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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10
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0answers
436 views

Proof of residue theorem (residue formula) for differential forms on curves over an arbitrary closed field.

I have been reading the book Algebraic Geometric Codes: Basic Notions by Tsfasman, Vladut and Nogin. They give a residue formula like this: Let $\mathbb{k}$ be an algebraically closed field and $X$ ...
5
votes
0answers
56 views

Inverse Fourier transform using Residues for a ratio of hyperbolic functions.

I'm new and glad to be here. I have a problem relating to an inverse Fourier transform. I have $$g(w)= \frac{\sinh{w(a-b)}}{w \cosh{wa}}$$ and want to find $$G(t)$$. I cannot find this in tables so I ...
4
votes
0answers
32 views

$ \int_\gamma e^{\frac{1}{z^2-1}}\sin{(\pi z)}dz $ on a closed curve of index $N$ with respect to the point $1$.

Let $\gamma$ be a closed curve in the right half plane that has index $N$ with respect to the point $1$. Find $$ \int_\gamma e^{\frac{1}{z^2-1}}\sin{(\pi z)}dz $$ This is a problem from an old ...
4
votes
0answers
93 views

Ugly-nice double series

I'm trying to evaluate the following ugly double sum (presented in raw notation as used in my calculations): $\sum _{m=1}^{\infty } \sum _{n=1}^{\infty } \frac{4 m \cos \left(\frac{2 \pi m ...
4
votes
0answers
73 views

Can we use a sum of residues to develop an asymptotic expansion for this unknown function?

In the course of solving a particular physical problem, I have derived a relationship between two unknown functions: $$ f(s) = \frac{s \sinh{\frac{\pi s}{2}}}{2 \pi i \beta} \int_{-c- i ...
4
votes
0answers
53 views

Integrating $\oint_\Gamma \cos(\log|z|)\cosh(\text{Arg}(z))\text{Arg}(z)e^{is(z-1)}dz$ using residue calculus.

I'm trying to use the residue calculus to evaluate $$\oint_\Gamma \cos(\log|z|)\cosh(\text{Arg}(z))\text{Arg}(z)e^{is(z-1)}dz,$$ where $s>0$, and where $\text{Arg}$ is the principal argument, ...
4
votes
0answers
564 views

Solve $\int \cos^{2n}\theta d\theta$

I am trying to solve the integral $\int_0^{2\pi} \cos^{2n}\theta d\theta$ using residues. I get the wrong answer so could you please say what I am doing wrong? We start with the substitution $z = ...
3
votes
0answers
113 views

Residue Integral: $\int_0^\infty \frac{x^n - 2x + 1}{x^{2n} - 1} \mathrm{d}x$

Inspired by some of the greats on this site, I've been trying to improve my residue theorem skills. I've come across the integral $$\int_0^\infty \frac{x^n - 2x + 1}{x^{2n} - 1} \mathrm{d}x,$$ where ...
3
votes
0answers
101 views

Is there a simple and fast way of computing the residue at an essential singularity?

Is there a simple and fast way of computing the residue at an essential singularity ? I mean if we have a pole of order $n$ at $c$ we can use the formula : $$\mathrm{Res}(f,c) = \frac{1}{(n-1)!} ...
3
votes
0answers
161 views

How to calculate this residue

How to calculate this residue $$Res\left(\frac{\ln z}{z(z+1)},0\right).$$ Is it $\infty$? And if this could not be calculated, then how to calculate $$\int_0^\infty \frac{x}{e^x+1}dx$$ by changing ...
3
votes
0answers
203 views

Turning a Line Integral into a Contour one

I'm trying to compute an integral appearing in the article "On Determinants of Laplacians on Riemann Surfaces" of D'Hoker and Phong (page 541). It is as following. Fix $B\in \mathbb{R}_+$ and let ...
2
votes
0answers
54 views

Can this integral similar to the Fourier transformation of $\delta$ function be calculated analytically?

