What is Residue: Definition and 247 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:







{\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. O

    I Calculate the residue of 1/Cosh(pi.z) at z = i/2 (complex analysis)

    I'm not sure if this should be in the calculus section or the anlaysis section. It's complex analysis related to integration around a contour. Can someone suggest a method to determine the residue of f(z) = ## \frac{1}{Cosh ( \pi z) } ## at the singular point z = i/2. Background...
  2. C

    Calculating Residue at -2: Math Methods w/ Arfken et al.

    This question is given as an example in the book by Arfken, Weber, Harris, Mathematical Methods - a Comprehensive Guide, Seventh Edition. It is solved as below attached in the image. Can someone point it out how they proceed with calculations ? I do not seem to get their calculation. I am...
  3. T

    Residue Theorem applied to a keyhole contour

    I'm really struggling with this one. A newbie to using the residue theorem. I'm trying to solve this by factorising the denominator to find values for z0 and I have: ##z=\frac{-\sqrt{2}+i\sqrt{2}}{2}## and ##z=\frac{-\sqrt{2}-i\sqrt{2}}{2}## I also know that sin(3π/8)=...
  4. U

    Complex Integration Along Given Path

    From plotting the given path I know that the path is a curve that extends from z = 1 to z=5 on the complex plane. My plan was to parametrize the distance from z = 1 to 5 as z = x, and create a closed contour that encloses z=0, where I could use Cauchy's Integral Formula, with f(z) being 1 / (z +...
  5. docnet

    Compute the residue of a function

    There is a typo. It should say ##h=\frac{f}{g}##. Attempt: ##f## and ##g## are holomorphic on ##\Omega##. Homomorphic functions form a ##\mathcal{C}^*## algebra, so ##h## is holomorphic on ##\Omega## where ##g\neq 0##. If ##z_0## is a removal singularity of ##h##, then ##Res(h,z_0)=0## by...
  6. L

    Can Cauchy's Residue Theorem be Used for Functions with Branch Cuts?

    First of all I am not sure which type of singularity is ##z=0##? \ln\frac{\sqrt{z^2+1}}{z}=\ln (1+\frac{1}{z^2})^{\frac{1}{2}}=\frac{1}{2}\ln (1+\frac{1}{z^2})=\frac{1}{2}\sum^{\infty}_{n=0}(-1)^{n}\frac{(\frac{1}{z^2})^{n+1}}{n+1} It looks like that ##Res[f(z),z=0]=0##
  7. R

    How to find the residue of a complex function

    Hi, I'm trying to find the residue of $$f(z) = \frac{z^2}{(z^2 + a^2)^2}$$ Since I have 2 singularities which are double poles. I'm using this formula $$Res f(± ia) = \lim_{z\to\ \pm ia}(\frac{1}{(2-1)!} \frac{d}{dz}(\frac{(z \pm a)^2 z^2}{(z^2 + a^2)^2}) )$$ then, $$\lim_{z\to\ \pm ia}...
  8. J

    Dishwasher and a white powder residue

    I put a knife at bottom of the dish washer, and this was the result after 2 hour cleaning and drying time (just rarely though). What is the white powder? Is it the salt? The soap? scales from hard water? Or dried rinse aid? I recently acquired dish washer. It seemed to clean the dishes and...
  9. M

    What does HC3 mean in 'His HC3'?

    My lecturer said it means: "3rd residue, C-terminal to Helix 'H'" but that makes no sense. If it's the 3rd residue then it can't be on the C-terminal, because the C-terminal is at the end of a massively long sequence of residues. Visa versa if it's on the C-terminal then it can't be the 3rd...
  10. Z

    Comp Sci Understanding Residue 3 Automata: A Guide to Moore Machines | Zulfi

    Hi, I am trying to create residue 3 automata using Moore. I can't understand why we go to state q1 when we have input 0 in q0 and similalry why we go to q2 when we have input zero in q1. I have attached an image of FA. Some body please guide me. Zulfi.
  11. Robin04

    Calculating the residue of a complex function

    The singularities occur at ##z = \pm i\lambda##. As ##\frac{d}{dz}(z^2+\lambda^2)^2|_{z=\pm i\lambda}=0##, these singularities aren't first order and the residues cannot be calculated with differentiating the denominator and evaluating it at the singularities. What is the general method to...
  12. Baibhab Bose

    A Understanding the Order of Poles in Complex Functions

    When The denominator is checked, the poles seem to be at Sin(πz²)=0, Which means πz²=nπ ⇒z=√n for (n=0,±1,±2...) but in the solution of this problem, it says that, for n=0 it would be simple pole since in the Laurent expansion of (z∕Sin(πz²)) about z=0 contains the highest negative power to be...
  13. T

    Finding residues with Laurent series.

