Legendres functions

This is one of the main reasons why these functions play a very important role in solving boundary value problems. Also, between Legendre s polynomials of different order, there is a simple recursive relationship  

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. 

Let us demonstrate that with help of Legendre s functions we can find a solution of Laplace s equation. As is well known, Laplace s equation has the following form in the spherical system of coordinates  

On the left hand side of Equation (1.165) it is natural to distinguish two terms  

At first glance it seems that they depend on the arguments R and 6, respectively, and Equation (1.165) can be represented as 

---

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... 

---