What is Laguerre: Definition and 28 Discussions

Edmond Nicolas Laguerre (9 April 1834, Bar-le-Duc – 14 August 1886, Bar-le-Duc) was a French mathematician and a member of the Académie des sciences (1885). His main works were in the areas of geometry and complex analysis. He also investigated orthogonal polynomials (see Laguerre polynomials). Laguerre's method is a root-finding algorithm tailored to polynomials. He laid the foundations of a geometry of oriented spheres (Laguerre geometry and Laguerre plane), including the Laguerre transformation or transformation by reciprocal directions.

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  1. appmathstudent

    I Rodrigues' Formula for Laguerre equation

    This is exercise 12.1.2 a from Arfken's Mathematical Methods for Physicists 7th edition :Starting from the Laguerre ODE, $$xy''+(1-x)y'+\lambda y =0$$ obtain the Rodrigues formula for its polynomial solutions $$L_n (x)$$ According to Arfken (equation 12.9 ,chapter 12) the Rodrigues formula...
  2. V

    MHB Where to Find Code for Computing Roots of Generalized Laguerre Polynomials?

    Hi - does anyone know of a program library/subroutine/some other source, to find the zeros of a generalised Laguerre polynomial? ie. LαN(xi)=0
  3. S

    MHB An expression resembling Laguerre

    This was posted to calculus forum. I suppose it should have been posted here. I am trying to find a closed form expression/or limit as $n\implies\infty$ of ${S}_{n}=\sum_{k=0}^{n}{n \choose k} {a}^{k} \sum_{j=0}^{n}{n \choose j}\frac{{b}^{n-j}{c}^{j}}{(k+j)!}$ where $a$ , $b$ and $c$ are...
  4. ognik

    Find the zeros of a generalised Laguerre polynomial

    Hi - does anyone know of a program library/subroutine - failing that some other source, to find the zeros of a generalised Laguerre polynomial? ie. ## L^{\alpha}_N (x_i) = 0 ##
  5. ognik

    MHB Zeros of generalised Laguerre polynomial

    Hi - does anyone know of a program library/subroutine/some other source, to find the zeros of a generalised Laguerre polynomial? ie. $ L^{\alpha}_N (x_i) = 0 $
  6. J

    Rodrigues’ formula of Laguerre

    Homework Statement I need to proof that Rodrigues’ formula satisfies Laguerre differential equation Homework Equations Rodrigues’ formula of Laguerre Laguerre differential equation The Attempt at a Solution first,I have to calculate = I tried to sum both terms and this is what I got...
  7. evinda

    MHB Solving the Laguerre Equation: 0 as Regular Singular Point

    Hello! (Wave) The differential equation Laguerre $xy''+(1-x)y'+ay=0, a \in \mathbb{R}$ is given. Show that the equation has $0$ as its singular regular point . Find a solution of the differential equation of the form $x^m \sum_{n=0}^{\infty} a_n x^n (x>0) (m \in \mathbb{R})$ Show that if...
  8. M

    Orthogonality of Associated Laguerre Polynomial

    I have a problem when trying to proof orthogonality of associated Laguerre polynomial. I substitute Rodrigue's form of associated Laguerre polynomial : to mutual orthogonality equation : and set, first for and second for . But after some step, I get trouble with this stuff : I've...
  9. S

    MHB Completeness of Laguerre polynomials

    How to establish the completeness of Laguerre polynomials?
  10. T

    Associated vs. Non-associated Laguerre Polynomials

    Homework Statement Could someone pls clarify if the value of x changes from just Laguerre polynomial to associated one? I am confused about the role of variable x. Homework Equations From what I have learned in the class, I understand that L1n(x) = d/dx Ln(x), n = 1, 2, 3... The Attempt at a...
  11. T

    What Are the Polynomials for 1s and 2s Orbitals?

    Homework Statement Show that L11(x) and L12(x) are precisely the polynomials for 1s and 2s orbitals. What is the role of variable x in each case? Homework Equations L1n(x) = d/dx Ln(x), n = 1, 2, 3... The Attempt at a Solution Because L1(x) = 1 - x L2(x) = 2 - 4x + x2: I did: L11(x) = d/dx...
  12. J

    Hydrogen Radial Equation: Recursion Relation & Laguerre Polynomials

    I'm in the first of 3 courses in quantum mechanics, and we just started chapter 4 of Griffiths. He goes into great detail in most of the solution of the radial equation, except for one part: translating the recursion relation into a form that matches the definition of the Laguerre polynomials...
  13. S

    MHB Relation between Hermite and associated Laguerre

    Please help me in in proving the relation between H2n(x) and Ln(-1/2)(x2) where Hn(x) is the Hermite polynomial and Ln(-1/2)(x) is associated Laguerre polynomial.
  14. K

