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. T

    Algorithm to find square root of a quadratic residue mod p

    I'm going through an explanation in a number theory book about Tonelli's algorithm to find the square roots of a quadratic residue modulo ##p## where ##p## is prime, i.e. I want to solve ##x^2 \equiv a \pmod{p}## with ##(\frac{a}{p}) = 1##. The book goes as follows: Let ##p - 1 = 2^s t##, where...
  2. J

    Residue Theorem for Laplace Transform

    I need to know what's the Residue Theorem for a Laplace Transform. Does anyone know the name or something, so I can search it? I couldn't find anything. For example, if I have this two equations: X(s).(s-1) = -Y(s)+5 Y(s).(s-4) = 2.X(s)+7 I know how to solve them using Simple Fractions, but...
  3. A

    Residue of f(z) involving digamma function

    Homework Statement Find the residue of: $$f(z) = \frac{(\psi(-z) + \gamma)}{(z+1)(z+2)^3} \space \text{at} \space z=n$$ Where $n$ is every positive integer because those $n$ are the poles of $f(z)$Homework EquationsThe Attempt at a Solution This is a simple pole, however: $$\lim_{z \to n}...
  4. A

    Find the residue of g(z) at z=-2 using Laurent Expansion

    Homework Statement Find the residue at z=-2 for $$g(z) = \frac{\psi(-z)}{(z+1)(z+2)^3}$$ Homework Equations $$\psi(-z)$$ represents the digamma function, $$\zeta(z)$$ represents the Riemann-Zeta-Function. The Attempt at a Solution I know that: $$\psi(z+1) = -\gamma - \sum_{k=1}^{\infty}...
  5. W

    Complex analysis: residue integration question

    I'm asked to evaluate the following integral: \int_{c} \frac{30z^2-23z+5}{(2z-1)^2(3z-1)}dz where c is the unit circle. This function has a simple pole at z=\frac{1}{3} and a second order pole at z=\frac{1}{2}, both of which are within my region of integration. I then went about computing the...
  6. kq6up

    Is the residue at ##2i## incorrect?

    Homework Statement Show that ##\int _{ 0 }^{ \infty }{ \frac{x^2dx}{(x^2+9)(x^2+4)^2} } =\frac{\pi}{200}##. Homework Equations ##Res=\frac{1}{n!}\frac{d^n}{dz^n}[f(z)(z-z_0)^{n+1}]## where the order of the pole is ##n+1##. The Attempt at a Solution Integreading of a semicircle contour one...
  7. kq6up

    What is the residue at ##z=0## for ##\frac{1}{z^3}+e^{2z}##?

    Homework Statement Find the residue of ##\frac{1-cos2z}{z^3}## at ##z=0## Homework Equations ##Res=\frac{1}{n!}\frac{d^n}{dz^n}[f(z)(z-z_0)^{n+1}]## Where the order of the pole is ##n+1## The Attempt at a Solution Differentiating ##(1-cos2z)z^3## twice, leaves me with zeros against every...
  8. kq6up

    Residue Theorem: Finding Residue at ##|z|=2##

    Homework Statement Find the residue of ##\oint { \frac { sinz }{ 2z-\pi } } dz## where ##\left| z \right| =2##[/B]Homework Equations ##f\left( z_{ o } \right) =\frac { 1 }{ 2\pi i } \oint { \frac { f\left( w \right) }{ w-z_{ o } } } dw## The Attempt at a Solution It seems to me that the...
  9. I

    Can the Residue Theorem be applied to these contour integrals?

    Hi, first post here. I'm having some trouble with contour integration. Basically here's the question: Contour Integral of ∫ 1+z dz (z-1)(z2+9) There are three cases: l z l = 2 l z+1 l = 1 l z-\iota l = 3 Is each case a straightforward application...
  10. A

    MHB What Are the Singularity and Residue of \( \frac{e^{z^2}}{(z-i)^3} \)?

