\equiv\sum_{k=1}^{n}\sum_{j=1}^{n}\int_{-\infty}^{\infty}\frac{w_{k}N(x-x_{k},B_{k})N(x-x_{j},v)}{\sum_{m=1}^{n}w_{m}N(x-x_{m},B_{m}+v)}dx $$ where ...
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1answer
39 views

Integrate with residue

I have this integral $$\int_0^{\infty}\frac{x^2}{(x^2+1)(x^2+4)}dx$$ and i want to use residues to compute it, how do i do that when the limits are $0$ to $\infty$? First of i define $$f(z):= ...
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1answer
89 views

Residue theorem for infinitely many singularities

The residue theorem is a standard result in complex analysis, I state it below so we are on the same page: note that $\overline{\mathbb{C}}$ is the extended complex plane (ie. $\simeq$ Riemann sphere) ...
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1answer
37 views

How to calculate this integral via residues.

I get into trouble in evaluating this integral: $$ C(a)=\frac{1}{i\beta}\int_\Gamma \cot\frac{\pi z}{\beta}\frac{1}{\sin^2\frac{z}{2}}dz $$ where the contour $\Gamma$ consists of two vertical lines, ...
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32 views

Pole and residue of the following function at infinity

I am confused about one particular problem regarding complex infinities. Suppose i have EXP[-z^2] with z being the complex number. Clearly it has poles at z=+i(infinity) and -i(infinity). . How to ...
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2answers
39 views

Finding the residue of $\frac{1}{z(z^2+4)(z+2i)}$ at $z=-2i$

I've literally tried every technique I know of and they all lead to explosions of the kind $1/0$. Generally speaking the residue at $c$ for a function can be calculated as: ...
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1answer
79 views

Definite Integral: $\int_0^{2 \pi} \frac{d\phi}{z + b \cos(\phi)}$

During my work, I stumbled upon this definite integral $$\int_0^{2 \pi} \frac{d\phi}{z + b \cos(\phi)} = \mathrm{sgn}(\Re(z))\frac{2\pi}{\sqrt{z^2-b^2}} \qquad z \in \mathbb{C}$$ which result I ...
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2answers
206 views

The meaning of the Imaginary value of the Residue while Evaluating a Real Improper Integral

When evaluating the improper integral $$\int_{0}^{\infty}\frac{x^{3}\sin\left(2x\right)}{\left(x^{2}+1\right)^{2}}\,dx$$ (which is an even function, so half of the $(-\infty,\infty)$ integral), I used ...
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1answer
91 views

Calculating the residue of a complex funciton with ln(z) at z=0

How can I calculate this residue: $$Res\left(\frac {z\ln(z)}{(z^2 +1)^3} , 0\right) $$ if it's possible at all. I know $0$ is a branch point for $\ln(z)$ and therefore isn't a pole, but when i plug ...
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1answer
50 views

How to solve this equation with implicit sum

I want to know how the authors of this arxiv paper (p. 10) solved the equation \begin{align} g\left(\lambda\right) ={}& ...
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1answer
93 views

My complex integral cancels at the end; how can I modify the integrand to prevent this?

$$\int_0^\infty \frac{x^a}{x^2 + b^2}$$ for $-1< a < 1$ and b>0 -- these constraints help with estimating the integral on the big circle and small circle of a keyhole contour that I chose to ...
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4answers
161 views

Can I use an upper semi-circle to integrate this function?

I'm trying to integrate $$\int_{-\infty}^{\infty} \frac{e^{iz}}{e^z + e^{-z}}dz$$ Do I have have to integrate this over a box, or can I use my first guess at a contour and use an upper semi-circle ...
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0answers
130 views

Contour integral mystery: why is the answer different from Maple/Matlab?

The mystery is that here is a fairly standard contour integral which can be done by the residue theorem. Yet when I tried to evaluate it using numerical softwares like Maple or Matlab, the answer is ...
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3answers
196 views

What is a residue?

