What is Residue: Definition and 248 Discussions

In number theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i.e., if there exists an integer x such that:





x

2


≡
q


(
mod

n
)

.


{\displaystyle x^{2}\equiv q{\pmod {n}}.}
Otherwise, q is called a quadratic nonresidue modulo n.
Originally an abstract mathematical concept from the branch of number theory known as modular arithmetic, quadratic residues are now used in applications ranging from acoustical engineering to cryptography and the factoring of large numbers.

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  1. R

    What is the Nontrivial Residue Integral for the Function sin(bx)/(e^pi x-1)?

    I found this integral in a book ("A course of modern analysis", Whittaker): \int_{0}^{\infty} \frac{sin(bx)}{e^{\pi x}-1} dx I tried to use residue theorem in the rectangular domain [0,R]x[0,i], with R-> \infty , but i couldn't do the integral in [0,i]
  2. T

    Complex Analysis Residue Problem

    Homework Statement The question asks me to find the integral from 0 to infinity of 1/(x^3 + 1), where I have to use the specific contours that they specify. Now I know that I need to use residues (in fact just one here) and the singular point is (1+sqrt(3)*i)/2. Once I can factor the (x^3...
  3. H

    Residue Calc: (z^2 e^z)/(1+e^2z)

    what is the residues in below function? (z^2 e^z)/(1+e^2z )
  4. M

    Residue of exp[i.kx] / [ 1 - k^2 ]: Find Answer & Fix Mistake

    Ive run into some residue problems, I can't seem to find a clear answer anywhere on this... I need to find the residue of exp[i.kx] / [ 1 - k^2 ], where k is my complex variable, and x is positive. I have poles at 1 and -1 in my integral. Now everywhere I look, a pole of order n is when...
  5. T

    Complex Integral Residue Theorem

    I have attached a pdf of my problem and attempted solution. I seem to be a factor of f'(z) out from the required solution, can anyone see where I've gone wrong?
  6. E

    Calc Residue of \frac{z \sin{z}}{\left( z - \pi \right)^3}

    Homework Statement Calculate the residu for the singularity of \frac{z \sin{z}}{\left( z - \pi \right)^3} Homework Equations R \left( a_0 \right) = \frac{1}{2 \pi i} \oint \frac{z \sin{z}}{\left( z - \pi \right)^3} dzThe Attempt at a Solution \pi is an essential singularity so the residue...
  7. P

    Improper integral using residue

    Homework Statement integral: \int\limits_0^\infty\frac{\mbox{d}x}{\left(x^2+1\right)\left(x^2+4\right)} The Attempt at a Solution normally i would do I=\frac12\int\limits_{-\infty}^\infty\frac{\mbox{d}x}{\left(x^2+1\right)\left(x^2+4\right)} and now count residues but is there any other...
  8. J

    Application of residue theorem

    Homework Statement 1) integral(0 to infinity) ((sqrt(x)*log(x))/(1+x^2))dx 2) integral(-infinity to infinity) (cos(pi*x)/(x^2-2x+2))dx Homework Equations The Attempt at a Solution I know I have to post all the steps and show the work, but in this case, I have more than 2 pages of...
  9. H

    Complex numbers - residue theorem

    Hi all, I'm trying to solve 4.15 from the attached file, can anyone help? I tried to use residue thm , i.e the integral of f over the curve gamma-r equals winding number of z0 over gamma-r and residue of z0 of f. I can't see how b-a relates to the winding number of z0. Can anyone help please?
  10. scupydog

    Could Oil from the Gulf of Mexico Be Causing Oily Residue on Windows in the UK?

    After some very strong winds last night here, in the U.K. There is an oily type residue which is visible on external glass surfaces ie windscreens. This effect has happened over a 25 mile radius as i know off. Could this be any thing to with hurricane tomas, lifting oil from the surface in the...
  11. G

    Please fix this residue integration

    \int_{0}^{2\pi} \frac{d\theta}{5 - 4sin\theta} = \oint_{C} \frac{dz/iz}{5 - \frac{2z}{i} - \frac{2}{iz}} = - \oint_{C} \frac{dz}{2z^{2} - 5iz - 2} = - \oint_{C} \frac{dz}{(4z - 8i)(49 - 2i)} z_{1} = 2i, z_{2} = \frac{1}{2} Res = \left|\frac{1}{4z - 2i} \right|_{z = 2i} = \frac{1}{6i} \Rightarrow...
  12. J

    Evaluate Integral with Residue Theorem: Residue Theorem Homework

    Homework Statement Evaluate \int_{0}^{2\pi} (cos^4\theta + sin^4\theta) d\theta by converting it to a complex integral over the unit circle and applying the Residue Theorem. Homework Equations The Attempt at a Solution First, I switch (cos^4\theta + sin^4\theta) to...
  13. N

    Variable exponent causing number of singularities to change for residue?