    Hello. Can you check this for me, please? Find the singularity of $\frac{e^{z^2}}{(1-z)^3}$ and find the residue for each singularity. My solution: There is a triple pole at z=i, therefore...
  11. M

    Integration using residue theorem (part 2)

    Hello. I need some explanation here. I got the solution but I don't understand something. Question: Find the integral using Residue Theorem. $$\int_{-\infty}^{\infty}\frac{dx}{(x^2+4)^2}$$ Here is the first part of the solution that I don't understand: To evaluate...
  12. A

    MHB Integration using residue theorem (part 2)

    Hello. I need some explanation here. I got the solution but I don't understand something. Question: Find the integral using Residue Theorem. $$\int_{-\infty}^{\infty}\frac{dx}{(x^2+4)^2}$$ Here is the first part of the solution that I don't understand: To evaluate...
  13. M

    Integration using residue theorem

    Hi. I have to use the residue theorem to integrate f(z). Can someone help me out? I am stuck on the factorization part. Find the integral $$\int_{0}^{2\pi} \,\frac{d\theta}{25-24\cos\left({\theta}\right)}$$ My answer: $$\int_{0}^{2\pi}...
  14. A

    MHB Integration using residue theorem

    Hi. I have to use the residue theorem to integrate f(z). Can someone help me out? I am stuck on the factorization part. Find the integral $$\int_{0}^{2\pi} \,\frac{d\theta}{25-24\cos\left({\theta}\right)}$$ My answer: $$\int_{0}^{2\pi}...
  15. A

    MHB Yes, your solution is correct. Good job!

    Hello. Can someone check if I got the answer right? $f(z)=\frac{e^{-2z}}{(z+1)^2}$ My solution: $f(z)=\frac{e^{-2z}}{(z+1)^2}$ $$Resf(z)_{|z=-1|}=\lim_{{z}\to{-1}}\frac{d}{dz}((z+1)^2\frac{e^{-2z}}{(z+1)^2})$$ $$\lim_{{z}\to{-1}}-2e^{-2z}=-2e^{2}$$
  16. M

    Singularity and residue theorem

    Hello. Can someone check if I got the answer right? Find the singularity and the residue. ##f(z)=\frac{e^{-2z}}{(z+1)^2}## My solution: ##f(z)=\frac{e^{-2z}}{(z+1)^2}## $$Resf(z)_{|z=-1|}=\lim_{{z}\to{-1}}\frac{d}{dz}((z+1)^2\frac{e^{-2z}}{(z+1)^2})$$...
  17. L

    Probe for Hydrogen Peroxide residue analysis

    Hi everyone, I've been reading articles and looking at many websites for methods to measure hydrogen peroxide residue concentration in lake water, waste water and ocean water as well as in cell extract in situ . But I'm not satisfied. sometimes they seem too complicated and may require a lot of...
  18. S

    MHB Using Residue Calculus For a General Cosine Angle

    Hi, I am supposed to use residue calculus to do the following integral $$\int_{0}^{2\pi}\frac{1}{a+b\cos( \theta) } \mathrm{d}\theta$$ for |b|<|a| i have paremetrise it on $$\gamma(0;1)$$ that is $$z=\exp(i\theta), 0\leq\theta\leq2\pi$$ and obtain the following...
  19. A

    A Definite Integral Using the Residue Theorem

    Homework Statement I'm trying to solve this definite integral using the residue theorem: \int _0^\pi \frac{d \theta}{ (2+ \cos \theta)^2} Homework Equations I got the residue theorem which says that \oint_C f(z)dz = 2 \pi i \ \ \text{times the sum of the residues inside C}...
  20. A

    MHB Is $f(z) = \sin(z)/z$ an analytic function on the complex plane?

    Find the integral $\displaystyle \int_C \dfrac{\sin(z)}{z} dz $ where $c: |z| = 1 $ Can I use Cauchy integral formula since sin(z) is analytic $\displaystyle\int_C \dfrac{\sin(z)}{z} dz = Res(f,0) = 2\pi i \sin(0) = 0$ I tired to compute it without using the formula $z(t) = e^{it} ...
  21. D

    Finding Residue of z/(1+z^n) for Homework

    Hello, I can't find the result to Homework Statement Have to prove that ∫x/(1+x^n) dx = π/n/sin(2π/n) so I'm trying to prove that by starting to find : 2πi*res(z/(1+z^n), exp(iπ/n)) but don't know what is res(z/(1+z^n), exp(iπ/n)) Thanks
  22. Math Amateur

    MHB Ideals of a Residue Class Ring- Ring Isomorphism

    I am reading R. Y. Sharp: Steps in Commutative Algebra. In Chapter 2: Ideals on page 32 we find Exercise 2.40 which reads as follows: ----------------------------------------------------------------------------------------------- Let I, J be ideals of the commutative ring R such that I...
  23. Math Amateur

    MHB Residue Class Rings (Factor Rings) of Polynomials _ R Y Sharp

    I am reading R Y Sharp: Steps in Commutative Algebra. In Chapter 3 (Prime Ideals and Maximal Ideals) on page 44 we find Exercise 3.24 which reads as follows: ----------------------------------------------------------------------------- Show that the residue class ring S of the ring of...
  24. mathbalarka

    MHB Why did the mathematician's friends think he was a therapist?