    Closed form expression of the roots of Laguerre polynomials

    The Laguerre polynomials, L_n^{(\alpha)} = \frac{x^{-\alpha}e^x}{n!}\frac{d^n}{dx^n}\left(e^{-x}x^{n+\alpha} \right) have n real, strictly positive roots in the interval \left( 0, n+\alpha+(n-1)\sqrt{n+\alpha} \right] I am interested in a closed form expression of these roots...
  15. S

    MHB Relation between Hermite and associated Laguerre

    Please help me in proving the following expression H_{2n}(x)=(-1)^n2^{2n}n!L_n^{-\frac{1}{2}}(x^2) where H_n is the Hermite polynomial and L_n^{-\frac{1}{2}} is the associated Laguerre polynomial.
  16. A

    Orthogonal properties of associated laguerre polynomial

    i need the derivation of orthogonal properties of associated laguerre polynomial (with intermediate steps). someone please tell me where can i get it (for easy understanding).
  17. R

    Solving Laguerre DEby translating it into an Euler equation

    Homework Statement Find the indicial equation and all power series solutions around 0 of the form xr Ʃan xn for: x y'' -(4+x)y'+2y=0 - apparently one of these solutions is a laguerre pilynomial Homework Equations the indicial equation is the roots of r(r-1) +p0r+q0 where p0=lim(x->0)(...
  18. D

    Fortran Implementing Generalized Laguerre Polynomials in Fortran

    Hi! Im trying to do some rather easy QM-calculations in Fortran. To do that i need a routine that calculates the generalized Laguerre polynomials. I just did the simplest implementation of the equation: L^l_n(x)=\sum_{k=0}^n\frac{(n+l)!(-x^2)^k}{(n-k)!k!} I implemented this in the...
  19. A

    Solve Laguerre Equation for -a/z Potential | Quantum Mechanics

    Homework Statement I am doing the quantum mechanics and meet the Schrodinger question : When the potential is given as U = -a/z ,the Schroedinger equation looks like [- hbar^2 /(2m)] d^2 / dz^2 Psi(z) - a/z Psi(z) = E Psi(z). And the thing here is that I couldn't solve this equation...
  20. K

    Associated Laguerre Polynomial

    Hello, (quick backgroun info) : I am a physics student who has gone through pre quantum type material and a little of quantum mechanics. I am working in a lab with fortan code based on Quantum field theory. Anyway I am working to change some pieces of this code to attempt to solve a...
  21. P

    Writing a polynomial in terms of other polynomials (Hermite, Legendre, Laguerre)

    Homework Statement The first 3 parts of this 4 part problem were to derive the first 5 Hermite polynomials (thanks vela), The first 5 Legendre polynomials, and the first 5 Laguerre polynomials. Here is the last part: Write the polynomial 2x^4-x^3+3x^2+5x+2 in terms of each of the sets of...
  22. V

    Power Series Solutions of Laguerre Differential Equation

    I was going through http://mathworld.wolfram.com/LaguerreDifferentialEquation.html" in Wolfram which gives brief details about finding a power series solution of the Laguerre Differential Equation. I was reading the special case when v = 0. I read earlier from Differential Equations by Lomen...
  23. R

    The associated Laguerre equation.

    Homework Statement Hello, I need to show that the radial part of the hydrogen wave function has the form \rho^{l+1} e^{-\rho} L_{n-l-1}^{2l+1} (2\rho) More specifically, I'm having trouble showing the L_{n-l-1}^{2l+1} (2\rho) part because what I get is L_{n+1}^{2l+1} (2\rho) . The...
  24. B

    Normalisation of associated Laguerre polynomials

    I'm looking right now at what purports to be the normalisation condition for the associated Laguerre polynomials: \int_0^\infty e^{-x}x^k L_n^k(x)L_m^k(x)dx=\frac{(n+k)!}{n!}\delta_{mn} However, in the context of Schroedinger's equation in spherical coordinates, I find that my...
  25. A

    Uses of Laguerre Differential Equ.

    Does anybody know what the use of the Laguerre Differential Equation would be? I am having a hard time finding what areas of physics this diff. eq. is used in. Thanks.
  26. L

    Problem with the Laguerre polynomials

    My task is to explicitly write down the first three Laguerre polynomials by using a power series ansatz. What should this ansatz look like? Should it be the Rodrigues representation L_n (x) = \frac{e^x}{n!} \frac{d^n}{dx^n} x^n e^{-x} ?
  27. L

    Solving Differential Equation with Laguerre Polynomials

    What differential equation does \phi_n (x) := e^{-x/2} L_n (x) solve? L_n is a Laguerre polynomial. Please give me a hint on this one. I haven't got a clue where to start.
  28. S

    Integrating Laguerre Polynomials - Fine structure hydrogen

    Hi I have the following problem: To calculate the fine structure energy corrections for the hydrogen atom, one has to calculate the expectation value for (R,R/r^m), where R is the solution of the radial part of the schroedinger equation (i.e. essentially associated laguerre polynomial) and...
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