    Homework Statement Determine the nature of the singularities of the following function and evaluate the residues. \frac{z^{-k}}{z+1} for 0 < k < 1 Homework Equations Residue theorem, Laurent expansions, etc. The Attempt at a Solution Ok this is a weird one since we've...
  14. Q

    Can poles on the contour still be used with the residue theorem?

    Hi, I am trying to prove that I have the correct value of an integral of the form \int_0^{2 \pi} f(\cos{\theta},\sin{\theta}) d\theta . I want to use the residue theorem, but I have one problem: all the literature I can find says that for contour integrals of this form, you can only use the...
  15. T

    MTBE production starting with a vacuum residue

    Hello all. Another year has begun and yet again I'm stumped with all this stuff. If anyone can provide me with sources, articles, or any type of information to help me with this design process. I uploaded a word document with a pretty nice intro I wrote with all the formulas and a...
  16. Z

    Is Using Distribution Theory Overkill for Differentiating Under the Integral?

    would it be valid (in the sense of residue theorem ) the following evaluation of the divergent integral ? \int_{-\infty}^{\infty} \frac{dx}{x^{2}-a^{2}}= \frac{ \pi i}{a} also could we differentiate with respect to a^{2} inside the integral above to calculate...
  17. T

    Using Residue Calculus to evaluate real integrals

    I need to calculate the definite integral of dx/(4+3cos(x)) from 0 to 2pi. I believe that the integral from 0 to 2pi of (f(e^ix))*i*(e^ix)dx is equivalent to the integral over the unit circle of f(z)dz. If that's true, then this problem boils down to finding f(z) so that...
  18. P

    Evaluating Real Integrals using Residue Calculus

    Homework Statement I = \int_{-\infty}^{\infty} \frac{{e}^{ax}}{1+{e}^{x}} dx \; \; 0 < a < 1 a) Show that the improper real integral is absolutely convergent. b) Integrating around the closed rectangle \boldsymbol{R} with corners -R, R, R+2\pi\iota, -R+2\pi\iota use residue calculus to...
  19. T

    Solve Simple Residue Question with Laplace Transforms | Limit as p->0

    Hi guys, I'm doing some Laplace transforms Apologies for not typing this out very well I don't know LaTex... where (1/p)((sinh(p^1/2)(1-x))/(sinh(p^1/2))) I need to work out the residue at p=0... It's been a while since I did this but you multiply through by p and then take the limit...
  20. D

    What is the residue of a branch cut?

    Homework Statement I'm finding the residues of the branch cut of \int^\infty_0 \frac{dx}{x^{1/4}(x^2+1)}dx Homework Equations The Attempt at a Solution I am trying to find the residue of i I am not sure how to handle lim z->i of \frac{1}{z^\frac{1}{4}(z+i)} Any nudges...
  21. A

    Finding the residue of a singularity

    Homework Statement The following function has a singularity at z=0 (e^z)/(1 - (e^z)) decide if its removable/a pole/essential, and determine the residue The Attempt at a Solution I played with the function and saw it can be re-written as: -1 /(z + z^2/2! + z^3/3! +...) In this...
  22. M

    Contour integration using residue theorem (quick question)

    Homework Statement The Attempt at a Solution So i have poles at: z=-1 of order 3, z=1 and z=2. For part i), no poles are located inside the contour, therefore the residue is 0. <--is that right to say, that since there are no poles inside the contour, the residue is zero?
  23. E

    Residue Calculus: Evaluating Poles and Contours

    Homework Statement The Attempt at a Solution So there are poles at: z=\pm2 and at z= -1 of order 4. Right? My query is, when evaluating these poles (using the residue theorem), is it right that for (i) Z = 1/2, no residues lie in that contour? for (iii), do all residues lie in the...
  24. Z

    Residue calculus and gauss bonnet surfaces

    I am not a mathematician but I have noticed how strangley similar the treatments of curvature and residues are when you compare the residues of residue calculus and the curviture of the gauss bonet forumlation of surfaces. Is there some generalization of things that contains both of these...
  25. L

    Residue of e^(az)/(1+e^z)^2 at I Pi

    Homework Statement I need to find the residue of e^(az)/(1+e^z)^2 at I Pi. For some reason this is such much harder than I thought it was going to be. Mathematica is not even helping :(. Homework Equations Cauchy's kth Integral formula. The Attempt at a Solution I made an...
  26. G

    What Are the Singularities and Residues of the Function z^-n(e^z-1)^-1?

