Questions on the evaluation of residues, on the evaluation of integrals using the method of residues or in the method's theory.

learn more… | top users | synonyms

4
votes
4answers
165 views

Integral $P\int_0^\infty \frac{x^{\lambda-1}}{1-x} dx$

I am trying to calculate the following principle value integral \begin{equation} P\int_0^\infty \frac{x^{\lambda-1}}{1-x} dx \end{equation} for $\lambda \in [0,1].$ I tried to turn this into a ...
3
votes
1answer
53 views

Prove that for an odd function, res(f, z0) = res(f, -z0)

I am having difficulty proving that, for an odd function, the residue function is symmetric. i.e $res(f, z_0) = res(f, -z_0)$ I am using the Laurent series expansion of a function ...
1
vote
1answer
39 views

integral using residual theorem

I have the following problem: $$\int_C\cot z\ \mathrm{d}z,\ \ \ C(t)=2e^{it},\ t\in[0,2\pi]$$ my solution is: $$\mathrm{res_0}\cot z=\frac{\cos(0)}{\cos(0)}=1$$ The circle only surrounds one ...
3
votes
1answer
88 views

Solve $\mathscr{F}^{-1} [ \cot{a \omega} \times \mathscr{F} \{ U(t) \sin{\omega_0 t} \} ] $ using contour integration

I wish to evaluate $y(t) = \mathscr{F}^{-1} [ \cot{a \omega} \times \mathscr{F} \{ U(t) \sin{\omega_0 t} \} ] $, where $\mathscr{F}$ represents the Fourier transform, and U(t) represents the ...
0
votes
1answer
27 views

Residue of $\frac{\cot{ax}}{x^2-b^2}$?

I am interested to find the residue of $$\frac{\cos{ax}}{(x^2-b^2)\sin{ax}}$$ at $x=b$. How would I go about doing this? I can see that the pole is second order, and so the formula $$\text{res} = ...
0
votes
1answer
36 views

Calculating residue of $(z^3+1)^{-3}$ at $z=\exp(i\pi/3)$

I'm struggling to calculate the residue of $(z^3+1)^{-3}$ at $z=\exp(i\cdot\pi/3)$, which is the pole for the given contour. Using the formula for residue has resulted in a prohibitive double ...
1
vote
1answer
81 views

Calculate the residue of $\cot\pi z$ at poles $z=n$

I'm having trouble calculating the residue of $f(z) =\cot\pi z$. The function has a simple pole for every integer n, and i'm, trying to find the residue at n. I know that by the residue theorem: ...
4
votes
0answers
40 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 ...
0
votes
1answer
24 views

Pole and residue of $f(z) = \frac{1}{1+z^n}$

Let $f(z) = \frac{1}{1+z^n}$ for $n \ge 3$. How can we compute poles and residue in first point over real axis? Firstly, we have to solve $1+z^n=0$. I think, that solution of this are $$z_k = e^{2 ...
2
votes
1answer
36 views

Calculating the residues of $\frac{\zeta^{\prime}{(s) x^{s}}}{\zeta(s)\cdot s}$

Calculating the poles of $\frac{\zeta^{\prime}{(s) x^{s}}}{\zeta(s)\cdot s}$, where x is a fixed real number. I am trying to calculate the poles of this function at the trivial zeros of $\zeta$, ...
2
votes
3answers
328 views

A nasty simple pole… A residue calculation.

We are asked to compute: Res$[\frac{z^n+1}{z^n-1},e^{2\pi ki/n}]$, where I am assuming $k\in\mathbb Z$. The only tools I am aware of to compute residues comes from relating the function to power ...
1
vote
3answers
52 views

Residue Calculation

I am stuck on what should be a trivial residue calculation. Any suggestions? Compute the residue of $\frac{e^{2iz}-1}{z^2}$ at $z=0$. Isn't it a double pole, i.e., shouldn't I be able to take the ...
1
vote
1answer
56 views

Integrating around pie-slice domain

We are asked to show $$\int_0^{\infty}\frac{\log(x)}{x^3+1}dx=-\frac{2\pi^2}{27}$$, and $$\int_0^{\infty}\frac{1}{x^3+1}dx=\frac{2\pi}{3\sqrt{3}}$$ By integrating around a pie slice with angle ...
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, ...
1
vote
1answer
72 views

Calculate integral using the residue theorem

I want to calculate the integral $$\int_C {{z^2-2z}\over{(z+1)^2(z^2+4)}}dz$$, where $C=\{z:|z|=4\}$ I want to use the Residue theorem to tackle this integral. Now, $f(z)$ has a pole of degree 2 at ...
2
votes
1answer
82 views

Calculating residues of multiple poles

I have $f(z)=\dfrac{z^2e^{iz}}{(z^2+1)^2}$, and I need its residue at $z=i$, which is a double pole. I tried expanding $f(w+i)$, where I got to $f(w+i)=w^{-2}\dfrac{(w^2+2iw-1)e^{iw-1}}{(w+2i)^2}$ and ...
3
votes
1answer
49 views

