Laguerre pseudospectral approximation to the thomasfermi. Laguerre, and are denoted in this case by in contrast to them, the are sometimes known as generalized laguerre polynomials. Generalized laguerre polynomial file exchange matlab. In this paper we demonstrated the experimental procedure to create higherorder laguerregauss modes with 95% purity for highpower, ultra stable laser beams. Laguerregaussian lg p modes u pr rl p2r 2w2 e r2w2 e i. Plug the above result into the radial equation to recover the associated laguerre equation for l. We propose an iterative method to solve the nonlinear thomasfermi equation based on laguerre pseudospectral approximation. Series solutions of laguerre and jacobitype differential equations. The qlaguerre matrix polynomials pubmed central pmc. In this paper, we propose a fractional generalization of the wellknown laguerre differential equation. Note that if, then the solution to the associated laguerre differential equation is of the form. Laguerre polynomials are obtained in a straightforward and simple way, through well known linear algebra methods based on sturmliouville theory. A linear homogeneous partial differential equation with.
Laguerre polynomials in spectral methods for nonlinear differential equations. Gauss laguerre integration since the common forms of funr decrease exponentially, it is reasonable to expand. Khader 2,3, mohamed adel1 1department of mathematics, faculty of science, cairo university, giza, egypt. Different beam shapes can be created as well, and recently laguerregauss beams have become of interest for their potential application in highprecision interferometry. Pdf in this paper, we derive a family of ordinary differential equations from the generating function of the laguerre polynomials. The solutions usually proceed by series expansion frobenius method, and. Note that in the special case lambda0, the associated laguerre differential equation is of the form. The generation of higherorder laguerregauss optical. Follow these steps to show that solutions to kummers equation 7. The laguerre polynomials have been extended to laguerre matrix polynomials by means of studying certain secondorder matrix differential equation.
The legendre, laguerre, and hermite equations have many real world practical uses which we will not discuss here. The laguerre polynomials are illustrated above for and, 2. Numerical solution for the fractional wave equation using. If no alpha is supplied, alpha is set to zero and this function calculates the normal laguerre polynomial. Laguerre differential equation from wolfram mathworld.
Some identities of laguerre polynomials arising from differential. How can i show that this jacobi polynomial can be expressed as the sum of these two legendre polynomials. Laguerre polynomials are used in quantum mechanics, for example, where the wave function for the hydrogen atom is given by the laguerre differential equation. Generalized laguerre functions and differential recursion. The laguerre polynomials are orthogonal with weight function. Laguerregen calculates the generalized laguerre polynomial ln, alpha this function computes the generalized laguerre polynomial ln,alpha. Orthogonality of laguerre polynomials from generating function. The set of all laguerre polynomials is dense in the space of functions whose square is integrable with weight on the interval laguerre polynomials are most frequently used under the condition. Each method is discussed with simple test codes in the accompanying file readme. Laguerre differential equation and its solutions, i. Generalized laguerre function and laguerre polynomials. Equation is a special case of the more general associated laguerre differential equation, defined by 2 where and are real numbers iyanaga and kawada 1980, p. Pdf some identities of laguerre polynomials arising from.
In this paper, we derive a family of ordinary differential equations from the generating function of the laguerre polynomials. If is a positive integer, then the series terminates and the solution is a polynomial, known as an associated laguerre polynomial or, if, simply a laguerre polynomial. The expansion coefficients, which are equal to tripleproduct integrals of generalized laguerre polynomials, are expressed in terms of a. The legendre, laguerre, and hermite polynomials3 proof. Laguerre polynomials, are found in many important physical problems.
Hermite and laguerre polynomials charles hermite 18221901 4. Pdf the solution of laguerres equation by using gtransform. Numerical solution for the fractional wave equation using pseudospectral method based on the generalized laguerre polynomials nasser h. Shen argues that laguerre numerical methods have unfairly acquired a poor reputation, owing to their misuse in the past. The boundary conditions quantize the energy of the hydrogen atom. Series solutions about an ordinary point if z z0 is an ordinary point of eq. Solutions to the laguerre differential equation with are called laguerre polynomials. The use of generalized laguerre polynomials in spectral. Its source code can be found in the file in the lib subdirectory of the idl distribution. What differential equation do the laguerre polynomials solve. We study a homogeneous partial differential equation and get its entire solutions represented in convergent series of laguerre polynomials. Solutions are written as infinite series of square.
Generalized laguerre polynomials collocation method. Laguerres equation mathematics the secondorder linear differential equation. A primer francesco pampaloni,1 jorg enderlein2 1embl, cell biology and biophysics programme, meyerhofstr. These codes are based on the seminal paper of shen 2000. We introduce two ordinary secondorder linear differential equations of the laguerre and. The mathematical expression for the electric field amplitude is a solution to the paraxial helmholtz equation. The expansion of products of generalized laguerre polynomials l. Just as for the hermite equation, solutions exist for noninteger. For certain special arguments, laguerrel automatically evaluates to exact values. The laguerre differential equation is a special case of the more general. This is an exercise in modern quantum mechanics by sakurai and napolitano. The gaussian beam is a transverse electromagnetic tem mode. A matrix corresponding to the laguerre differential operator is found and its eigenvalues are obtained. Pdf we consider the solution of laguerres equation by using gtransform, a generalized laplacetyped transform, and additionally, we find.
Laguerre s equation is equivalent to the statement that l m x is the eigenfunction with eigenvalue m of the second order di erential operator l x d2. Laguerrel can be evaluated to arbitrary numerical precision. Laguerre functions and differential recursion relations p. Gaussian, hermitegaussian, and laguerregaussian beams. The orthogonality with respect to the laguerre measure may be proved as follows. Different from the direct use of pseudospectral methods for solving the problem in the literature, we represent the solution of the thomasfermi equation as sum of two parts due to its singularity at the origin. There is a very explicit sense, to be detailed in a future lecture, in which this is a best fit to the function in this interval. David department of chemistry university of connecticut storrs, connecticut 062693060 dated. This equation has nonsingular solutions only if n is a nonnegative integer. Laguerre spectralpseudospectral library file exchange. In this paper, certain secondorder matrix qdifference equation is investigated and solved. Most special functions arise as solutions to \di cult odes, meaning ones not solvable by exponentials or polynomials. Its solution gives a generalized of the qlaguerre polynomials in matrix variable. Laguerres equation is equivalent to the statement that l m x is the eigenfunction with eigenvalue m of the second order di.
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