ECE 6382 Fall 2016 David R. Jackson Notes 24 Legendre Functions

Helmholtz Equation Recall the solution of the Helmholtz equation (wave equation) in spherical coordinates Separation of variables: where

Solution for the H Function To simplify this, let and denote

Solution for the H Function (cont.)

Solution for the H Function (cont.) Canceling terms, we have Multiplying by y, we have This is the associated Legendre equation.

Associated Legendre Functions The solutions to the associated Legendre equation are represented as Associated Legendre function of the first kind. Associated Legendre function of the second kind. n = “order”, m = “degree” If m = 0, Eq. (8) is called the Legendre equation, in which case Legendre function of the first kind. Legendre function of the second kind.

Associated Legendre Functions (cont.) Hence: To be as general as possible: n   m  w

Associated Legendre Functions (cont.) Relation to Legendre functions (when w = m = integer): These also hold for n . For m  w (not an integer) the associated Legendre function is defined in terms of the hypergeometric function.

Properties of Legendre Functions Rodriguez’s formula (for  = n): Legendre polynomial (a polynomial of order n)

Properties of Legendre Functions (cont.) Note: This follows from these two relations:

Properties of Legendre Functions (cont.) (see next slide) The Q functions all tend to infinity as

Properties of Legendre Functions (cont.) Lowest-order Qn functions:

Properties of Legendre Functions (cont.) Negative index:

Plots of Legendre Functions P0(x)

Series Forms of Legendre Functions

Legendre Functions with Non-Integer Order infinite series infinite series N = largest integer less than or equal to . Both are valid solutions, which are linearly independent for   n (See next side for a proof.)

Properties of Legendre Functions (cont.) Proof that a valid solution is Let Then Or (letting t  x) Hence, a valid solution is

Properties of Legendre Functions (cont.) and are two linearly independent solutions. Valid independent solutions: or

Properties of Legendre Functions (cont.) In this case we must use

Properties of Legendre Functions (cont.) Summary of z-axis properties (x = cos ( ))

Properties of Legendre Functions (cont.) y x z

Generating Function

Recurrence Relations

Recurrence Relations (cont.)

Wronskians

Recurrence Relations for Associated Legendre Functions

Special Values of the Associated Legendre Functions

Orthogonalities

The Spherical Harmonics and Their Orthogonalities

Spherical Harmonic Expansion