Keyhole Domain Residue problem with logarithm

Show $$\int^{\infty}_{0}\frac{\log(x)}{x^a(x+1)}dx = \frac{\pi^2\cos(\pi a)}{\sin^2(\pi a)}, \ 0<a<1$$ I have tried tackling this by using a keyhole domain, where $\gamma_{\epsilon}$, where ...
0
votes
1answer
70 views

Application of Residue Theorem and limits

I am trying the following problem from Fisher's Complex Variables book: If $f$ is analytic on a plane except at poles $\gamma_1, \cdots \gamma_N$ and none of them are integers and ...
0
votes
1answer
20 views

Look for the group members $\left ( (\mathbb Z/24\mathbb Z)^*,\underset{24}{\odot} \right )$ and calculate their orders.

Look for the group members $\left ( (\mathbb Z/24\mathbb Z)^*,\underset{24}{\odot} \right )$ and calculate their orders. The elements are $\overline1,\overline2,\ldots,\overline{23}$. Have the ...
3
votes
1answer
47 views

Residue of $\frac{1}{(1-z)^3}$ at $z=1$

I know there is a singularity of $z=1$ but I am a bit confused on how to find the residue at that point since if we have that $f(z)=\frac{g(z)}{h(z)}$ with $g(z)=1$ and $h(z)=(1-z)^3$ then $g(z)$ has ...
0
votes
1answer
31 views

Application of Residues

So in applying the residue theorem to solve improper real integrals, we agree to take our semicircles to be as large or as small as necessary such that all the poles we wish to work with lie inside ...
5
votes
3answers
126 views

Find the Fourier Transform of $\dfrac{x}{x^4+4}$

I have a problem here that becomes quite difficult to manage. I have to find the fourier transform of: $$f(x)=\frac{x}{x^4+4}$$ I'm sure there will be many ways to do this and I'll post my method ...
2
votes
2answers
96 views

Riemann zeta, why are the residues either zero or one?

One more question, probably equally simple to answer but I don't know how this is true either: Why is the residue of Riemann zeta zero - trivial or non-trivial: $$\text{residue}\left(\zeta ...
1
vote
0answers
74 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.
5
votes
1answer
33 views

Meromorphic and even

I would like to do the following exercise : Let $f$ be a meromorphic function and $\mathcal{P}$ the set of its poles. We also assume that $f$ is even ($\forall z \in \mathbb{C}, \; ...
1
vote
1answer
53 views

Deducing Laplace Formulas

I have to compute the followings integrals $\forall\; b\in \mathbb{C},\; \text{Re} \;b \gt0,p\gt 0$ $$ \int_{-\infty}^\infty \frac{e^{ipx}}{x-ib}$$ $$ \int_{-\infty}^\infty \frac{e^{ipx}}{x+ib}$$ ...
0
votes
1answer
248 views

Is f(z)=1/z truly an analytic function

For an analytic function $f(z)$, we have $$\frac{\partial f}{\partial \bar{z}}=0.$$ Consider the function $f(z)=\frac{1}{z}$, which, at first sight, is a bona fide analytic function. However, we can ...
2
votes
1answer
45 views

I want to compute $\int_0^\infty \frac{x^t}{1+x^2}dx \; \forall t \in (-1,1)$ using residue theroem.

I want to compute $$\int_0^\infty \frac{x^t}{1+x^2}dx \qquad \forall t \in (-1,1)$$ using residue theroem. I consider $$f(z) = \frac{z^t}{1+z^2}$$ I find two pole of order 1 in $z=i$ and $z=-i$ with ...
3
votes
0answers
93 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)!} ...
1
vote
1answer
70 views

Complex contour integral with sign function:$-i \int \limits_{-\infty}^\infty \frac{{\rm sgn}(x)^2 ~x~ e^{i x}}{1+ax^2} dp$

I am trying to evaluate the integral: $-i \int \limits_{-\infty}^\infty \frac{{\rm sgn}(x)^2 ~x~ e^{i x}}{1+ax^2} dx$ with sgn$(x)$ the sign function and $a$ positive real. Naively applying the ...
2
votes
2answers
67 views

compute the integral using residue theory

I am trying to compute an integral in an example in my complex analysis textbook: $$\int_{-\infty}^\infty {xsinx\over x^4+1}dx$$ The book gives some startup hints, but I don't quite follow, I set ...
0
votes
0answers
20 views

computing integral using residue theory [duplicate]

I want to compute the integral $\int_{-\infty}^\infty {x^4\over {1+x^8}}dx$ by using residue theory. I find the zero of $Q(x)$ is $i^{1/4}$. Do I have to factor the denominator into 8 different ...
2
votes
1answer
66 views

Find the Laurent series of the function and give the residue

I have $f(z)={z^2\over {z^2-1}}$. I want to find the Laurent series of $f$ and the residue at the point $z_0=1$. Can I say that $f$ has a pole of order 2 at $z=1?$ Or is that only used when the ...
3
votes
1answer
13 views

$a_{-1}$ term of a function on the punctured plane?