    Homework Statement z^-n(e^z-1)^-1 , z not equal to zero locate the singularities and evaluate the residue. Homework Equations The Attempt at a Solution i don't have an idea about when z is not equal to zero because i think that only singularity point is z=0 hence if there...
  27. P

    Residue of a ratio of polynomials

    Homework Statement The problem is to find the inverse laplace of \frac{s^2-a^2}{(s^2+a^2)^2} I am supposed to use the residue definition of inverse laplace (given below) The poles of F(s) are at ai and at -ai and they are both double poles. Homework Equations f(t) =...
  28. K

    Integrate Sin(u) over [0, ∞] with Residue Calculus

    \intu^-B sin(u) du, 0<B<2 integrating from 0 to infinity. What is really throwing me off is the condition, I'm not sure why it's there or really what to do with it. Can I just solve this the same way I'd solve sin(x)/x?
  29. K

    Primitive roots & Reduced residue system

    Let p be a prime. a) If gcd(k,p-1)=1, then 1^k, 2^k,..., (p - 1)^k form a reduced residue system mod p. b) If 1^k, 2^k,..., (p - 1)^k form a reduced residue system mod p, then gcd(k,p-1)=1. ================================= I proved part a by first showing that each of 1^k, 2^k,..., (p -...
  30. Z

    Residue Theorem Application for Integrating Functions with Poles of Higher Order

    Homework Statement Calculate the integral I(k)= \int_{-\infty}^{\infty} \frac{dx}{(x^{2}+1)^{k}} with 'k' being a real number Homework Equations the integral equation above The Attempt at a Solution from the residue theorem , there is a pole of order one at 2+ix=0 , my problem is the...
  31. R

    Congruence of the sum of terms in a reduced residue system

    Homework Statement Prove that if {r1,r2,...rx} is a reduced residue system mod m (where x=\phi(m), m>2), then r1 + r2 + ... + rx= 0 mod m. Homework Equations The Attempt at a Solution I've been able to prove it pretty simply for odd m and for m=2k where k is odd, but for m with higher powers...
  32. S

    Complete residue system Question

    Hi i am doing self-study of number theory as it looks interesting and enlightening. Can someone help because I encounter a problem here.. Suppose A = {a1,a1,,,,,,,ak} is a complete residue system modulo k. Prove that for each integer n and each nonnegative integer s there exists a congruence...
  33. F

    Solving Cauchy Residue Theorem for p(t) in Complex Analysis Homework

    Homework Statement p(t) = integral[-inf,+inf] ( x/sinh(x) exp (i t x) dx) Homework Equations singularity @ x = n*pi*i where n = +-1, +-2, +-3,... Near n*pi*i one can write sinh(x) ~ (x - n*pi*i) The Attempt at a Solution I apply the cauchy residue theorem. For a positive...
  34. J

    Integration using residue theory

    I'm trying to evaluate the integral: \int_0^{2\pi} \frac{d\theta}{24-6sin\theta} using calculus of residues. I've tried this so far: Let z=e^{i\theta} so d\theta=\frac{dz}{iz}. Also, using the exponential definition of sine, sin\theta=\frac{z^2-1}{2iz} This gives messy...
  35. B

    Evaluating Integrals using the Residue THM

    Homework Statement integral |z|=1 of sinz/z2dz Homework Equations Rule #1 if f(z) has a simple pole at z0, then Res[f(z),z0] = lim(as z goes to z0) (z - z0)*f(z) Rule #2 if f(z) has a double pole at z0, then Res[f(z),z0] = lim(as z goes to z0)d/dz (z - z0)2*f(z) Rule #3 If...
  36. B

    Residue Theorem: Evaluating Integral |z|=1 (sin(z)/z^2)dz

    Sorry I don't have equation editor, for some reason every time I install it on Microsoft Word it never appears... Homework Statement Calculate the residue at each isolated singularity in the complex plane e^(1/z) Homework Equations #1 Simple pole at z0 then, Res[f(z), z0] = lim...
  37. V

    Use residue theory to establish the following integral

    Homework Statement Use residue theory to establish the result: \int^{\pi}_{0}\frac{dx}{A + Bcosx} = \frac{\pi}{\sqrt{A^2 - B^2}} The Attempt at a Solution So I've gotten to the point that the above integral = \frac{1}{2} \oint^{2\pi}_{0} \frac{-2i}{Bz^{2} + 2Az + B} dz...
  38. F

    What Is the Residue Number Representation of Negative Numbers?

