Residue Definition and 243 Threads

Homework Statement Find Res(0;f) for f(z) = \frac{e^{4z} - 1}{sin^2(z)}. Homework Equations Residue Theorem The Attempt at a Solution If there's a nice (z-a)n singularity in the denominator, then I can simply use the Residue Theorem. However, I'm skeptical on what I'm doing: The...

Homework Statement In order to use cauchy's residue theorem for a question, I need to put ##f(x)=\frac{z^{1/2}}{1+\sqrt{2}z+z^2}## Into the form ##f(x)=\frac{\phi(z)}{(z-z_0)^m}##. Where I can have multiple forms of ##{(z-z_0)^m}## on the denominator, e.g...

I'm going through an explanation in a number theory book about Tonelli's algorithm to find the square roots of a quadratic residue modulo ##p## where ##p## is prime, i.e. I want to solve ##x^2 \equiv a \pmod{p}## with ##(\frac{a}{p}) = 1##. The book goes as follows: Let ##p - 1 = 2^s t##, where...

I need to know what's the Residue Theorem for a Laplace Transform. Does anyone know the name or something, so I can search it? I couldn't find anything. For example, if I have this two equations: X(s).(s-1) = -Y(s)+5 Y(s).(s-4) = 2.X(s)+7 I know how to solve them using Simple Fractions, but...

Homework Statement Find the residue of: $$f(z) = \frac{(\psi(-z) + \gamma)}{(z+1)(z+2)^3} \space \text{at} \space z=n$$ Where $n$ is every positive integer because those $n$ are the poles of $f(z)$Homework EquationsThe Attempt at a Solution This is a simple pole, however: $$\lim_{z \to n}...

Homework Statement Find the residue at z=-2 for $$g(z) = \frac{\psi(-z)}{(z+1)(z+2)^3}$$ Homework Equations $$\psi(-z)$$ represents the digamma function, $$\zeta(z)$$ represents the Riemann-Zeta-Function. The Attempt at a Solution I know that: $$\psi(z+1) = -\gamma - \sum_{k=1}^{\infty}...

I'm asked to evaluate the following integral: \int_{c} \frac{30z^2-23z+5}{(2z-1)^2(3z-1)}dz where c is the unit circle. This function has a simple pole at z=\frac{1}{3} and a second order pole at z=\frac{1}{2}, both of which are within my region of integration. I then went about computing the...

Homework Statement Show that ##\int _{ 0 }^{ \infty }{ \frac{x^2dx}{(x^2+9)(x^2+4)^2} } =\frac{\pi}{200}##. Homework Equations ##Res=\frac{1}{n!}\frac{d^n}{dz^n}[f(z)(z-z_0)^{n+1}]## where the order of the pole is ##n+1##. The Attempt at a Solution Integreading of a semicircle contour one...

Homework Statement Find the residue of ##\frac{1-cos2z}{z^3}## at ##z=0## Homework Equations ##Res=\frac{1}{n!}\frac{d^n}{dz^n}[f(z)(z-z_0)^{n+1}]## Where the order of the pole is ##n+1## The Attempt at a Solution Differentiating ##(1-cos2z)z^3## twice, leaves me with zeros against every...

Homework Statement Find the residue of ##\oint { \frac { sinz }{ 2z-\pi } } dz## where ##\left| z \right| =2##[/B]Homework Equations ##f\left( z_{ o } \right) =\frac { 1 }{ 2\pi i } \oint { \frac { f\left( w \right) }{ w-z_{ o } } } dw## The Attempt at a Solution It seems to me that the...

Hi, first post here. I'm having some trouble with contour integration. Basically here's the question: Contour Integral of ∫ 1+z dz (z-1)(z2+9) There are three cases: l z l = 2 l z+1 l = 1 l z-\iota l = 3 Is each case a straightforward application...

Hello. Can you check this for me, please? Find the singularity of $\frac{e^{z^2}}{(1-z)^3}$ and find the residue for each singularity. My solution: There is a triple pole at z=i, therefore...

[SIZE="3"]Hello. I need some explanation here. I got the solution but I don't understand something. Question: Find the integral using Residue Theorem. $$\int_{-\infty}^{\infty}\frac{dx}{(x^2+4)^2}$$ Here is the first part of the solution that I don't understand: To evaluate...

Hello. I need some explanation here. I got the solution but I don't understand something. Question: Find the integral using Residue Theorem. $$\int_{-\infty}^{\infty}\frac{dx}{(x^2+4)^2}$$ Here is the first part of the solution that I don't understand: To evaluate...

Hi. I have to use the residue theorem to integrate f(z). Can someone help me out? I am stuck on the factorization part. Find the integral $$\int_{0}^{2\pi} \,\frac{d\theta}{25-24\cos\left({\theta}\right)}$$ My answer: $$\int_{0}^{2\pi}...

Hi. I have to use the residue theorem to integrate f(z). Can someone help me out? I am stuck on the factorization part. Find the integral $$\int_{0}^{2\pi} \,\frac{d\theta}{25-24\cos\left({\theta}\right)}$$ My answer: $$\int_{0}^{2\pi}...