I want to calculate the following integral: $$\int_{-\infty}^{+\infty}dk\ \exp\left[i\big(kx-\sqrt{k(k-b)}\big)\right]$$ where $x$ and $b$ are both real. If $b=0$, the integral reduces to the Fourier ...
2
votes
0answers
80 views

What is the value of the integral $ \int_{-\infty}^{\infty}\frac{\sin(2x)}{x^3}dx$?

I tried to evaluate the integral $$ \int\limits_{-\infty}^{\infty}\frac{\sin(2x)}{x^3}dx$$ using residues but the answer comes out to be a negative value, $-2 \pi$, which seems strange. Any help on ...
2
votes
0answers
33 views

Calculate a complex integral using residues

Let $f(z)= \frac{2(e^\frac{1}{z}-1)(\sin^2z)}{z^3}$. Calculate $\int\limits_{\partial B_+(O,1)} f(z)\operatorname{d}z$ Could someone confirm my solution? Solution? I try to calculate the ...
2
votes
0answers
26 views

Solution of gaussian integral with hyperbolic cotangent

I was wondering if the integral $$I=\int_{-\infty}^{\infty}d\omega \omega e^{-(\omega/a)^2}\coth(\frac{b\omega}{2})\cos(\omega c)$$ where $a,b,c>0$ can be solved using complex countour ...
2
votes
0answers
30 views

complex integral with non integer power

I want to calculate this integral ...
2
votes
0answers
116 views

The inverse laplace transform of $p^{-3/2}e^{-\sqrt{pa}}(\cos(\sqrt{ap})+\sin(\sqrt{ap}))$ can be written in Fresnel integrals?

I used the Residue theorem to solve this problem. But, I could not obtain the solution given by $$\mathscr{L}^{-1}\left( { p^{-3/2}e^{-\sqrt{pa}}\over{2\sqrt{2}}} [\cos(\sqrt{ap})+\sin(\sqrt{ap})] ...
2
votes
0answers
103 views

Finding the limit of sum $\sum \frac{1}{n^4}$

I'm trying to use the residue theorem to find the limit of $$\sum_{n=1}^{\infty} \frac{1}{n^4}.$$ So I am considering the function $$f(z) = \frac{\pi \cos(\pi z)}{\sin (\pi z)z^4}$$ on a square ...
2
votes
0answers
54 views

Integral on the circle

It's a standard fact that to calculate integrals of the form $$\int_{0}^{2\pi}\mathcal{R}(\cos(\theta),\sin(\theta)) \ d\theta$$ with $\mathcal{R(x,y)}$ a rational function in two variables without ...
2
votes
0answers
173 views

The solution of the contour integral for $\epsilon =+1$

I understand the solution for $\epsilon =-1$. And I am trying the solve this question for $\epsilon =+1$. This is important for me. I want really to learn perfectly because I am continuously seein' ...
2
votes
0answers
107 views

Find the inverse laplace transform: $\frac{1}{{{{({s^2} + 1)}^3}}}$

Find the inverse Laplace transform: $$x(t) = {L^{ - 1}}\left[ {\frac{1}{{{{({s^2} + 1)}^3}}}} \right]$$ with $x(t=0)=0$. I did: $${\left[ {{\mathop{\rm R}\nolimits} {\rm{es}}\frac{{{e^{st}}{{(s - ...
2
votes
0answers
76 views

Evaluate $\int_0^\tau \frac{t\sin(t z)}{z\cos(t z)-\sin(tz)}\text{d}t$

I'm trying to evaluate the following definite integral. Mathematica gives me a complicated expression which I think I can simplify, but I was wondering if there was a "nice" way to evaluate it. ...
2
votes
0answers
60 views

Integrating form over a path on projective algebraic curve

Let $X$ be an algebraic projective curve in $\mathbb{C}P^2$ given by $$ X = \left\{ w \in \mathbb{C}P^2 \mid w_0^2 = w_1 w_2 \right\}. $$ I have a differential form on $X$ defined by $$ ...
1
vote
0answers
20 views

