Residue Definition and 249 Threads

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...

Homework Statement it ask me to integral from 0 to 2pi for 1/[5+4sin(deta)]Homework Equations i knoe i nid to find out the pole bu convert sin(deta) into exponential fomat The Attempt at a Solution after i solve it until 2z^2+5iz-2 i stucked,can i juz use [-b+-squarot[b^2-4ac]]/2a formula to...

Homework Statement I'm trying to evaluate the integral I(a)=\int\frac{cos(ax)}{x^{4}+1} from 0 to ∞ Homework Equations To do this I'm going to consider the complex integral: J=\oint\frac{e^{iaz}}{z^{4}+1} Over a semi-circle of radius R in the upper half plane, then let R-->∞...

Homework Statement Find the value of ∫ x^2/(((x^2 + 4)^2)(x^2+9)) dx. With the limits from 0 to infinity. Homework Equations The Attempt at a Solution I know that this is an even function so we can calculate the value of the integral from -∞ to ∞ and just divide by 2. The upper...

I need to find the residue of \frac{e^{2/z}}{1+e^z} at z = \pi i I've been scribbling over numerous papers trying to figure this out. So far I've tried to expand the denominator \frac{e^{2/z}}{(1-e^z + \frac{e^{2z}}{2!} - ...)} I think maybe I've expanded that incorrectly...

Homework Statement evaluate the given trigonometric integral ∫1/(cos(θ)+2sin(θ)+3) dθ where the lower limit is 0 and the upper limit is 2π Homework Equations z = e^(iθ) cosθ = (z+(z)^-1)/2 sinθ = (z-(z)^-1)/2i dθ = dz/iz The Attempt at a Solution after I substitute and...

Homework Statement I need to compute \int_0^\infty \frac{dx}{x^3+a^3}.Homework Equations If f = g/h, then Res(f, a) = \frac{g(a)}{h'(a)}. The Attempt at a Solution In the first I've used a semicircular contour in the upper plane that is semi-circular around the pole at -a. So I calculate the...

Homework Statement I need to solve this integral for a>0: \int _0^{\infty }\frac{\text{Sin}[x]}{x}\frac{1}{x^2+a^2}dx The Attempt at a Solution Using wolfram mathematica, I get that this integral is: \frac{\pi -e^{-x} \pi }{2x^2}=\frac{\pi (1-\text{Cosh}[a]+\text{Sinh}[a])}{2...

Homework Statement Integrate from zero to infinity; f(x)=\sqrt(x)log(x)/(x^2+1) Homework Equations Branch cut makes log(z)= ln|z|+i Arg(z) Poles are at +/- i and Res(z=i) is \pi/4 e^(i \pi/4) I'll need to close the contour; probably as an annullus in the top half of the...

Hi, I am trying to evaluate the integral from 0 to infinity of x^2/[(x^2+9)(x^2+4)^2 dx (really really sorry, I don't know how to use latex yet but as of today I will begin learning!) I chose to evaluate it around a semi-circle of radius R, R>3 (y>0 quadrants) and have found that there are...

Homework Statement I am curious, if I,J, and M are ideals of the commutative ring R, and M/I\subseteqJ/I, then M\subseteqJHomework Equations M/I = { m+I : m is in M} J/I = { j+I : j is in J} I\subseteqR is an ideal if 1.) if a and b are in I then a+b is in I 2.) if r is in R and a is in I...

Homework Statement To find the integral by Cauchy Residue Theorem and apply substitution method. Homework Equations To show: ∫^{2∏}_{0}\frac{cosθ}{13+12cosθ}=-\frac{4∏}{15} The Attempt at a Solution The solution I have done is attached. It is different as what the question wants me...

Hi, I want to know how you find the residue of z=-i for the function exp(1/(z+i)). Clearly, the function has an essential singularity at z=-i so the good ol' formula for the residue for a pole of order m, doesn't really work here. What do I do? :)

Homework Statement how to compute the residue of 1/(cosh(z)^n) at z=ipi/2? Homework Equations b]3. The Attempt at a Solution [/b] i tried to use cosh(z)=-isinh(z-ipi/2) and taylor expansion of this.then from expansion of 1/(1+z) and some algebraic manipulations i tried...

Homework Statement Calculate the residue of sinz/z^4 Homework Equations The residue for a pole of order m at 0 is given my: lim z->0 [1/(m-1)!dm-1/dzm-1[(zmf(z)]] The Attempt at a Solution Clearly the pole has order 3: So we get: Re(0) = limz->0[1/2d2/dz2[sinz/z]] I get that the...

Homework Statement ∫ from infinity to 0 of x^2/(x^2+1)(x^2+16) Homework Equations The Integral will be the sum of the residues times 2πi The Attempt at a Solution The only singularities that matter are at i and 4i and they are simple poles. So x^2/(x^2+1)/(2x) and I plug in 4i...

