Legendre differential equation

Equation (E.17) is the associated Legendre differential equation. Orthogonality... 

Following the strategy used to solve the Hermite and Legendre differential equations, we begin with a function... 

This is the Legendre differential equation, which has the following general solution for n N ... 

The Legendre polynomials define the angular dependency of spherical harmonics for rotational symmetry. Yet for the general case we have to refer to Eq. (C.9) with non-zero constant m, which leads to the associated Legendre differential equation,... 

In this appendix we recall the basic properties of the solutions to the Bessel and Legendre differential equations and discuss some computational aspects. Properties of spherical Bessel and Hankel functions and (associated) Legendre functions can be found in [1,40,215,238]. 

This equation is characterized by regular singularities at the points x = 1 and at infinity. For m = 0, there are two linearly independent solutions to the Legendre differential equation and these solutions can be expressed as power series about the origin x = 0. In general, these series do not converge for... 

Let us notice that due to orthogonality of Legendre s polynomials many functions can be represented as a series, which is similar to Equation (1.162), and this fact is widely used in mathematical physics. Now, we will derive the differential equation, one of the solutions of which are Legendre s functions. 

The third and last terms on the left-hand side may be eliminated by means of equation (E.5) to give Legendre s differential equation... 

The solutions of differential equations often define series of related functions that can be obtained from simple generating functions or formulae. As an example consider the Legendre polynomials... 

Formulae of this type are especially useful when looking for solutions of more complicated differential equations related to the simpler classical ones. Two important examples are the associated Legendre and associated Laguerre (18) equations, respectively... 

The next step is to differentiate Legendre s equation n times. Using the Leibniz [6] theorem one gets... 

If we substitute from (18.8 into Legendre s equation (171. ) wc find that II7n 1 satisfies the differential equation... 

Legendre s Associated Functions. We saw in example 1 of Chapter 1 that the solution of Laplace s equation in spherical polar coordinates reduces to the solution of the ordinary differential equation,... 

Dennis and Walker (D3) expanded and Z as a series of Legendre functions in the position coordinates. Equations (5-1) and (5-2) were reduced to a set of ordinary differential equations, solved numerically. This approach is inconvenient for high Re since the number of terms which must be included becomes prohibitive. Solutions to the steady equations were obtained for Re < 40 (D3) and for impulsively started motion for Re < 100 (D4). 

In quantum mechanics and other branches of mathematical physics, we repeatedly encounter what are called special functions. These are often solutions of second-order differential equations with variable coefficients. The most famous examples are Bessel functions, which we wiU not need in this book. Our first encounter with special functions are the Hermite polynomials, contained in solutions of the Schrodinger equation. In subsequent chapters we will introduce Legendre and Laguerre functions. Sometime in 2004, theU.S. National Institute of Standards and Tec hnology (NIST) will publish an online Digital Library of Mathematical Functions, http / /dlmf. nist. gov, including graphics and cross-references. 

A linear combination of the bound and continuum atomic wavefiinction was used to approximate the continua (10). Wavefunctions in the spherical potential were separated into the spherical harmonics and the radial wavefunctions. The spherical harmonics are expressed in terms of the associated Legendre functions. The differential equation for the radial wavefunction R at position r is... 

Only numerical solutions of the VERSE model can be obtained [65]. The partial differential equations are discretized by application of the method of orthogonal collocation on fixed finite elements. Equation 16.59 is divided into 50 or 60 elements, each with four interior collocation points. Legendre polynomials are used for each element. For Eq. 16.62, only one element is required. It is described by a Jacobi polynomial with two interior collocation points. The resulting set of ordinary differential equations, with their initial and boundary conditions and the chemical equations, are solved using a differential algebraic system solver (DASSL) [65,66]. 

The terms in P and P, 1 may be replaced by PPt, from 19-4, and there then results the differential equation for the Legendre polynomials... 

To obtain the normalization integral for the associated Legendre functions we proceed as follows.1 By differentiating Equation 19-7 and multiplying by (1 — z2) we obtain... 

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