Zeta functions and their poles

consider a zeta function as follows $f(x):=\sum\limits_{m=1}^{\infty}\sum\limits_{n=0}^{\infty}\frac{1}{\left(a\cdot m+n+\frac{1}{2}\right)^{2x}}$, for $a>0$ and $\Re(x)>1$. How can I construct ...
1
vote
0answers
28 views

Higher order residues using series

For example if we have any $f(z)$ with a singularity at $z=0$ but it is of order 12 Instead of using the limit definition, suppose $f(z)$ is in the form of: $$f(z) = a_0z + a_1z^2 + a_2z^3 + ... + ...
1
vote
0answers
22 views

calculating the residue of a complex function in an integral

I wonder how to solve this integral: $$P(q, \omega) = \frac{-2i}{(2\pi)^4}\times\int\frac{\mathrm d\mathbf{k}~\mathrm d\omega' e^{i\omega'\eta/h}}{\left[\omega' - E(\mathbf k) + ...
1
vote
0answers
34 views

Validity of Laurent series's principal part calculation

I need to calculate the principal part of the Laurent expansion of $f$ around a given $z_0$ in an annulus of the form $\{z\in \mathbb{C}:0<|z-z_0|<r$} and then use this to find $Res(f,z_0)$ ...
1
vote
0answers
102 views

Residue with half order pole?

I'm having issues evaluating the following integral using Cauchy's residue theorem. $$\int_{-\infty}^{\infty} \frac{e^{ix}}{\sqrt{x^2 - 1}} dx $$ Here's what I have tried. We have to make a ...
1
vote
0answers
48 views

Calculate integral when I have a pole on the unit circle

I'm having trouble integrating the following integral $$\int_0^{2\pi} \frac{d\theta}{3 -2\cos\theta + \sin\theta}$$ I make the substitutions $z= e^{i\theta}$ , $\sin\theta = ...
1
vote
0answers
101 views

obtaining inverse z-transform by different methods

how can i obtain the z-transform of $X(z) = \frac{z+1}{(z-1)(z+2)^2}$ by: 1) Partial fraction expansion, 2) residue theorem, and 3) direct division method any help is appreciated.
1
vote
0answers
82 views

Inverse Laplace Transform using Jordan's Lemma?

Following is the question that i am trying to solve: "Consider a second order linear ODE $x\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}+(3-2x)y=0$ A) Find the solution employing Laplace integrals by ...
1
vote
0answers
52 views

k-differentials and their residues.

I am a theoretical physics student currently working on dualities in quantum field theories, which apparently are very well described by complex geometry. I have a question concerning this. In one of ...
1
vote
0answers
28 views

possible results of integral along closed path after defining branch of sqrt

Prove that one can define a branch of the function $\sqrt{1-z^2}$ in every region $D\subset \mathbb{C}$ such that the points $-1$ and 1 belong to the same connected component of the complement of $D.$ ...
1
vote
0answers
102 views

Using Residue theorem to evaluate integral with complex poles

I'm trying to evaluate the integral of a function $$\frac{\psi_m^T\cdot P\cdot\phi_m\cdot\exp(-\gamma\xi i)}{(\gamma_m-\gamma)\cdot B_m}$$ with respect to $\gamma$, where $\psi_m$ is a $1\times n$ ...
1
vote
0answers
207 views

Improper integral equal to -pi with square root and Cauchy principal value

I'd like to know if the following proof for the value of $I$ is correct, and if there is a simpler solution to it. Also, I will probably encounter more improper integrals like this in the future, and ...
1
vote
0answers
262 views

Laplace transform of a product of functions

While trying to compute the Laplace transform of a certain product, part of the calculation leaves me with a Bromwich integral which has the form: ...
1
vote
0answers
218 views

Residue of $\mathrm{exp} (z+1/z)/((z-z_1)(z-z_2))$ at z=0

The ultimate aim is to solve the following integral: \begin{equation} \label{eq:Icos1} \begin{aligned} I = \int\limits_{0}^{2\pi} \frac{\mathrm{exp}\left(c \cos(\varphi)\right)\mathrm{d}\varphi}{a - ...
0
votes
0answers
27 views

Showing that an integral of a curve in $\mathbb{C}$ vanishes when the parameter approaches infinity

I'm trying to solve a problem where you have to use the residue theorem in order to get the value of a certain integral, but I cannot go on from this point: I need to show that $\int_{0}^\pi ...
0
votes
0answers
26 views

Residues of a digamma based function.