Homework Statement I need to find the residue of \frac{e^{iz}}{(1+9z^{2})^{2}} so I can use it as part of the residue theorem for a problem. Homework Equations Laurent Series R(z_{0}) = \frac{g(z_{0})}{h^{'}(z_{0})} The Attempt at a Solution I tried using the laurent series but...

Let $f(z)=\frac{z^{100}}{z^{102}+1}$. Prove that the sum of the residues of f is 0. (hint: consider the integral of f around a circular contour centrered at zero)

Greetings everyone, I have a question about calculating the residue of the following function at z=0 point. I am new to complex analysis so I would like to hear all the possible methods. I tried to expand the function I was dealing with in Laurent series and was unable to do so. How can I...

Homework Statement Convert to suitable contour integral and use Cauchy's Residue Theorem to evaluate it ##\int_0^{2\pi} \frac{\sin \theta d\theta}{5-4\sin \theta}## Homework Equations Res ##f(z)_{z=z_0} = Res_{z=z_0} \frac{p(z)}{q(z)}=\frac{p(z_0)}{q'(z_0)}## The Attempt at a...

Folks, I am trying to understand calculating residues. http://www.wolframalpha.com/input/?i=residue+of+1%2F%28z%5E2%2B4%29%5E2+at+z%3D2i How is that answer determined? I mean (2i)^2=-4 and hence denominator is 0...? Thanks

For the contour $|z| = 2$ $$ \int_C\frac{z + 1}{z^2 + 1}dz = \int_C\frac{z + 1}{(z + i)(z - i)}dz = 2\pi i\sum\text{Res}_{z = z_j}\frac{z + 1}{z^2 + 1} $$ Let $g(z) = z^2 + 1$. The zeros of $g$ occur when $z = \pm i$. $g'(\pm i)\neq 0$ so the poles are simple for $1/g$. Let $f(z) = \dfrac{z +...

I am tying to find all the residue of $\dfrac{\sin z}{z^n}$. I am think can I do this but I am not sure where to start. Should I use the Weierstrass product definition of sin z?

Hi all, i have an integral equation of ∫1/[(z+ia)^2 *(z-ia)]*exp(-p^2[(A+iBz)/(C(z^2+a^2))])*exp(-ikbz)dz from the limit of 0 t0 l I tried perform residue theorem but due to the 1/[(z+ia)(z-ia)] factor in the exponential term complicated it...I also tried incorporate L'Hopita rules in...

Homework Statement calculate the residue of the pole at z=i of the function f(z)=(1+z^2)^-3 State the order of the pole Homework Equations I know the residue theorem and also the laurent series expansion but I'm having trouble applying these The Attempt at a Solution I...

Homework Statement Prove that if p is prime, then if A is a quadratic nonresidue mod p and B is also a quadratic nonresidue mod p, then AB is a quadratic residue mod p. Homework Equations A is a nonresidue means A = x^2 (mod p) has no solutions The Attempt at a Solution I already...

Homework Statement Im not sure if i understood correctly how to calculate the residue for functions with essential singularities like: f(z)=sin(1/z) h(z)=z*sin(1/z) j(z)=sin(1/z^2) k(z)=z*(1/z^2) Homework Equations So, according to what I've read, when we have a functions with an essential...

Homework Statement Compute the integral: ∫ x2/(x4-4x2+5) Homework Equations Uses Residue theorem. The Attempt at a Solution So I found the zeroes of x4-4x2+5 to be 2+i and 2-i, and therefore the one that is of relevance is 2+i since it is in the upper half-plane. Then I used...

$$ \int_0^{\infty}\frac{(\log x)^2}{1 + x^2}dx = \frac{\pi^3}{8} $$ I have no idea what to do with this integral. I can't see it is even and do 1/2 the integral from -infinity to infinity since log -x doesn't make sense.

Homework Statement The original problem is to find the integral from 0 to 2\pi of \frac{1}{2\pi}e^{cosθ}cos(2θ) Homework Equations I have it down to two equations which I believe I have to find the residues of and add the results, multiply by 2\pii and I will have my answer. The...

Edit: Never mind I found my error, moderator can lock this.Homework Statement Evaluate the integral \int_0^{\pi} \frac{dt}{(a+cost)^2} for a > 1. Homework Equations \int_0^{\pi}\frac{dt}{(a+cost)^2} = \pi i\sum_{a\epsilon \mathbb{E}}Res(f;\alpha) Where \mathbb{E} is the open unit disk, and...

I have read the chapter on Residue Theorem in Complex Analysis by Serge Lang but don't quite understand how to do the problems. Can someone walk me through the problem (see below) so I can see a better example? Find the residue at 0 for $$ \frac{e^z}{z^3} $$ I see we have pole of order 3 at...

$$ \int_0^{2\pi}\frac{\bar{z}}{z^2}dz $$ How would this be integrated?