Hello. Can someone check if I got the answer right? $f(z)=\frac{e^{-2z}}{(z+1)^2}$ My solution: $f(z)=\frac{e^{-2z}}{(z+1)^2}$ $$Resf(z)_{|z=-1|}=\lim_{{z}\to{-1}}\frac{d}{dz}((z+1)^2\frac{e^{-2z}}{(z+1)^2})$$ $$\lim_{{z}\to{-1}}-2e^{-2z}=-2e^{2}$$

Hello. Can someone check if I got the answer right? Find the singularity and the residue. ##f(z)=\frac{e^{-2z}}{(z+1)^2}## My solution: ##f(z)=\frac{e^{-2z}}{(z+1)^2}## $$Resf(z)_{|z=-1|}=\lim_{{z}\to{-1}}\frac{d}{dz}((z+1)^2\frac{e^{-2z}}{(z+1)^2})$$...

Hi everyone, I've been reading articles and looking at many websites for methods to measure hydrogen peroxide residue concentration in lake water, waste water and ocean water as well as in cell extract in situ . But I'm not satisfied. sometimes they seem too complicated and may require a lot of...

Hi, I am supposed to use residue calculus to do the following integral $$\int_{0}^{2\pi}\frac{1}{a+b\cos( \theta) } \mathrm{d}\theta$$ for |b|<|a| i have paremetrise it on $$\gamma(0;1)$$ that is $$z=\exp(i\theta), 0\leq\theta\leq2\pi$$ and obtain the following...

Homework Statement I'm trying to solve this definite integral using the residue theorem: \int _0^\pi \frac{d \theta}{ (2+ \cos \theta)^2} Homework Equations I got the residue theorem which says that \oint_C f(z)dz = 2 \pi i \ \ \text{times the sum of the residues inside C}...

Find the integral $\displaystyle \int_C \dfrac{\sin(z)}{z} dz $ where $c: |z| = 1 $ Can I use Cauchy integral formula since sin(z) is analytic $\displaystyle\int_C \dfrac{\sin(z)}{z} dz = Res(f,0) = 2\pi i \sin(0) = 0$ I tired to compute it without using the formula $z(t) = e^{it} ...

Hello, I can't find the result to Homework Statement Have to prove that ∫x/(1+x^n) dx = π/n/sin(2π/n) so I'm trying to prove that by starting to find : 2πi*res(z/(1+z^n), exp(iπ/n)) but don't know what is res(z/(1+z^n), exp(iπ/n)) Thanks

I am reading R. Y. Sharp: Steps in Commutative Algebra. In Chapter 2: Ideals on page 32 we find Exercise 2.40 which reads as follows: ----------------------------------------------------------------------------------------------- Let I, J be ideals of the commutative ring R such that $$ I...

I am reading R Y Sharp: Steps in Commutative Algebra. In Chapter 3 (Prime Ideals and Maximal Ideals) on page 44 we find Exercise 3.24 which reads as follows: ----------------------------------------------------------------------------- Show that the residue class ring $$ S $$ of the ring of...

Why did the mathematician name his dog "Cauchy?" Because he left a residue at every pole.

Homework Statement So guys..the title says it! I need to find the residue of cot(z) at z=0. Homework Equations For this situation, since the pole order is 1 Residue=\lim_{z \to z_{0}}(z-z_{0})f(z) The Attempt at a Solution So here's what I am doing in steps: First, the...

Hi, Homework Statement I was wondering whether any of you could kindly explain to me how the equation in the attachment was derived. I mean, how could I have known that it could be separated into these two fractions? Homework Equations The attachment also specifies the integration to be...

\[ \int_0^{\infty}\frac{\cos(mx)}{(x^2 + a^2)^2}dx = \frac{\pi}{4a^3}e^{-am} (1 + am) \] The integral is even so \[ \frac{1}{2}\text{Re}\int_{-\infty}^{\infty}\frac{e^{imz}}{(z + ia)^2(z - ia)^2}dz. \] Since the singularity is of order two, I believe I need to use \[ \int\frac{f'}{f} =...

\[ \int_0^{\infty}\frac{x\sin(mx)}{x^2 + a^2}dx = \frac{\pi}{2}e^{-am} \] The inetgral is even so \[ \frac{1}{2}\int_{-\infty}^{\infty}\frac{x\sin(mx)}{x^2 + a^2}dx. \] We can also write \(x^2 + a^2\) as \((x + ai)(x - ai)\). Should I also write \(\sin(mx) = \frac{1}{2i}(z^m - 1/z^m)\)? I...

Does the Dirac delta have a residue? It seems like it might, but I don't know how to attack it, since I really know very little about distributions. For example, the Dirac delta does not have a Laurent-expansion, so how would you define its residue?

Homework Statement Calculate the integral ∫\frac{1}{(a+bcos^2(ϕ))^2}dϕ from 0 to 2π a,b >0 Homework Equations Residue theorem Cauchy's integral formula The Attempt at a Solution The first thing I did was attempt to find the poles of the integral and use residue theorem to solve the...

