Legendre Polynomials, Pn(x)

synonyms: zonal harmonics

Differential Equation: wpe3C.gif (1462 bytes)

DE in Sturm-Liouville Form: wpe3D.gif (1451 bytes)

Examples: wpeC.gif (2097 bytes)

Note the parity of Legendre polynomials: Pn(-x)=(-1)nPn(x); polynomials with even n are even and those with odd n are odd.

Rodrigues Formula: wpe3A.gif (1335 bytes)

Generating Function: wpe3B.gif (1483 bytes)

Recursion Formula: (n+1)Pn+1(x) = (2n+1)xPn(x) - nPn-1(x)

Note the recursion formula is useful to determine successive Legendre polynomials once P0=1 and P1=x are known. Also, this formula confirms that even parity applies to even n and that odd parity applies to odd n.

A further recurrence relation: wpe57.gif (1436 bytes).

Orthogonality: wpeA.gif (1352 bytes)

Normalization integral and normalized polynomials: wpeD.gif (1787 bytes)


Associated Legendre Functions.
Plm(x) (l=0,1,2,...; m=0,1,2,...l)

Rodrigues Formula: wpe59.gif (1571 bytes)

Generating Function:

Examples:

Recursion Formula:

Differential Equation: wpe58.gif (1972 bytes)

 


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Created or up-dated 07/20/99   by R.D. Poshusta
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