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. Z

    Complex residue integral

    Homework Statement i need to calculate the inverse Mellin transform \oint ds {x^{-s}}\frac{1}{\Gamma(s)cos(\pi s/2)} Homework Equations I can use Cauchy's integral theorem, The Attempt at a Solution i know that Gamma function has poles at -1,-2,-3, ... and that the cosine...
  2. B

    F(z) = [1-cosh(z)] / z^3. its pole, order & residue

    [b]1. Show that f(z) = [1 - cosh(z)] / z3 has a pole as its singular point. Determine its order m & find the residue B. [b]2. lim |f(z)| tends to \infty as z tends to singular point bm + bm+1(z-z0)+...+ b1(z-z0)m-1 + \sum^{\infty}_{n = 0}an(z-z0)n= (z-z0) m f(z) [b]3. f(z) = [1...
  3. L

    Finding the Residue of f'/f at 0 with a Zero of Multiplicity k

    Homework Statement Let f has a zero of multiplicity k at 0. Find the residue of f'/f at 0 The Attempt at a Solution I'm kind of get stuck on this one. I got only this far: Since f has a zero of multiplicity k at 0, then f(z) = (z^k)g(z) :( Thanks a lot for helping!
  4. C

    Very easy and quick residue calculus question.

    Homework Statement \oint_{\left|z\right|=3/2} \frac{e^{\frac{1}{z-1}}}{z} dz Homework Equations Using residue theorem, since there are two singularities withing the domain, evaluate residues at each singularity, and multiply by 2\pi i The Attempt at a Solution Here is the...
  5. ~christina~

    Evaporation of organic solvent and leftover residue

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  6. L

    Finding the residue of e^(-1/z^2) at z=0

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  7. L

    Solving the Pole and Residue at a for f

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  8. J

    When to use the residue at infinity

    why is it that we can use the residue at infinity on the methods of contour integration example on wikipedia for the last one? we can only use the residue at infinity when it is a rational function, i.e. the ratio of two polynomials, if I'm wrong when i say this, why?
  9. Q

    Simple residue theorem question.

    Homework Statement So I know how to evaluate the integral from 0 to 2pi of 1/2+cos theta. However, the question I am being asked to do has me calculate this integral from 0 to pi. I am not sure what adjustment is necessary to get the integral i am given (from 0 to pi) to the form I know how...
  10. A

    Find Residue at z=1 for Essential Singularity Integral - Homework Help

    Homework Statement Hey guys. So, I've got this integral: http://img18.imageshack.us/img18/5742/scan0017.jpg And I want to find the residue at z=1. I know it's an essential singularity point. I tried to calculate the residue, but I'm completely not sure about my solution. Can I...
  11. S

    Residue Theorem: Show r1+r2 = Res(f1+f2, z0)

    Homework Statement If f1 and f2 have residues r1 and r2 at z0. show that the residue of f1+ f2 is r1 + r2 The Attempt at a Solution Res(f1, z0) = limz-->z0 (z-z0)f1(z) = r1 Res(f2, z0) = limz-->z0 (z-z0)f2(z) = r2 now calculate Res(f1+f2, z0) = limz-->z0 (z-z0)(f1(z)+ f2(z))...
  12. C

    How Do You Calculate Probabilities with Non-Coprime Residue Classes?

    Suppose I had a lot of residue classes and I wanted to find the probability that a random integer (mod the product of the moduli) was in at least one of the classes. How could I calculate that? If the moduli were pairwise coprime, it would be easy: start an accumulator at 0 and for each class...
  13. A

    Solving an Integral Using Residue Theorem

    Homework Statement Hey guys. So I got this integral I need to solve, of curse using the residue theorem. The thing is, that I don't understand the curve. I know that whenever Z^2 = integer, this function has a singularity point because e^(2*pi*i*n) = 1. But again, I'm not sure what this...
  14. N

    Do I have to use residue theory here?