Please refer to attached material. For the first question, I have tried looking at examples and have noted that the bounds have been provided in a manner: like |z|=1 (as given in part ii) I am not sure how to get transform the given |z-pi|=pi in such a format, although i suspect it would be...

Really need help for this one. Cheers! Homework Statement Question: calculate function z/(1-cos z) integrated in ac ounterclockwise circular contour given by |z-2pi|= 1 Homework Equations The Attempt at a Solution Clearly the pole in the given contour is 2pi. But the problem is: if it's a...

Find Residue at $z =0 $ of $$f(z) = \Gamma(z) \Gamma(z-1) x^{-z}$$ Try to find Residues for $ z=-n $

I have a doubt on this following procedure using the residue theorem: Initially we have ψ(k,t)=\frac{1}{2\pi}\int_{L_{\omega}}\frac{S(k,\omega)}{D(k,\omega)}e^{-i\omega t}d\omega Then the author said using the residue theorem, we have ψ(k,t)=-iƩ_{j}\frac{S(k,\omega_j(k))}{\partial D/ \partial...

I've been studying for a test and have been powering through the recommended problems and have stumbled upon a problem I just can't seem to figure out. $$\int_{0}^{\infty} \frac{logx}{1+x^{2}} dx$$ (Complex Variables, 2nd edition by Stephen D. Fisher; Exercise 17, Section 2.6; pg. 167)...

Compute the least residue of 3^215 (mod 65537) (65537 is prime). I've tried to use Euler's theorem, Fermat's little theorem and Wilson's theorem, but nothing seems to work, please help.

Hello guys, I just want to confirm about this problem ..Find the residue of this function: f(z)=e1/z/(1-z) Thx in advance.

Homework Statement Hi, I want to calculate the residue of this expression: Homework Equations I know that the residue of a function with a pole of k-th order is given by this: The Attempt at a Solution I know that the function has infinite number of poles at k*∏, for k=-∞ to...

Homework Statement Let C be a regular curve enclosing the distinct points w1,..., wn and let p(w)= (w-w1)(w-w2)...(w-wn). Suppose that f(w) is analytic in a region that includes C. Show that P(z)= (1/2\pii)∫(f(w)\divp(w))\times((p(w)-p(z)\div(w-z))\timesdw is a polynomial of degree n-1...

Homework Statement Considering the following integral, I = \int^\infty_{-\infty} \frac{x^2}{1+x^4} I can rewrite it as a complex contour integral as: \oint^{}_{C} \frac{z^2}{1+z^4} where the contour C is a semicircle on the half-upper plane with a radius which extends to infinity. I can...

Homework Statement Find the Laurent series for the given function about the specified point. Also, give the residue of the function at the point. $$ \frac{z^2}{z^2 - 1}, z_0 = 1 $$ Homework Equations A Laurent expansion is comparable to a power series, except that it includes negative...

Homework Statement f(z) = \frac{1}{ \exp{ \frac {z^2 - \pi/2}{ \sqrt{3} } } + i } Find the residue of f(z) at z_0 = \frac{ \sqrt(\pi) }{2 } ( \sqrt(3) - i ) Homework Equations The Attempt at a Solution I was able to verify that the given z_0 is a singularity, and...

∫(dθ)/(a+bcosθ)^2 Homework Equations I'm trying to find the above integral (from 0-2pi) using Cauchy's Residue theorem. After closing the contour and re-writing the integrant, I know that I have singularity at (-a/b)+(√(a/b)^2-1)- (double pole or is it??). The Attempt at a Solution...

Using contour integration and the residue theorem, evaluate the following "Fourier" integral: F_1(t) := \int_{-\infty}^\infty \frac{\Gamma sin(\omega t)}{\Gamma^2 + (\omega +\Omega )^2} dw with real-valued constants \Gamma > 0 and \Omega. Express your answers in terms of t, \Gamma and \Omega...

Consider the function $$f(z)=\frac{e^{\frac{1}{z-1}}}{e^z -1}$$ $z_0=1$ is an essential singularity, hence $$f(z)=\displaystyle\sum_{-\infty}^{+\infty}a_n(z-1)^n$$ near to $z_0=1$ and i want to find $a_{-1}$. I can write $$f(z)=\frac{\sum\frac{1}{n!(z-1)^n}}{e\cdot...

How can i compute $Res(f,z_k)$ where $$f(z)=\frac{z-1}{1+cos\pi z}$$ and $z_k=2k+1, k\neq 0$?

Does the Dirac delta fuction have a residue? Given the close parallels between the sifting property and Cauchy's integral formula + residue theory, I feel like it should. Unfortunately, I have no idea how they tie together (if they do at all).

So I ran into residue theorem recently and found it to be pretty amazing, and have been trying to get some of the more fundamental aspects of Laurent series and contour integrals down to make sure I understand it properly, but there's still one big aspect that keeps confusing me majorly...