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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13
votes
0answers
766 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
89 views

Evaluate $\int_{-\infty}^{\infty}\frac{\sqrt{a+ix}}{a^2+x^2}\,dx$ using residue calculus

I'm asked to evaluate $$\int_{-\infty}^{\infty}\frac{\sqrt{a+ix}}{a^2+x^2}\,dx$$ $\mathbb{R}\ni a>0$, using residue calculus (where $\sqrt{\cdot}$ is the PV $\sqrt{}$). My approach is as follows: ...
5
votes
0answers
171 views

Integrating $\int_{-\infty}^\infty \frac{1}{1 + x^4}dx$ with the residue theorem

Calculate integral $$\int\limits_{-\infty}^{\infty}\frac{1}{x^4+1} dx$$ with residue theorem. Can I evaluate $\frac 12\int_C \dfrac{1}{z^4+1} dz$ where $C$ is simple closed contour of the upper ...
5
votes
0answers
151 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 ...
5
votes
0answers
760 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 = e^{...
4
votes
0answers
71 views

inverse Laplace transform by finding residues of essential singularities

I want to find the inverse Laplace transform of $$F(s)=\exp\Big(-\sqrt{2s}\tanh(\sqrt{2s})\Big).$$ Despite the square roots, $F$ doesn't have any branch points since $$\sqrt{2s}\tanh(\sqrt{2s})=\frac{\...
4
votes
0answers
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 $\sum_{...
4
votes
0answers
54 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
119 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 x}{T_x}\...
4
votes
0answers
96 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 \infty}^{-c+i\...
4
votes
0answers
64 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
59 views

Use Residue Theorem to Sum Series

Show that $$\sum_{n=- \infty}^{\infty} \frac{1}{(3n-1)^2} = \frac{4 \pi^2}{27}$$ I'm pretty sure I need to use the Residue Theorem to sum the series, but I'm unsure how to apply it. Here's what I ...
3
votes
0answers
63 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 ...
3
votes
0answers
69 views

Find $\int_0^{2\pi} \frac{d\theta}{2\pi\cos^{2n}(\theta)} \ n=1,2,3,\dots$ via Residue Theorem

So the question is as follows: Use the Residue Theorem to calculate $$\int_0^{2\pi} \frac{1}{2\pi\cos^{2n}(\theta)} d\theta \quad\quad n=1,2,3,\dots.$$ Now I believe the first step would be to use the ...
3
votes
0answers
71 views

Computing an integral using residues

I am trying to find an integral: $$\int_{-\infty}^{+\infty}\frac{e^{-\sqrt{(x^2 + 1)}}}{(x^2 + 1)^2}\,\mathrm dx$$ I went about applying contour integral over a semicircle with diameter along $ x = +...
3
votes
0answers
88 views

Finding residues of rational functions with extremely large powers

$h(z)=\frac{5z^{2015} + 7z^{2010} - 38z^5 + z^4 - 2z^3 + 5}{7z^{2016} + 2z^{2013} - 6z^8 + 3z^7 + z^5 - 4z^2 - z + 111}$ Find the sum of the residues of h at its poles in $C$ How do I find the ...
3
votes
0answers
135 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
201 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
312 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
35 views

Compute $\int_{0}^{\infty} \frac{x^{1/3}}{(x^2 + 1)^2} dx$

So I want to compute $\int_{0}^{\infty} \frac{x^{1/3}}{(x^2 + 1)^2} dx$ using complex analysis, Cauchys theorem and the residue theorem. What I did was the following: define $g(z) = e^{1/3(\ln|z| + i\...
2
votes
0answers
47 views

Evaluating Improper Integrals with Residues - don't think I'm calculating the residues properly

I have to evaluate the integrals $\displaystyle \int_{-\infty}^{\infty}\frac{dx}{x^{2}+p^{2}}$, for $p > 0$, and $\displaystyle \int_{-\infty}^{\infty} \frac{dx}{(x^{2}+p^{2})^{2}}$, for $p > 0$ ...
2
votes
0answers
74 views

Sum of Gamma Function Residues

I was exploring Cauchy's residue theorem with the gamma function and came across an interesting identity. Consider $$\int_{C_R} \Gamma(z) \, dz $$ Over the complex plane where $C_R$ is the curve ...
2
votes
0answers
41 views

$\frac{\omega}{2\pi}\int_0^{\frac{2\pi}{\omega}}\frac{\sin^2\theta \cos^2\omega t}{(1+\beta\cos\theta \sin\omega t)^5} dt$

