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ECE 6382 Fall 2016 David R. Jackson Notes 24 Legendre Functions
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Helmholtz Equation Recall the solution of the Helmholtz equation (wave equation) in spherical coordinates Separation of variables: where
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Solution for the H Function
To simplify this, let and denote
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Solution for the H Function (cont.)
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Solution for the H Function (cont.)
Canceling terms, we have Multiplying by y, we have This is the associated Legendre equation.
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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.
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Associated Legendre Functions (cont.)
Hence: To be as general as possible: n m w
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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.
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Properties of Legendre Functions
Rodriguez’s formula (for = n): Legendre polynomial (a polynomial of order n)
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Properties of Legendre Functions (cont.)
Note: This follows from these two relations:
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Properties of Legendre Functions (cont.)
(see next slide) The Q functions all tend to infinity as
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Properties of Legendre Functions (cont.)
Lowest-order Qn functions:
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Properties of Legendre Functions (cont.)
Negative index:
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Plots of Legendre Functions
P0(x)
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Series Forms of Legendre Functions
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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.)
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Properties of Legendre Functions (cont.)
Proof that a valid solution is Let Then Or (letting t x) Hence, a valid solution is
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Properties of Legendre Functions (cont.)
and are two linearly independent solutions. Valid independent solutions: or
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Properties of Legendre Functions (cont.)
In this case we must use
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Properties of Legendre Functions (cont.)
Summary of z-axis properties (x = cos ( ))
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Properties of Legendre Functions (cont.)
y x z
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Generating Function
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Recurrence Relations
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Recurrence Relations (cont.)
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Wronskians
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Recurrence Relations for Associated Legendre Functions
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Special Values of the Associated Legendre Functions
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Orthogonalities
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The Spherical Harmonics and Their Orthogonalities
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Spherical Harmonic Expansion
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