# 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. ### 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. ### MHB Laguerre polynomials Roots

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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### MHB Completeness of Laguerre polynomials

How to establish the completeness of Laguerre polynomials?
10. ### 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. ### Associated Laguerre polynomials .

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