    Homework Statement \begin{subequations} \begin{eqnarray} s + \kappa + g^{2} \int_{-\infty}^{\infty} d \Delta'\; {\cal \rho}(\Delta')\, \frac{1}{s+\gamma+i\Delta'} &=& s + \kappa + g^{2} \int_{-\infty}^{\infty} d \Delta'\; \frac{1}{(s^{2}+\omega^{2})}\...
  15. C

    Ionisation chamber - doubt about current nature and residue

    Homework Statement This is from Advanced Physics by Adams and Allday, section 8, practice exam questions, question 25. An α-source with an activity of 150 kBq is placed in a metal can. A 100 V d.c. source and a 109 Ώ resistor are connected in series to the can and the source. This...
  16. A

    Compelx variable question, the Residue Thm and applications

    Hello, I am stuck with two complex variable questions. Among others, I can't find residues when I have the poles, and these two questions are getting me out of hope. Maybe one of the geniuses on this site knows how to drive those 2 problems to an answer: 1) Computing the following...
  17. S

    Definite Integral using Residue Thm

    Homework Statement Calculate the integral [ z^4/(1 + z^8) ] over negative infinity to positive infinity. Homework Equations Residue Theorem. Specifically for real-valued rational functions (on the real axis) where the denominator exceeds the degree of the numerator by at least two or...
  18. S

    Evaluate the integral using the residue theorem and its applications.

    Homework Statement the integral from negative infinity to positive infinity: z^4/(1 + z^8)dz Homework Equations The residue theorem: <http://en.wikipedia.org/wiki/Residue_theorem>. The Attempt at a Solution I found the 8th roots of z^8 = -1, which are e^(πiz), where z =1/8, 3/8...
  19. F

    What is the application of the Residue Theorem in finance?

    Hi everyone. I'm a brazilian mathmatician that didn't studied complex analysis. I study finance and now I'm needing to study that. In a paper of Lewis (2001) I found an expression that I couldn't understand. Does anyone can help me with that? They say they use the Residue theorem but I...
  20. M

    Solving Residue & Integral Problems: Step-by-Step Guide

    hi all, my first post; had a minor headache with this problem lol. PROBLEM 1: Finding Residue: ----------------- find Res(g,0) for g(z) = z^{-2}coshz My Attempt/Solution: ----------------- I know coshz = 1 + \frac{x^2}{2!} + \frac{z^4}{4!} ... so now z^{-2}coshz = z^{-2} (1 +...
  21. V

    Integrals over an infinite range (using residue theorem)

    I am taking a short course of complex variables and am trying to understand how to evaluate the integral: \int\frac{dx}{1+x^{2}} (1) where the integration is from -infinity to +infinity. To do this, we must, apparently consider: \oint\frac{dz}{1+z^2} (2). The closed loop is a countour...
  22. M

    Residue Theorem: Theory & Application

    the Residue theorem states that : \oint {f(z)dz} = 2\pi i\sum Res f(z) and the summation is taken for all the poles of f(z) enclosed by the counter at which the integration is performed . now i have read somewhere that \oint \frac{f(z)dz}{z^{n+1}} = 2\pi i\sum Res f(z) [tex]a^{n}
  23. P

    Finding Inverse Laplace Transforms with Residue Method

    Homework Statement Inverse laplace transforms F(s)=\frac{5s-2}{s^{2}(s-1)(s+2)} Homework Equations Residue technique The Attempt at a Solution F(s)=\frac{5s-2}{s^{2}(s-1)(s+2)} = \frac{k1}{s^{2}} + \frac{k2}{s-1} + \frac{k3}{s+2} I solved for K1,K2, and K3, which all came to...
  24. P

    Multi-dimensional residue theorem

    Hi, I'm wondering if a generalization of the residue theorem/formulae to several complex variables could be just as helpful as in the one-dimensional case. For example if you were to calculate \int_\mathhb{R}{\frac{dk}{2\pi}\frac{e^{-ikx}}{1+k^2}} One way would be to observe that the...
  25. P

    Residue of ArcTan: Is the Residue Theorem Applicable?