    Homework Statement Use an appropriate Laurent series to find the indicated residue for ##f(z)=\frac{4z-6}{z(2-z)}## ; ##\operatorname{Res}(f(z),0)## Homework Equations n/a The Attempt at a Solution Computations are done such that ##0 \lt \vert z\vert \lt 2##...
  14. B

    Calculating Integral Using Residue Theorem & Complex Variables

    Homework Statement I have never formally studied complex analysis, but I am reading this paper: http://adsabs.harvard.edu/abs/1996MNRAS.283..837S wherein section 2.2 they make use of the residue theorem. I am trying to follow along with this (and have looked up contour integration, cauchy's...
  15. J

    Residue at poles of complex function

    Homework Statement Homework Equations First find poles and then use residue theorem. The Attempt at a Solution Book answer is A. But there's no way I'm getting A. The 81 in numerator doesn't cancel off.
  16. J

    MHB The Residue Theorem To Evaluate Integrals

    I have no idea if this is in the right direction. I know I am going to need the summation of the residues to use the theorem. I found the residues using the limit, but do I need to change these using the euler formula? We are supposed to be working problems at home and I am getting a bit lost...
  17. Noisy Rhysling

    Disintegrator Residue: Effects on Rock & Danger to OH

    Our Hero has a disintegrator, invented by Dr. Someguy, and he fires it at rock blocking his path through a cave system. Would the disintegrated material be loose particles of dust or would it disappear? Disappearing would imply energy release of some quantity? Loose dust would be a possible...
  18. S

    I Understanding Isolated Singularities in the Residue Theorem

    Hello! Why do the singularities in the Residue Theorem must be isolated? If we have let's say a disk around ##z_0##, ##D_{[z_0,R]}## where all the points are singularities for a function ##f:G \to C## with the disk in region G, but f is holomorphic in ##G-D_{[z_0,R]}##, we can still write f as a...
  19. D

    A Dirac Delta and Residue Calculus

    I'm an undergraduate student, so I understand that it may be difficult to provide an answer that I can understand, but I have experience using both the Dirac delta function and residue calculus in a classroom setting, so I'm at least familiar with how they're applied. Whether you're integrating...
  20. C

    Residue Theorem: Finding the Integral of z^3e^(-1/z^2) over |z|=5

    Homework Statement use the residue theorem to find the value of the integral, integral of z^3e^{\frac{-1}{z^2}} over the contour |z|=5 The Attempt at a Solution When I first look at this I see we have a pole at z=0 , because we can't divide by zero in the exponential term. and a pole of...
  21. C

    What is the residue of f(z) = e^(-2/z^2) using a laurent series?

    Homework Statement Use a laurent series to find the indicated residue f(z)=e^{\frac{-2}{z^2}} Homework EquationsThe Attempt at a Solution So I expand the series as follows 1-\frac{2}{z^2}+\frac{2}{z^4} ... my book says the residue is 0 , is this because there is no residue term ? the...
  22. K

    Is Solid Residue Present in Pure Water?

    Hi everyone! If I have a sample that didn’t present conductivity and left a solid evaporation residue but its density is approximately 1 g/mL (1,08 g/mL), can I still say it is pure water? I suppose pure water shouldn’t have any kind of solid residue. Thanks!
  23. binbagsss

    Elliptic functions, residue computation, same zeros and poles

    Homework Statement Hi, I am trying to understand the attached: I know that if two functions have zeros and poles at the same point and of the same order then they differ only by a multiplicative constant, so that is fine, as both have a double zero at ##z=w_j/2## and a double pole at...
  24. mercenarycor

    Contour integration with a branch cut

    Homework Statement ∫-11 dx/(√(1-x2)(a+bx)) a>b>0 Homework Equations f(z0)=(1/2πi)∫f(z)dz/(z-z0) The Attempt at a Solution I have absolutely no idea what I'm doing. I'm taking Mathematical Methods, and this chapter is making absolutely no sense to me. I understand enough to tell I'm supposed...
  25. B