    Why did the mathematician name his dog "Cauchy?" Because he left a residue at every pole.
  25. D

    What is the residue of cot(z) at z=0?

    Homework Statement So guys..the title says it! I need to find the residue of cot(z) at z=0. Homework Equations For this situation, since the pole order is 1 Residue=\lim_{z \to z_{0}}(z-z_{0})f(z) The Attempt at a Solution So here's what I am doing in steps: First, the...
  26. A

    Relation between residue integration and the Dirac Delta function

    Homework Statement OK so I'm doing a course on Signals and Systems and I'm taking inverse z transforms using residue integration. One particular formula in complex integration made me think a bit. \oint{\frac{f(z)}{z-z_0} dz} = 2\pi jf(z_0) This looks eerily similar to the definition...
  27. P

    Complex integration and residue theorem.

    Hi, Homework Statement I was wondering whether any of you could kindly explain to me how the equation in the attachment was derived. I mean, how could I have known that it could be separated into these two fractions? Homework Equations The attachment also specifies the integration to be...
  28. D

    MHB What is the residue of a complex integral with a double pole at infinity?

    \[ \int_0^{\infty}\frac{\cos(mx)}{(x^2 + a^2)^2}dx = \frac{\pi}{4a^3}e^{-am} (1 + am) \] The integral is even so \[ \frac{1}{2}\text{Re}\int_{-\infty}^{\infty}\frac{e^{imz}}{(z + ia)^2(z - ia)^2}dz. \] Since the singularity is of order two, I believe I need to use \[ \int\frac{f'}{f} =...
  29. D

    MHB Residue Calculus: Solving Integrals with Sinusoidal Functions

    \[ \int_0^{\infty}\frac{x\sin(mx)}{x^2 + a^2}dx = \frac{\pi}{2}e^{-am} \] The inetgral is even so \[ \frac{1}{2}\int_{-\infty}^{\infty}\frac{x\sin(mx)}{x^2 + a^2}dx. \] We can also write \(x^2 + a^2\) as \((x + ai)(x - ai)\). Should I also write \(\sin(mx) = \frac{1}{2i}(z^m - 1/z^m)\)? I...
  30. F

    Residue of Dirac delta function?

    Does the Dirac delta have a residue? It seems like it might, but I don't know how to attack it, since I really know very little about distributions. For example, the Dirac delta does not have a Laurent-expansion, so how would you define its residue?
  31. S

    Calculating integral using residue

    Homework Statement Calculate the integral ∫\frac{1}{(a+bcos^2(ϕ))^2}dϕ from 0 to 2π a,b >0 Homework Equations Residue theorem Cauchy's integral formula The Attempt at a Solution The first thing I did was attempt to find the poles of the integral and use residue theorem to solve the...
  32. N

    MHB Residue theorem to evaluate integrals

    Please refer to attached material. For the first question, I have tried looking at examples and have noted that the bounds have been provided in a manner: like |z|=1 (as given in part ii) I am not sure how to get transform the given |z-pi|=pi in such a format, although i suspect it would be...
  33. L

    Application of Cauchy's residue theorem

    Really need help for this one. Cheers! Homework Statement Question: calculate function z/(1-cos z) integrated in ac ounterclockwise circular contour given by |z-2pi|= 1 Homework Equations The Attempt at a Solution Clearly the pole in the given contour is 2pi. But the problem is: if it's a...
  34. alyafey22

    MHB Find Residues for f(z) at $z=-n$

    Find Residue at $z =0 $ of f(z) = \Gamma(z) \Gamma(z-1) x^{-z} Try to find Residues for $ z=-n $
  35. J

    Residue Theorem Applied to Calculating ψ(k,t)

    I have a doubt on this following procedure using the residue theorem: Initially we have ψ(k,t)=\frac{1}{2\pi}\int_{L_{\omega}}\frac{S(k,\omega)}{D(k,\omega)}e^{-i\omega t}d\omega Then the author said using the residue theorem, we have ψ(k,t)=-iƩ_{j}\frac{S(k,\omega_j(k))}{\partial D/ \partial...
  36. T

    Application of Residue Theorem to Definite Integrals (Logarithm)