I was wondering how we can find the residues of the digamma function, From: Integral Calculation Find the residues of: $$f(z) = \frac{(\gamma + \psi(-z))^2}{(z+1)(z+2)^3}$$ The answer is in the ...
0
votes
0answers
11 views

Proving susceptibility in Lorentz Model satisy Kramers-Kronig relations

My instructor asked me to prove that the real and imaginary parts of the electric susceptibility derived from Lorentz Model satisfy the Kramers-Kronig relations using the residue theorem. The problem ...
0
votes
0answers
60 views

Calculate $\int_0^3 \frac{x^{\frac{3}{4}} (3-x)^{\frac{1}{4}}}{5-x}\,dx$ using principal branch

I would like to calculate the following integral $$ I = \int_0^3 \frac{x^{\frac{3}{4}} (3-x)^{\frac{1}{4}}}{5-x}\,dx $$ using contour integration but using principal branch of the function, i.e. ...
0
votes
0answers
37 views

A Contradiction of Riemann Zeta Residues

We can show (1+2+3...+n)^2 = 1^3 + 2^3 + ... +n^3, which holds for any finite n, shouldn't this imply Z(-1)^2 = Z(-3)? However, this does not hold if we look at the residues of the zeta function ...
0
votes
0answers
22 views

How do I compute the following residue

Not really sure how to tackle this problem. Compute the following residue: 3^3 + 2 mod 5 Any hints in the right direction would be appreciated.
0
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0answers
25 views

Complex analysis, poles and singularities and boundedness

So I am on the following problem: Prove that an isolated singularity of $f(z)$ is removable as soon as either $\text{Re}f(z)$ or $\text{Im}f(z)$ is bounded above or below. The hint is to use a ...
0
votes
0answers
28 views

Guidance or advice with the determination of the type of singularities

I need to determine the type of singularity in $f$ at $z_0=0$ and calculate the Residue at that point. $$ \begin{align} f(z)=(z^2+z) \cos\left(\frac{1}{z}\right) \end{align} $$ I know that ...
0
votes
0answers
103 views

Proper way to set up “Pac-Man” contour integral

I'm trying to evaluate $$ \int_0^\infty \frac{x^a}{1+x} \: dx, \: -1<a<0 $$ using contour integrals. Actually, I have found the correct answer using a "Pac-Man" contour and residues. My only ...
0
votes
0answers
59 views

Laurent-Series Expansion and Residues

I was asked to find a Laurent-series expansion for $f(z) = z^{-1} \sinh(z^{-1})$ about the point 0, and to classify the singularity at 0. Now, that wasn't too bad - with some manipulation of the ...
0
votes
0answers
67 views

Computing the contour integral of $\frac{\log(z)}{z^2 +a^2}$.

I'm still a bit insecure when it comes to complex analysis and I wondered if you guys could take a look at my solution to this problem. Let $a > 0 $ and define $$f(z) = \frac{\log(z)}{z^2 +a^2}$$ ...
0
votes
0answers
25 views

What if a line integral is independent of the function?

This concerns the proof that the $n$ times punctured plane has $\mathbb Z ^n $ as his second homotopy group. We choose for each puncture $a_i$ a loop $\sigma_i$ which circles it once counterclockwise, ...
0
votes
0answers
102 views

Convergence of improper real integrals using the residue theorem

Let $f:\mathbb{R} \to \mathbb{R}$ be a funktion that satisfies $$ |xf(x)| \to 0 \qquad (|x| \to \infty) $$ If $f$ can be extended to a holomorphic function with finitely many singularities $S$ and the ...