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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4
votes
1answer
89 views

Hint to compute the following integral

Can someone give a hint on how to solve the following integral? $$ \int_{0}^{2N\pi} \frac{-R\left(\xi t - r\right)\cos\left(t\right) + \xi R\sin\left(t\right)} {\left[R^{2} + \left(\xi t - ...
4
votes
1answer
147 views

Complex Integral using Residues

This is the question: Find the integral using residue theorem. $$\int_0^{2\pi}{d\theta \over1+8\cos^2\theta} $$ I solved it like this : $$\int_0^{2\pi}{d\theta \over1+8\cos^2\theta}=\int_0^{2 \pi} ...
3
votes
1answer
137 views

is this trig integral doable using contour integration?.

Is it possible to evaluate $\displaystyle \int_{0}^{\pi}\frac{x\cos(x)}{1+\sin^{2}(x)}dx=\frac{-{\pi}^{2}}{4}+ln^{2}(\sqrt{2}-1)$ by using residues?. I attempted it by considering $\displaystyle ...
2
votes
1answer
48 views

Residue of a 1-form in a Riemann Surface does not depend of the chart

Let's suppose that $X$ is a Riemann Surface, $\omega$ is a meromorphic 1-form in $X$ and let $p$ be a pole of $\omega$ of order $M$. I want to show that the residue of $\omega$ at $p$, defined by $$ ...
1
vote
1answer
19 views

Conformal mapping, Residue

I have solved the second question till $z=0$ is a simple pole... But it's residue is mentioned as $2$. How come the laurent series expansion helps in finding $a_{-1}$, is there any other way to ...
1
vote
1answer
153 views

Locate poles and calculate residue

Let $a = \sqrt{\pi}e^{\pi i/4} = (1+i)\sqrt{\pi/2} $ Consider the function $$ f(z) = \frac{e^{-z^2}}{1+e^{-2az}} $$ I have already shown that $f(z) - f(z+a) = e^{-z^2}$ if that helps at all. The ...
13
votes
0answers
325 views

Integrating $\int_0^\infty\frac{\log (1+z^2)}{e^z-1}dz$ using residue calculus.

I've been looking at how to integrate the following definite integral using the residue calculus, but can't seem to get my thoughts together. I know the $\log$ term is a multivalued function and the ...
9
votes
0answers
359 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 $X$ ...
4
votes
0answers
38 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
89 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
59 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
96 views

Number of zeros equal number of linearly independent analytic functions

I'm trying to read this paper and I'm stuck on a particular point. The authors are constructing an analytic function $f(z)$ which have to satisfy the following boundary conditions: ...
3
votes
0answers
92 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
42 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, ...
3
votes
0answers
102 views

How to calculate this residue

how to calculate this residue $$Res(\frac{Ln z}{z(z+1)},0).$$ 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 variables ...
3
votes
0answers
174 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 ...
3
votes
0answers
506 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 = ...
2
votes
0answers
53 views

inverse laplace transform of $$p^{-3/2}e^{-\sqrt{pa}}(\cos(\sqrt{ap})+\sin(\sqrt{ap}))$$

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
69 views

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

I'm trying to use the reside theorem to find the limit of $\sum \frac{1}{n^4}$. So I am considering the function $f(z) = \frac{\pi \cos(\pi z)}{\sin (\pi z)z^4}$ on a square contour. Now I am ...
2
votes
0answers
43 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
132 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
90 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
73 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
59 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
48 views

Classification of Singularities and Residues

I'm kind of stumped on a question here. I've been asked to determine and classify the singularities of; $$f(z) = \frac{z^3}{(1+z)^3}$$ To me, it's pretty obvious that a singularity will occur when ...
1
vote
0answers
45 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
34 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
62 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
47 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
25 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
96 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
173 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
230 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
205 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 - ...
1
vote
0answers
77 views

What is suitable contour shape for $\int_0^\infty\dfrac{b^2+2ab+k}{b(b^2+ab+l)}e^{bx}~db$

$\int_0^\infty\dfrac{b^2+2ab+k}{b(b^2+ab+l)}e^{bx}~db$ . What kind of contour is suitable for this integral?
0
votes
0answers
37 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
36 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
53 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
23 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
95 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 ...
0
votes
0answers
33 views

Residue at Infinity Question

So, I'm trying to determine a residue of the following at infinity: $$f(z)=\frac{z^6}{(1+z)^3}$$ How would I even begin to approach this? Here are my thoughts; I'm mostly looking for a confirmation ...
0
votes
0answers
41 views

Polynomial Expansion in a Complex Improper Integral

I want to evaluate the integral below by using the Residue theorem. $$\int_{-\infty}^{\infty}\frac{\exp(-iwt)w}{(w-ic)\sqrt{w^2-aw+b}}dw$$ There are branch cut points due to the square rooted term ...
0
votes
0answers
47 views

Integration by Residue Theorem - Is This Integral equal to Zero?

$$ \int_{-\infty}^{+\infty}\frac{\sin(x)}{2x^{5}-3jx^{3}+2x}dx=0 $$ ...
0
votes
0answers
33 views

Integration of a complex function having square rooted denominator

How to evaluate this integral ? $$\int_{-\infty}^{\infty}\frac{1}{(w-ia)\sqrt{w^2-ibw+b^2c}}dw$$ where $a$, $b$ and $c$ are real and greater than zero. Does the residue theorem work on it? Please ...
0
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0answers
16 views

A question on Residues and a family of functions

I have the following: $$\alpha(t)=\int_{\left|z\right|=1}z^{k}\frac{f^{\prime}_{t}(z)}{f_{t}(z)}\,dz$$ where $f_{t}(z)$ is a family of entire functions depending on $t \in \Delta$ and $k \geq 0$. By ...
0
votes
0answers
80 views

Pole on path of integration.

Upon evaluating $$\int_0^\infty \frac{1}{1+z^5}dz$$ using the Residue Theorem, why isn't the pole at $z=e^{\pi i}$ taken under consideration in the summation of the residues? We were taught that ...
0
votes
0answers
36 views

perturbative series expansion of integral via complex integration

Define for real $x>0$ and $\epsilon>0,$ the function $$ f(x,\epsilon):= \int_{\epsilon}^\infty \frac{\mathrm{d}s}{s} \frac{1}{\sinh^2 s/2} e^{-sx}. $$ Question: is it possible to compute ...
0
votes
0answers
46 views

inverse mellin transform involving $ \zeta(s) $

what is the inverse mellin transform of $ \frac{\zeta (s)}{\zeta (1-s)} $ aplying cauchy theorem only the trivial zeros contribute to the integral so i beleieve that the inverse mellin transform is ...
0
votes
0answers
210 views

Compute a Real Integral using Residue Theorem

My question is related to the computation of the integral: $$I(z)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\frac{e^{-iz~p}}{p^2}dp,$$ where $z\in \mathbb R$. Using the Residue Theorem I found the ...
0
votes
0answers
47 views

How I can prove that singularities are outside or within the region?

I can not find a way to justify rl as the singularities $z = -1 + i$ and $z = 1 + i$ are inside the semicircle. Can you help me please? Consider $~\int\limits_C\frac{z^3e^{iaz}}{z^4+4}\,\mathrm ...