I've heard of residues in complex analysis, contour integration, etc. but all I really know it to be is the $c_{-1}$ term in the Laurent series for a function. Is there some sort of intuition on what ...
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1answer
93 views

Inverse Laplacetransform of rational function with multiple pole

I have to calculate the inverse Laplacetransorm of this function using Residue calculus $$ \frac{s^4 + 6s^3 - 10s^2 + 1}{s^5} $$ but I can't find any Residue rule that would solve this. Can you show ...
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1answer
75 views

Finding the residues of poles

Consider the equation $\mathcal{F}(\lambda)=0\ \ \ \forall\ \lambda = \lambda_{n},\ n \in \mathbb{N}$. I understand that the expression $\frac{d}{d\lambda}\ ...
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0answers
66 views

Multivariate Residue Theorem

Let $G(s,t)$ be a complex valued function in two variables that converges absolutely for $Re(s), Re(t)>1$. Suppose we can analytically continue $G$ in such a way that $$G(s,t) = f(s)g(t)H(s,t)$$ ...
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2answers
47 views

Two different answers with Laplace

Find the solution for the equation $$ -u'' + u = \delta'(t)$$ for which it "disappears" for $t<0$ By using residuals! So I used Laplace transformation for this. $$Y(-s^2 + 1) = ...
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101 views

Why does this example of global residue theorem not work?

This question is related to and inspired by a previous question What is the residue obtained from this integral? , but note that the appearing functions are slightly different. Consider the following ...
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1answer
51 views

What is the residue obtained from this integral?

Consider the following integral in two complex variables $z_1$ and $z_2$: $$\frac{1}{(2\pi i)^2}\oint_{{|z_1|=\epsilon}\atop{|z_2|=\epsilon}}dz_1 dz_2\frac{1}{z_1 ...
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2answers
82 views

A Residue problem [duplicate]

Preparing myself for qualifying exam, I found this problem in residues in some previous qualifying exam, and I am stuck and don't know how to solve it, any helps? ...
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3answers
144 views

What is a “Contour Integral” and how do I evaluate one?

A very general question, I apologize, but as you read this, hopefully you get what I'm asking. Recently, Bernoulli Numbers have caught my eye, for I am studying infinite series' and it is a part of ...
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3answers
127 views

Residue Problem

I am trying to find residues for all singularities of the function: $$f(z)= \frac{\tanh z}{z^2}$$ Here is what I did: $$f(z)= \frac{\sinh z}{z^2\cosh z}$$ when $$\cosh z=0$$ then $z_k =i( ...
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1answer
105 views

A Tough Problem about Residue

I tried my best to solve this problem from what I learned in residues, but the solution seems very far from what I was doing!! Is there any way other than using Laurent series expansion? Here is the ...
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1answer
69 views

The residue of $\frac{f(z)}{g(z)^2}$

Let $f$ and $g$ be analytic near $a$, and suppose $g$ has a simple zero at $a$. Find a formula for the residue of $\frac{f(z)}{g(z)^2}$ at $a$ in terms of $f(a)$ and the derivatives of $f$ and $g$ at ...
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1answer
45 views

Complex integration, showing that arc integral vanishes

I stumbled upon the following integral in QFT: $$\int_{\mathbb{R}} \frac{k e^{ikr}}{k^2+m^2} dk$$ If We turn this into complex variable integral along semi-circle arc in the upper complex plane: ...
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2answers
87 views

my computation of a real integral still has an imaginary number in it,

I have four residues that I have found. I multiplied each by $2\pi i$, using the Residue Theorem. But my final answer still has an $i$ in it. Needless to say, it is not the right answer, since the ...
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39 views

Is there a faster way to compute the residues of this function,

$$\frac{z^2log(z)}{1+z^4}$$ I have that this function has simple poles at $$e^{i\pi/4},e^{i3\pi/4},e^{i5\pi/4},e^{i7\pi/4}$$ which are the zeroes of the denominator (1+$z^4$). The computation of ...
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0answers
58 views