    Hi Folks, Does it make sense to speak of the residue of the arctan function at z=\pm i? Or the residue of the natural logarithm at z=0 ..? The problem probably is that these functions are not holomorphic in however a small disk around the singularity... So am I right in assuming that...
  26. C

    Several problems on series representations, residue theorem

    First question pertains to the Residue Theorem We are to use this theorem to evaluate the integral over the given path... There is one problem from this section that I am stuck on. An example in the book evaluates \int_{\Gamma} e^{1/z} dz for \Gamma any closed path not passing through...
  27. T

    Finding the Residue of an Expression with a Constant

    How do you get the residue of this expression: \frac{1}{(e^z - k)} where k is a constant.
  28. S

    Evaluating integrals using the residue theorem

    Hi, I have trouble evaluating simple integrals like \int_{-\infty}^{\infty} \frac{dx}{\sqrt{x^2 + 2}(x^2+1)} = \frac{\pi}{2} I'd like to calculate the integral closing the integration loop in the upper half-plane enclosing the pole at +i. The residue is - i / 2 and hence 2 \pi i ( - i \ 2 )...
  29. P

    Do Residues and Singularities Define Functions Like tanh(z) and tan(z)?

    Homework Statement Do residues exist only for holomorphic function? Classify the singularity and calculate the residue of tanh(z) and tan(z) The Attempt at a Solution For both Essential isolated singularity because the numerator has an infinite number of terms. Residue = 0.
  30. L

    Is this the answer for what is the residue of f(z)=e^z/[(z-1)^2(z+2)]

    Homework Statement This was an exam question I had today: Give the location and order of each pole of f(z)=e^z/[(z-1)^2(z+2)] and evaluate the reside at these points Homework Equations Res(f(z)) = g(p-1)(a) / (p-1)! The Attempt at a Solution g(z) = e^z/(z+2) g'(z) =...
  31. M

    Residue theorem for integral of real sinusodial function

    I've seen a few examples but don't understand how the contour is chosen. We use the substitution z=e^{i \theta} If the integral is over -pi to pi, or over 0 to 2*pi, then the contour is the unit circle centred on the origin. My questions: 1.) Why? 2.) What would the contour be...
  32. L

    How to use the residue theorem when I have an exponential as the pole?

    Homework Statement Q. Use residues and the contour shown (where R > 1) to establish the integration formula \int^{\infty}_{0} \frac{dx}{x^3+1} = \frac{2\pi}{3 \sqrt{3}} The given contour is a segment of an arc which goes from R (on the x-axis) to Rexp(i*2*pi/3) Homework Equations...
  33. T

    What exactly is a residue - what are its applications[complex analysis]

    In https://www.amazon.com/dp/0139078746/?tag=pfamazon01-20 - residues is introduced as an exercise at the end of a chapter and that's it! (or it may resurface in a later chapter), My question is that saff and snider looks at it as the numerator of the partial fraction exapansion of a...
  34. D

    What is the residue of exp(1/z) at z=0?

    Hi. I'm trying to find the residue of \exp{\frac{1}{z}} at z=0 since it is a pole, so I can integrate the function over the unit circle counterclockwise. I expanded this function in Laurent Series to get \exp{\frac{1}{z}} = 1 + \frac{1}{1!z} + \frac{1}{2!z^2}+ ... So in this case the...
  35. L

    ReEvaluating real integrals using residue calculus

    Hi, I recently came to know that some real integrals become easier to evaluate using the techniques of residue calculus from complex analysis. I thought that this would be a good tool to pickup apart from the standard techniques of by parts and substitution that we are taught. I have a few...
  36. A

