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

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Presentation on theme: "ECE 6382 Fall 2016 David R. Jackson Notes 24 Legendre Functions."— Presentation transcript:

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

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

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

4 Solution for the H Function (cont.)

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

6 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.

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

8 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.

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

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

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

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

13 Properties of Legendre Functions (cont.)
Negative index:

14 Plots of Legendre Functions
P0(x)

15 Series Forms of Legendre Functions

16 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.)

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

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

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

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

21 Properties of Legendre Functions (cont.)
y x z

22 Generating Function

23 Recurrence Relations

24 Recurrence Relations (cont.)

25 Wronskians

26 Recurrence Relations for Associated Legendre Functions

27 Special Values of the Associated Legendre Functions

28 Orthogonalities

29 The Spherical Harmonics and Their Orthogonalities

30 Spherical Harmonic Expansion


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