Hi, I'd like to fit a straight line to some data which is noisey with gaussian noise with some st dev. Using least squares, I can estimate the slope and intercept. I'd like to know the uncertainty in these numbers. I can find the residue, I believe this is a measure of the variance of the...

The problem Find Res(f,z1) With: f(z)=\frac{z}{(z^2+2aiz-1)^2} The attempt at a solution The singularities are at A=i(-a+\sqrt{a^2-1}) and at B=i(-a-\sqrt{a^2-1}) With the normal equation (take limit z->A of \frac{d}{dz}((z-A)^2 f(z)) for finding the residue of a pole of order 2, my attempt...

I've tried this using the definition of a contour integral and Cauchy's residue theorem but get conflicting answers. \displaystyle f(z) = \frac{z+1}{z(2z-1)(z+2)} We can parametrise the contour \gamma (the unit circle) by \gamma(t) = e^{it} for t \in [0, 2\pi ] So by the definition of a...

In the field of rationals \mathbb{Z}_{(p)} (rationals in the ring of the p-adic integers), how is it possible to prove the residue field \mathbb{Z}_{(p)}/p\mathbb{Z}_{(p)} is equal to \mathbb{Z}/p\mathbb{Z} ? I've narrowed it down to \mathbb{Z}_{(p)}/p\mathbb{Z}_{(p)} = \left\{ a/b\in\mathbb{Q}...

Homework Statement X(z)=z/(z-.7)^2 Use the residue method to calculate x(k) Hello, I have this exercise that I am trying to complete. I am not really familiar with the residue method and am having a hard time finding a good example on line. The section in my book does not describe it...

Homework Statement http://imageshack.us/photo/my-images/593/aaayl.png/ http://imageshack.us/photo/my-images/593/aaayl.png/ Homework Equations The Attempt at a Solution I already solved part b but I put it up there for part c. I know that there is a singularity at a=cost...

Hi Folks, I worked out a couple of problems on finding the Cauchy Principal Value, and I would like to check whether my solutions are correct and also take the opportunity to ask a couple of general questions about the residue theorem, contour integration, and the Cauchy principal value. The...

Evaluate the following integral using the residue theorem: Any hint?

The question asks to show using the residue theorem that \intcos(x) / (x2 +1)2 dx = \pi / e (the terminals of the integral are -∞ to ∞ but i didnt know the code to write that) I found the singularities at -i and +i so i think we change the function inside the integral to cos(z) / (z2...

The question asks to show using the residue theorem that \int cos(x)/(x2+1)2 dx = \pi/e (the terminals of the integral are -\infty to \infty but i didnt know the code to write that) I found the singularities at -i and +i so i think we can then say \intcos (z) / (z+i)2(z-i)2 dz...

Homework Statement I=\int_{-\infty}^{\infty} { dx \over {5x^2+6x+5}}Homework Equations The residue theorem.The Attempt at a Solution I can't use the residue theorem since the denominator has real zeros. How should I solve this?

Homework Statement Use suitable contours in the complex plane and the residue theorem to show that integral from -infinity to +infinity of [1/(1+(x^4))] dx=pi/(sqrt(2)) Fix R > 1, and consider the counterclockwise-oriented contour C consisting of the upper half circle of radius R...

I forgot why the next statement is true and it's bugging me endlessly... If p is prime such that p =1 mod 4 then (-1) = x^2 mod p. Now in Ashe's Algebriac Number theory notes (book?), he says that ((\frac{p-1}{2})!)^2= -1 mod p I am quite stumped as how to show this, he argues we...

Homework Statement Hello friends, I'm a student of mechanical engineering and I have a problem with computing residues of a complex function. I've read some useful comments. Now I ve got some ideas about essential singularity and series expansion in computing the residue. However, I still...

Homework Statement Calculate \stackrel{Res}{z=0}(\frac{z^5}{(z^2-4)(z-4))} Homework Equations I've learned enough Latex for one day, thank you very much. Wikipedia: http://en.wikipedia.org/wiki/Residue_theorem http://en.wikipedia.org/wiki/Cauchy%27s_integral_formula The...

Homework Statement f(z)=cot(pi*z)/(z^2+1) Homework Equations The Attempt at a Solution now I want to get the residue at z=i, I know the definition of f(z)'s residue form but when I try to get the expansion of cot(pi*z) at z=i, I used a lot of method like use sin*csc this form...

http://www2.imperial.ac.uk/~bin06/M2...nation2008.pdf Solutions are here. http://www2.imperial.ac.uk/~bin06/M2...insoln2008.pdf My first question is about 3(ii), the proof of Cauchy's integral formula for the first derivative. The proof here uses the deformation lemma (from second...