    I was wondering what the residue number representation of -1 was? (Or for negative numbers in general) Thanks, flouran
  39. Y

    Integral using Residue Theorem

    Homework Statement the integral of 1/(1+x^4) from -infinity to +infinity Homework Equations Residue theorem. The Attempt at a Solution 1/(1+z^4) so z^4 = -1 I know I should be using the residues at z = -sqrt(i) and z= i*sqrt(i) I am getting a complex number as an...
  40. S

    Meaning of the complex residue

    Just spent the last few months working on an undergrad course in complex analysis and have a couple of things that aren't clear to me yet. One of them is the meanings of the residue of a complex function. I understand how to find it from the Laurent series and using a couple of other rules and I...
  41. J

    Need help with finding residue of a simple function

    Need help with finding residue of a "simple" function Hello, I'm trying to find the residue z=0 of f(z) = (1 + z)e^(3/z) I understand this is a essential singularity. I know the answer is 15/2 but I can't seem to find the solution. I've tried this so far: f(z) = (1 + z) ( (3/z) +...
  42. H

    Calculating Residue of cos(z)/z in Contour Integral

    Homework Statement So the problem at hand is to calculate the contour integral \oint cos(z)/z around the circle abs(z)=1.5 . Homework Equations The integral is going to follow from the Cauchy-Integral Formula and the Residue theorem. The problem I am having is figuring out what the...
  43. P

    Contour Integral and Residue Theorem

    Homework Statement Show that: For a = 0 \int_{0}^{\infty} \frac{cos{ax}+x sin{ax}}{1+x^2} dx = \frac{\pi}{2} For a > 0 \int_{0}^{\infty} \frac{cos{ax}+x sin{ax}}{1+x^2} dx = \pi e^{-a} For a < 0 \int_{0}^{\infty} \frac{cos{ax}+x sin{ax}}{1+x^2} dx = 0 Homework Equations Residue...
  44. G01

    Complex Integration / Residue Problems

    Homework Statement Hi everyone. I'm currently taking a graduate math physics course and complex integrals are beating the crap out of me. Some of my questions may be relatively basic. Forgive me, I'm trying to teach myself and am regretting not taking a course on complex analysis as an...
  45. J

    Trouble with use of the Residue Theorem

    I have just learned the residue theorem and am attempting to apply it to this intergral. \int_{0}^{\infty}\frac{dx}{x^3+a^3}=\frac{2\pi}{3\sqrt{3}a^2} where a is real and greater than 0. I want to take a ray going out at \theta=0 and another at \theta=\frac{2\pi}{3} and connect them with an...
  46. O

    Understanding and Calculating Residues for 1/(1+z²+z⁴)

    Homework Statement computing the residue; 1/(1+z²+z⁴) Homework Equations Can someone explain to me what a residue is and how to calculate it! Is it simply the discontinuities of the function? The Attempt at a Solution
  47. F

    Residue Problem: Seeking Alternative Solutions

    The problem (#3) can be found here:http://img198.imageshack.us/i/img002lf.jpg/" It would be helpful if someone could look over my solution (found here:http://img525.imageshack.us/i/img001of.jpg/" ) and also it would be helpful if anyone had a different approach to this problem. Thanks
  48. I

    Contour integral, taylor and residue theory question

    Homework Statement http://img243.imageshack.us/img243/4339/69855059.jpg I can't seem to get far. It makes use of the Exponentional Taylor Series: Homework Equations http://img31.imageshack.us/img31/6163/37267605.jpg The Attempt at a Solution taylor series expansions for cos...
  49. C

    Colored Burn Residue - High Voltage Electricity & Materials

    Hello. I am working on a project that employs the use of high voltage electricity to burn specific dielectric bodies: woods, plastics. My question is sort of unrelated to the goal of the project, but I was wondering if there was a way to burn a dielectric with high voltage so that the burn mark...
  50. Z

    Can Cauchy's residue theorem be used for functions with poles at infinity?

    my question is , let us have the following complex integral \oint f(z)dz where f(z) has a simple pole at z=\infty then by Residue theorem \oint f(z)dz =2\pi i Res(z,\infty,f(z) or equal to the limit (z-\infty )f(z) with 'z' tending to infinity
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