    Calculating Residue of a Complex Function: A GRE Problem

    In the complex plane, let C be the circle |z| = 2 with positive (counterclockwise) orientation. Show that: \int _C \frac{dz}{(z-1)(z+3)^2} = \frac{\pi i}{8} This isn't homework, it was a problem in one of the practice GREs. It looks like a straightforward application of the residue...
  37. P

    Residue of z/cos(z): Explanation & Calculation

    The definition of a residue is the coefficient of the -1 power in the Laruent series. If I do z/cos(z) by long division, I get a series starting with z so z^-1 never occurs hence has a coefficient of 0. But why does it have a non zero residue, namely z/sin(z) at each z when cos(z)=0?
  38. quasar987

    What Are the Residues and Poles of a Function?

    Here I must evaluate \frac{1}{2\pi i}\int_C \frac{f(\zeta)}{w(\zeta)}\frac{w(\zeta)-w(z)}{\zeta -z}d\zeta where f is holomorphic on the hole complex plane, where w(z) is a polynomial of degree n with all of its zeroes distinct (i.e. all n have multiplicity 1), and where C is a closed curve...
  39. G

    Exploring Residue in Complex Analysis: Origins, Significance, and Applications

    Please anyone tell me about when, why and how did the concept of residue in complex analysis started?
  40. benorin

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    Ok, the quick and dirty: The given Theorem: If f is analytic everywhere in the finite complex plane except for a finite number of singular points interior to a positively oriented simple closed contour C, then \int_{C}f(z)dz=2\pi i \mbox{Res}\left[ \frac{1}{z^2}f\left( \frac{1}{z}\right)...
  41. T

    Improper integral using the Residue Theorem

    I'm trying to find \int\limits_0^{ + \infty } {\frac{{\sin x}} {{x^3 + x}}dx} Since the function is even, I can compute it as \frac{1} {2}\int\limits_{ - \infty }^{ + \infty } {\frac{{\sin x}} {{x^3 + x}}dx} To use the residue theorem, I construct a large semi-circle C with center O and...
  42. T

    Residue theory can this be right

    if f(z) = e^z/(z^2 - 1) does Res[f,-1] = -1/2e and Res[f,1] = e/2
  43. T

    Anyone familiar with residue theory

    from my understanding we use residue theory when we have poles. The question i have is if f(z) = 1/(1-Z^2) has two poles at 1, -1 each of order 1 then does Res[f(z),-1] = lim as z -> -1 of (z+1)(f(z)) = -1/2 if we have a pole of order 1 then Res[f(z),z0] = lim as z -> z0 of (z -...
  44. E

    Residue at a pole of non-integer order

    Hi, Does anyone know a straightforward way to calculate a residue at at a pole of non-integer order. I'm trying to find the residue of \frac {e^{ipx}}{(p - i \kappa)^\eta} at p = i \kappa where \eta is a positive non-integer. Thanks. I have reason to suspect it's zero, but I'd need...
  45. B

    Is There a Solution for x^4 ≡ 2 (mod p) When p ≡ 1 (mod 4)?

    show that x^4 == 2( mod p) has a solution for p==1(mod 4) iff p is of the form A^2+64B^2, where A,B are integers I let x^2=M then the conguence is reduced to M^2==2( mod p) but any # squared == 0 or 1 ( mod4 ) so p must be == 1(mod4)... but I'm not sure what to do now.. any hints/ or...
  46. J

    Can the Residue Theorem be Extended to Higher Dimensions?

    Due to the functional form of typical Lagrangian densities that arise in particle physics, field theorists run into integrals having integrands that are fractions with polynomial denominators when they calculate propagators and Green‘s functions. That is where talk of “poles” and “contour...
  47. A

    Residue Formula for Exponential Fraction with Constant Terms

    Is there a simple formula for the residue of \frac{e^{ik}}{\prod_j (k-is_j)} where s_j are constants ?
  48. 1

    Understanding Reduced Residue Systems Mod 7: Can You Help?

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