I'm going to write out the whole problem as it is given to me (bad grammar and all) even though some of the info may be irrelevant to finding a solution. A charge $e$ moving along a straight line ...
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 ...
2
votes
0answers
37 views

Evaluating the residue of $(1 - e^{-z})^n$ at $z = 0$ with $n \in \mathbb{Z}$

For $n \in \mathbb{Z}$, I want to find a way to determine the residue of the function $f(z) := (1 - e^{-z})^n$ at $z = 0$. I must admit that I haven't come too far yet. In case that $n ≥ 0$, $f$ ...
2
votes
0answers
34 views

Integral of the principal value of a hypergeometric function

I am looking to write the hypergeometric function $${}_2F_1\left(1,1,2+\epsilon, -\frac{\alpha}{\beta}\right) = \int_0^1\,dt\,\frac{(1-t)^{\epsilon}}{1-tz + i\delta},$$ where $z=-\alpha/\beta$ and $0&...
2
votes
0answers
35 views

Is this integral automatically zero?

If I integrate $\int e^{iz}\,dz$ for z complex, along the positive real line, then is the imaginary part of the integral $i\int \sin(x)\,dx$ automatically equal to zero (integration only along the ...
2
votes
0answers
151 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 ...
2
votes
0answers
116 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 ...
2
votes
0answers
29 views

Residue of function (two functions multiplied)

I need to know if I'm doing this exercise correctly: $f(z)$ has only one singularity, which is at $z=0$, and it's a pole of order 7 $f(z)=-f(-z)$ $f(z)$ is analytic in $z=\infty$ $g(z)$ is analytic ...
2
votes
0answers
261 views

Example of an analytic continuation for a function in integral form

Given $f(z) = \int_{-\infty}^\infty \frac{exp(-t^2)}{z-t}\,dt$, where $Im(z)>0$. Find an analytic continuation to the region $Im(z)<0$. Firstly the solution said that there is a branch cut on ...
2
votes
0answers
113 views

Residue theorem definite trigonometric integration

I am trying to solve this integration $$\int_{0}^{\pi} e^{cos(\theta)} \tan^{3}(\theta)d\theta$$ putting $$z=e^{i\theta}$$ $$\int_{\gamma} e^{(\frac{z^{2}+1}{2z})}\frac{{(z^{2}-1)}^3}{z(z-i)^{3}(z+i)^...
2
votes
0answers
75 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
47 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
54 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
48 views

complex integral with non integer power

I want to calculate this integral $$I\stackrel{\text{def}}{=}\int_{-\infty}^{\infty}\frac{\mathrm{exp(-\mathrm{i}\mathbf{q}\mathbf{r}})}{-\mathrm{i}\omega+\mathbf{q}^{3-\eta}(-\mathrm{i}\omega)^{\eta/...
2
votes
0answers
202 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
200 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
61 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
889 views

inverse Laplace transform of $e^\sqrt{as}$

I am trying to find the inverse Laplace transform of $e^\sqrt{as}$ for $a>0$. So we need to solve $\oint_B dz \: e^\sqrt{az} e^{z t}$ (Bromwich contour), but not sure how to start. How do we even ...
2
votes
0answers
277 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
124 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 - i)}^...
2
votes
0answers
82 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
307 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: $$\mathcal{G}(s)=\int_{\gamma-i\infty}^{\gamma+i\...
2
votes
0answers
63 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 $$ \Omega_{\...
1
vote
0answers
12 views

Relationship between the residues $Res(g\circ\varphi,z_0)$ and $Res(g,w_0)$

Let $\varphi:U\rightarrow\mathbb{C}$ be holomorphic with $\varphi'(z_0)\neq 0$ for some $z_0\in U$. Let $g$ be another function having a pole of order $1$ in $w_0=\varphi(z_0)$. What is the ...
1
vote
0answers
44 views

Show that $\int_{C} \frac{e^{az}}{\sin(\pi z)} \,dz \rightarrow 0 \quad as \quad R \rightarrow \infty$

Given the straight line in the complex plane: $b+iR$ to $b+1+iR$ where $0<b<1$ and $|Im(a)|<\pi$, show the following: $$\int_{C} \frac{e^{az}}{\sin(\pi z)} \,dz \rightarrow 0 \quad as \quad ...
1
vote
0answers
71 views

Finding a closed form for $\sum_{n=-\infty}^{n=+\infty}\frac{1}{n^{2k}+a^{2k}}$

I am trying to find a closed form for $S=\sum_{n=-\infty}^{n=+\infty}\frac{1}{n^{2k}+a^{2k}}$, $k \in \mathbb{N^{*}}$, $a>0$ I don't even bother to look for a closed form with an odd exponent, ...
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
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 ...