Functions Legendre

With the substitution x = cos 9, the associated Legendre equation transforms to 

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 

but if n is a positive integer, one of the series breaks off after a finite number of terms and has a finite value at the poles. The polynomial solutions are called Legendre polynomials and are denoted by Pn x). For m 0, the solutions to (A.13) which are finite at the poles x = 1 are the associated Legendre functions. If m and n are integers, the associated Legendre functions are defined as 

In our analysis we use the associated Legendre functions with positive values of the index m. For m 0, the normalized associated Legendre functions are given by 

Similarly, the normalized angular functions tt and r are related to the angular functions tt and by the relations 

In the preceding sections we introduced several integrals and orthogonality relations of associated Legendre functions, which we shall prove in the following  

To find the integrals P P ,P , P P P VsinS necessary in the orthogonality relations (7.40), (7.64), we consider the general expansion 

Multiplication of Eq. (7.78) with P (cos9) and integration over the solid angle df2 = dcosdd ji yields 

We obtain two integrals of the type given in Eq. (7.81) with v = 1 and find 

In order to treat the integral (7.74) containing the Green s function yoiJ k-sl), we first turn from spherical to rectangular coordinates. Assuming the x-axis parallel to the direction S = 0 as shown in Fig. 29 we have 

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Let ) ( ) represents the (21 + l)th derivative of the (n + l)th Laguerre polynomial (20) and P7 (cos ) is Ferrers associated Legendre function of the first kind, of degree l and order m. Yim Zm thus constitutes a tesseral harmonic (21). The p s are in this form orthogonal and normalized, so that they fulfill the conditions... 

The functions Ztmn , = cos 0) are a generalization of associated Legendre functions and the coefficients of the series V(mn are given by... 

Substitution into the Rodriques formula gives the associated Legendre functions in the form... 

To normalize a function such as P m(cos0) it is necessary to equate the integral / [P,m( cos 9)]2dx = 1. Starting from the Rodriques formula and integrating by parts it can be shown that the normalized associated Legendre functions are ... 

The variation of Pn(ft) with ft for a few values of n is sliown in Fig. 5. Since, in most physical problems, the Legendre polynomial involved is usually / tl(cos 0) we have shown in Fig. fi the variation of this function with 0. Numerical values may be obtained from Tables of Associated Legendre Functions (Columbia University Press, 194-5),... 

Neumann s Formula for the Legendre Functions. Let us now consider the integral... 

Recurrence Relations for the Function Recurrence relations for the Legendre function of the second kind can be derived from Neumann s formula (18.1) anti the corresponding recurrence relations for the Legendre polynomials Pn(fi). From the recurrence relation (It.2) and Neumann s formula we have... 

The Use of Legendre Functions in Potential Theory, In potential theory we have frequently to determine solutions of Laplace s equation [7ayj = 0 which satisfy certain prescribed boundary conditions. If we have a problem in which the natural boundaries are spheres with centre at the origin of coordinates it is natural to employ polar coordinates r, 0, polar axis, y> will not depend on

= y>(r, 0). It then follows from example 1 of Chapter I that... 

The Legendre functions of the second kind, Qn(cos 0), which are absent from problems involving spherical boundaries, enter into the expressions for potcntiul functions appropriate to the space between two coaxial cones. If 0 < a < Q < fi < n we must take a solution of the form (20.1). Suppose, for example, that rp = 0 on 0 = a, ami yj = iToc r" on Q = ( then we must have... 

To illustrate the use of the solution (20.1) and of some of the properties of Legendre functions we shall now consider the problem in which nn insulated conducting sphere of radius a is placed with its centre ut the origin of coordinates in nn electric field whose potential is known to be... 

Use of Associated Legendre Functions in Wave Mechanics. To illustrate the use of associated Legendre functions in wave mechanics, we shall consider one of the simplest problems in that subject — that of solving Sehrodinger s equation... 

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