Our instructor said if f is analytic on the punctured complex plane (missing origin), then $$a_{-1}=\oint_{|z|=r} f(z) dz$$ This may be an obvious question, but what happened to the $\frac{1}{2\pi i}$ ...
2
votes
2answers
63 views

Find $\int_\Gamma\frac{2z+j}{z^3(z^2+1)}\mathrm{d}z$ where $Γ:|z-1-i| = 2$

pls, some ideas for integral solution (residue theory)? $$\int_\Gamma\dfrac{2z+j}{z^3(z^2+1)}\mathrm{d}z$$ Where $Γ:|z-1-i| = 2$ is positively oriented circle. Thx, for help!
0
votes
1answer
23 views

Residue Calculation Problem

I need to find the residue of the following equation at $z = 0$. $$ \frac{\cot(z)\cot(hz)}{z^3}$$ My attempt is as follows: The residue will be the coefficient of $1/z$ in the Laurent Series ...
0
votes
1answer
82 views

Residue theorem in evaluating complex integrals?

It's been a while since I used residue theorem to evaluate anything. I remember that whenever we have a real valued function, we can use residue theorem to evaluate its integral with associated ...
0
votes
2answers
119 views

Calculating residue of $z\sin{ \frac {z+1}{z-1}}$

Let $f=z\sin{ \frac {z+1}{z-1} }$. Calculate the residue of $f$ in $z=1$. I think $f$ has an essential singularity at $z=1$ so the only way I can proceed is with Laurent series. I've defined ...
1
vote
1answer
34 views

complex residue involving exponent of quotient of polynomials

I was trying to work out an integral and came to trying to find the complex residue of $$R = \exp\left(\frac{Ax^2 + Bix}{Dx + 1}\right)$$ at $x = -D^{-1}$. I used partial fractions to get: $$ = ...
1
vote
1answer
80 views

$\int e^{\cos(x)} \cos(nx)\ dx$ using the residue theorem

I am trying to evaluate the following integral using the residue theorem: $$\int_0^{2\pi} e^{ \cos(\theta)} \cos(n\theta) d\theta$$ I have already evaluated $\int_0^{2\pi} e^{e^{-i\theta}} e^{i ...
4
votes
1answer
150 views

Residue theorem with exponential and trig functions

The following integral should be doable using the residue theorum: $$\frac1{2\pi}\int_{0}^{2\pi}e^{\cos\theta}\cos(n\theta) \,d\theta$$
0
votes
0answers
31 views

Contour Integral of $I= \int_{-i\infty}^{i\infty}\frac{a^{z+1}}{1+z} dz$ [duplicate]

I'm trying to evaluate the following integral: $I= \int_{-i\infty}^{i\infty}\frac{a^{z+1}}{1+z} dz$ $0<a<1$ I've integrated from 0 to $ i\infty $ then from ...
4
votes
0answers
90 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 ...
1
vote
1answer
122 views

Compute $\int_0^{\pi/2} \frac{1}{(\sinh t)^{2}+(\sin{\theta)^{2}}} d\theta$ by residues

We want to compute $\int_0^{\pi/2} \frac{1}{(\sinh t)^{2}+(\sin{\theta)^{2}}} d\theta$ with $t>0$ using residues. The first thing I want to do is using $z=e^{i\theta}$ to transform the integral ...
1
vote
1answer
109 views

Calculation of a Residue

Does anyone know of a good way to calculate the residue at zero of the following function? I was able to calculate it with the higher order pole formula for residues and then used Mathematica to find ...
0
votes
0answers
98 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 ...
1
vote
2answers
101 views

Infinite sums and integrals using residues

I have no idea how to solve these two, any help? $\mathtt{i)}$ $$\frac{1}{2\pi i}\int_{a-i\infty}^{a+i\infty}\frac{e^{tz}}{\sqrt{z+1}}dz$$ $$ a,t\gt0$$ $\mathtt{ii)}$ $$ \sum_{n=1}^\infty ...
2
votes
0answers
45 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 ...
1
vote
1answer
45 views

The definition of Residue

In Wikipedia the definition of a residue of a function $f$ in a point $a$ is a unique value $R$ such that $f(z)-\frac{R}{z-a}$ has an anti derivative in a punctured disk $0<|z-a|<\delta$. How is ...
3
votes
1answer
134 views

prove $\int_{0}^{2\pi}(1-\cos x)^n\cos nx dx=(-1)^n\frac{\pi}{2^{n-1}} $

prove $$\int_{0}^{2\pi}(1-\cos x)^n\cos nx dx=(-1)^n\frac{\pi}{2^{n-1}} $$ I tried with $2\cos x =z+\frac1z$ then use residue theorem but I faced some troubles my try is : $2\cos x =z+\frac1z$ ...