Integral of ratio of complex polynomials

Let $p(z),q(z) \in \mathbb{C}[z]$ two polynomials with coefficients in $\mathbb{C}$ s.t. $deg(p) = m$, $deg(q) = n$ and $n \ge m +2$. I need to show that $$ \lim_{R \to \infty} \int_{|z| = R} ...
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2answers
90 views

$\int_0^\infty \frac{\log(x)}{x^2+\alpha^2}$ using residues

I'm trying to find $\int_0^\infty \frac{\log(x)}{x^2+\alpha^2}dx$ where $\alpha>0$ is real. My approach was to take an integral along the real line from $1/R$ to $R$, around the circle ...
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2answers
60 views

Is there a simpler way to compute the residue of a function at a pole of order 3?

The function $$\frac {1}{z^2(e^{i2\pi z}-1)}$$ has a triple pole at z = 0. To compute the residue of f at z = 0, I can compute the Laurent expansion of f about z = 0, and then read off the ...
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1answer
173 views

Computing residues of $\cot(\pi z)/z(z+1)$ with symmetries

I would like to know if there is a quick way of computing the residues of $$f(z) = \frac{\cot \pi z}{z(z+1)}$$at the points $z = 0$ and $z = -1$. They are double poles. Expanding this in Laurent ...
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2answers
51 views

Help evaluating residue with simple poles

I am having a bit of trouble evaluating $$\sum_{k=1}^3{ \rm Res}\left(\frac{\log(z)}{z^3+8};z_k\right)$$ where $z_1=2e^{i\pi}$, $z_2=2e^{i\pi/3}$ and $z_3=2e^{i5\pi/3}$. I know that each $z_k$ is a ...
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2answers
489 views

Compute this integral, using a method other than the Residue Theorem,

$\int_0^\infty$ $\frac{1}{1+x}$$\frac{dx}{\sqrt{x}}$ Part (a) asks to compute the integral by means of the residue at x = -1. I have done this just now, and the answer is $\pi$. Part (b) asks, "can ...
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1answer
40 views

What does complexification mean in complex analysis, .e.g., in residue calculus,

I've learned complexification formally in a graduate linear algebra class. But what does the word mean in the setting of complex analysis? If I consider a real integral on the positive half line, ...
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129 views

A particular integral: $\int_{-\infty}^{+\infty}\frac{\sin(\pi x)}{\prod_{k=-n}^{n}(x-k)}\,dx$

I have to show summability, then compute the following integral: $$\int\limits_{-\infty}^{+\infty} \frac{\sin(\pi\,x)}{\prod_{k = - n}^n (x - k)}\,dx = \frac{(-4)^n}{(2\,n)!}\,\pi $$ for every $n\in ...
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62 views

How to calculate this Ei(x)-involved definite integral?

I want to solve the integral attached below by means of residue theorem. I tried the common integration ways and seeked references(e.g, Rjadov, et. al). Finally, I decided to solve this integral by ...
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52 views

How to calculate this residue which has a pole of order n-r?

So I have this complex integral: $$ \oint \frac{dz}{2\pi}\frac{e^{iz(br-(n-r)a)}}{\left(1-(1-q)e^{-ik_{1}-iza}\right)^{n-r}}$$ b,r,q,a,n are all constants in this context. However I'm not entirely ...
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1answer
72 views

Reciprocals of theta functions

I've spent the last few months with partial fraction expansions, and thought to create a function with simple poles over a lattice of zeros, like that of any of the Jacobi theta functions... but I ...
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1answer
72 views

Find the residue of $e^{\frac{1}{z^2-1}}\sin(\pi z)$ at $z=1$

I'm dealing with the following problem (from an old qualifying exam): Let $\gamma$ be a closed curve in the right half-plane that has index $N$ with respect to the point 1. Find $$ ...
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1answer
40 views

How can I calculate the singularities and residues of…?

$$\frac{e^z}{z^3(z-1)}+\frac{1}{z^3}$$ I have problems specially for $z=0$ Can anyone show me how to do it?