    A Residue Proof of Fourier's Theorem Dirichlet Conditions

    Whittaker (1st Edition, 1902) P.132, gives two proofs of Fourier's theorem, assuming Dirichlet's conditions. One proof is Dirichlet's proof, which involves directly summing the partial sums, is found in many books. The other proof is an absolutely stunning proof of Fourier's theorem in terms of...
  26. arpon

    Complex Integration using residue theorem

    Homework Statement [/B] ##C_\rho## is a semicircle of radius ##\rho## in the upper-half plane. What is $$\lim_{\rho\rightarrow 0} \int_{C_{\rho}} \frac{e^{iaz}-e^{ibz}}{z^2} \,dz$$Homework Equations If ##C## is a closed loop and ##z_1, z_2 ... z_n## are the singular points inside ##C##...
  27. dykuma

    Contour integral using residue theorem

    Homework Statement Find the solution of the following integral Homework Equations The Attempt at a Solution I applied the above relations getting that Then I was able to factor the function inside the integral getting that From here I should be able to get a solution by simply finding the...
  28. B

    Proving Least Positive Residue Using m-Cycle Permutations

    Homework Statement Prove that if ##\sigma## is the ##m##-cycle ##(a_1 ~ a_2 ... a_m)##, then for all ##i \in \{1,...,m\}##, ##\sigma^{i}(a_k) = a_{k+i}##, where ##k+i## is replaced by its least positive residue ##\mod m##. Homework EquationsThe Attempt at a Solution My question is...
  29. DCN

    Residue Theorem with real zero

    Homework Statement Find \int_{0}^{\infty} \frac{\cos(\pi x)}{1-4x^2} dx Homework Equations The residue theorem The Attempt at a Solution The residue of this function at $$x=\pm\frac{1}{2}$$ is zero. Therefore shouldn't the integral be zero, if you take a closed path as a hemisphere in the...
  30. P

    Calculating the Residue of ##\frac{1}{(x^4+1)^2}## at Double Poles

    Homework Statement How would I calculate the residue of the function ##\frac{1}{(x^4+1)^2}## Homework EquationsThe Attempt at a Solution So I have found that the poles are at ##z=e^{\frac{i \pi}{4}}## ##z=e^{\frac{3i \pi}{4}}## ##z=e^{\frac{5i \pi}{4}}## ##z=e^{\frac{7i \pi}{4}}## I tried...
  31. P

    Find the residues of the following function + Cauchy Residue

    Homework Statement Find the residues of the function f(z), and compute the following contour integrals. a) the anticlockwise circle, centred at z = 0, of radius three, |z| = 3 b) the anticlockwise circle, centred at z = 0, of radius 1/2, |z| = 1/2 f(z) = 1/((z2 + 4)(z + 1)) ∫Cdz f(z) Homework...
  32. saybrook1

    What are the orders of the poles and the residue for sin(1/z)/cos(z)?

    Homework Statement Hello guys, I need to find the orders of each pole as well as the residue of the function sin(1/z)/cos(z). Homework Equations I imagine that this is a simple pole so I will either find the Laurent series and get the coefficient of (z-z_0)^{-1} or use the simpler limiting...
  33. D

    Goodness of fit, Residual STD, chi square

    Homework Statement Hello, I am using CasaXPS to model synthetic peak models for X-ray photoelectron spectroscopy data. I am fitting. The software has a lot of manuals online but they do not explain how they yield a Residual Standard Deviation, after each fit iteration. Most software use...
  34. polygamma

    MHB Residue Calc: $\cot^n(z)$ at $z=0$

    Show that the residue of $\cot^{n}(z)$ at $z=0$ is $\sin \left( \frac{n \pi}{2}\right)$, $n \in \mathbb{N}$.
  35. I

    Finding Laurent Series and Residues for Complex Functions

    Homework Statement Find four terns of the Laurent series for the given function about ##z_0=0##. Also, give the residue of the function at the point. a) ##\frac{1}{e^z-1}## b) ##\frac{1}{1-\cos z}## Homework Equations The residue of the function at ##z_0## is coefficient before the...
  36. M

    Finding Residue of Complex Function at Infinity

    Hello everyone, I have a problem with finding a residue of a function: f(z)={\frac{z^3*exp(1/z)}{(1+z)}} in infinity. I tried to present it in Laurent series: \frac{z^3}{1+z} sum_{n=0}^\infty\frac{1}{n!z^n} I know that residue will be equal to coefficient a_{-1}, but i don't know how to find it.
  37. H

    Translating from Japanese: residue in capacitor?