    I've been studying for a test and have been powering through the recommended problems and have stumbled upon a problem I just can't seem to figure out. $$\int_{0}^{\infty} \frac{logx}{1+x^{2}} dx$$ (Complex Variables, 2nd edition by Stephen D. Fisher; Exercise 17, Section 2.6; pg. 167)...
  37. T

    MHB Finding the Least Residue of 3^215 (mod 65537)

    Compute the least residue of 3^215 (mod 65537) (65537 is prime). I've tried to use Euler's theorem, Fermat's little theorem and Wilson's theorem, but nothing seems to work, please help.
  38. A

    Residue of Function f(z)=e1/z/(1-z): Guide and Explanation

    Hello guys, I just want to confirm about this problem ..Find the residue of this function: f(z)=e1/z/(1-z) Thx in advance.
  39. V

    Calculating the residue of complicated expression

    Homework Statement Hi, I want to calculate the residue of this expression: Homework Equations I know that the residue of a function with a pole of k-th order is given by this: The Attempt at a Solution I know that the function has infinite number of poles at k*∏, for k=-∞ to...
  40. N

    Complex analysis proof with residue theorem, argument principle

    Homework Statement Let C be a regular curve enclosing the distinct points w1,..., wn and let p(w)= (w-w1)(w-w2)...(w-wn). Suppose that f(w) is analytic in a region that includes C. Show that P(z)= (1/2\pii)∫(f(w)\divp(w))\times((p(w)-p(z)\div(w-z))\timesdw is a polynomial of degree n-1...
  41. S

    Solve integral using residue theorem

    Homework Statement Considering the following integral, I = \int^\infty_{-\infty} \frac{x^2}{1+x^4} I can rewrite it as a complex contour integral as: \oint^{}_{C} \frac{z^2}{1+z^4} where the contour C is a semicircle on the half-upper plane with a radius which extends to infinity. I can...
  42. S

    Finding the Laurent series and residue of a function

    Homework Statement Find the Laurent series for the given function about the specified point. Also, give the residue of the function at the point. $$ \frac{z^2}{z^2 - 1}, z_0 = 1 $$ Homework Equations A Laurent expansion is comparable to a power series, except that it includes negative...
  43. O

    Finding a Laurent series / residue problem

    Homework Statement f(z) = \frac{1}{ \exp{ \frac {z^2 - \pi/2}{ \sqrt{3} } } + i } Find the residue of f(z) at z_0 = \frac{ \sqrt(\pi) }{2 } ( \sqrt(3) - i ) Homework Equations The Attempt at a Solution I was able to verify that the given z_0 is a singularity, and...
  44. D

    Solving Second Pole Residue ∫(dθ)/(a+bcosθ)^2

    ∫(dθ)/(a+bcosθ)^2 Homework Equations I'm trying to find the above integral (from 0-2pi) using Cauchy's Residue theorem. After closing the contour and re-writing the integrant, I know that I have singularity at (-a/b)+(√(a/b)^2-1)- (double pole or is it??). The Attempt at a Solution...
  45. S

    Using Contour Integration and the Residue Theorem

    Using contour integration and the residue theorem, evaluate the following "Fourier" integral: F_1(t) := \int_{-\infty}^\infty \frac{\Gamma sin(\omega t)}{\Gamma^2 + (\omega +\Omega )^2} dw with real-valued constants \Gamma > 0 and \Omega. Express your answers in terms of t, \Gamma and \Omega...
  46. P

    MHB Residue at essential singularity

    Consider the function $$f(z)=\frac{e^{\frac{1}{z-1}}}{e^z -1}$$ $z_0=1$ is an essential singularity, hence $$f(z)=\displaystyle\sum_{-\infty}^{+\infty}a_n(z-1)^n$$ near to $z_0=1$ and i want to find $a_{-1}$. I can write $$f(z)=\frac{\sum\frac{1}{n!(z-1)^n}}{e\cdot...
  47. P

    MHB Compute Residues for Periodic Function with Multiple Poles?

    How can i compute $Res(f,z_k)$ where $$f(z)=\frac{z-1}{1+cos\pi z}$$ and $z_k=2k+1, k\neq 0$?
  48. elfmotat

    Dirac Delta Function: Does it Have a Residue?

    Does the Dirac delta fuction have a residue? Given the close parallels between the sifting property and Cauchy's integral formula + residue theory, I feel like it should. Unfortunately, I have no idea how they tie together (if they do at all).
  49. M

    Confusion regarding Residue Theorem

    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...
  50. D

    Findinf residue of 1/5+4sin(deta)

    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...
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