# 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:

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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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##

31. ### 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...
32. ### 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...
33. ### 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...
34. ### 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...
35. ### MHB Residue Calc: $\cot^n(z)$ at $z=0$

Show that the residue of $\cot^{n}(z)$ at $z=0$ is $\sin \left( \frac{n \pi}{2}\right)$, $n \in \mathbb{N}$.
36. ### 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...
37. ### 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.
38. ### 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...
39. ### 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...
40. ### 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...
41. ### 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...
42. ### 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...
43. ### 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 \$...
44. ### 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?
45. ### 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...
46. ### 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...
47. ### 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...
48. ### 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}...
49. ### 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...
50. ### 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...