    So, I have made an experiment of a low pass filter, and made a bode plot out of it. My experiment was done 100 percent in Japanese and this means I cannot understand completely the experiment I have done. So, there is a statement there saying the higher the frequency of the input voltage, the...
  38. Z

    Calculating integrals using residue & cauchy & changing plan

    Homework Statement \int_{0}^{2\pi} \dfrac{d\theta}{3+tan^2\theta} Homework Equations \oint_C f(z) = 2\pi i \cdot R R(z_{0}) = \lim_{z\to z_{0}}(z-z_{0})f(z) The Attempt at a Solution I did a similar example that had the form \int_{0}^{2\pi} \dfrac{d\theta}{5+4cos\theta} where I would change...
  39. D

    Contour integration & the residue theorem

    When one uses a contour integral to evaluate an integral on the real line, for example \int_{-\infty}^{\infty}\frac{dz}{(1+x)^{3}} is it correct to say that one analytically continues the integrand onto the complex plane and integrate it over a closed contour ##C## (over a semi-circle of radius...
  40. ognik

    MHB Please check Residue theorem excercise

    Show $ \int_{-\infty}^{\infty}\frac{1}{x^4 - 2 cos 2 \theta + 1} \,dx = \frac{\pi}{2sin \theta} $ I know I want to use the residue theorem for $ \int_{0}^{\pi}\frac{1}{z^4 - 2 cos 2 \theta z^2 + 1} \,dz $, and have found the 4 poles ($ \pm e^{\pm i \theta } $). I chose the upper semi-circle...
  41. ognik

    MHB Please check find residue problem

    Hi - I get a different answer from the book, but please also review for correct mathematical language & notation ... find residues for $ f(z) = \frac{sin(\frac{1}{z}) }{z^2 + a^2} $ There are 2 simple poles, $ \pm ia $ ; also I note that $ {z}_{o}^2 = -a^2 $ which proves useful for...
  42. ognik

    MHB Find coefficiant of Laurent series, without using residue

    Hi - I admit to struggling a little with my 1st exposure to complex analysis and Laurent series in particular, so thought I'd try some exercises; always seem to help my understanding. A function f(z) expanded in Laurent series exhibits a pole of order m at z=z0. Show that the coefficient of $...
  43. M

    Residue on steel after caustic cleaning

    After cleaning aluminum from steel I use with sodium hydroxide, there a brown chalky layer I want to clean off. Other than sanding, I cannot remove it. Is there a way to chemically remove it without affecting the steel?
  44. J

    MiracleGro urea residue removal?

    I use large quantities of Miracle Gro fertilizing large pots of annual flowers. Anyone know a way to dissolve the white residue that builds up on stone or tile after repeated applications (I assume this is the insoluble urea in the product.). I usually wash down these surfaces after the stuff...
  45. M

    Use Residue Theorems or Laurent Series to evaluate integral

    Homework Statement Evaluate the integral using any method: ∫C (z10) / (z - (1/2))(z10 + 2), where C : |z| = 1 Homework Equations ∫C f(z) dz = 2πi*(Σki=1 Resp_i f(z) The Attempt at a Solution Rewrote the function as (1/(z-(1/2)))*(1/(1+(2/z^10))). Not sure if Laurent series expansion is the...
  46. S

    Cleaning crystal meth residue from car

    I had a roommate that is a regular user of meth. I recently found out that smoking meth leaves a residue behind that can be absorbed through the skin, which worries me because I let him borrow my car and he smoked meth in it before. I'm kind of a hypochondriac when it comes to health issues like...
  47. K

    2nd order pole while computing residue in a complex integral

    Hello, I am trying to understand how to get the residue as given by wolfram : http://www.wolframalpha.com/input/?i=residue+of+e^{Sqrt[x^2+%2B+1]}%2F%28x^2+%2B+1%29^2 The issue I am facing is - since it is a second order pole, when I try to different e^{\sqrt{x^+1}} I get a \sqrt{x^+1}...
  48. Shackleford

    Find Res(0;f) for f (z)

    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...
  49. S

    Finding poles for cauchy's residue theorem.

    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...