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BUILT-IN MATHEMATICA SYMBOL
LegendreP
LegendreP[n, x]
gives the Legendre polynomial 
.
LegendreP[n, m, x]
gives the associated Legendre polynomial 
.
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- Explicit formulas are given for integers n and m.
- The Legendre polynomials satisfy the differential equation
. - The Legendre polynomials are orthogonal with unit weight function.
- The associated Legendre polynomials are defined by
. - For arbitrary complex values of n, m, and z, LegendreP[n, z] and LegendreP[n, m, z] give Legendre functions of the first kind.
- LegendreP[n, m, a, z] gives Legendre functions of type a. The default is type 1.
- The symbolic form of type 1 involves
, of type 2 involves 
, and of type 3 involves 
. - Type 1 is defined only for
within the unit circle in the complex plane. Type 2 represents an analytic continuation of type 1 outside the unit circle. - Type 2 functions have branch cuts from
to 
and from 
to 
in the complex 
plane. - Type 3 functions have a single branch cut from
to 
. - LegendreP[n, m, a, z] is defined to be Hypergeometric2F1Regularized[-n, n+1, 1-m, (1-z)/2] multiplied by
for type 2 and by 
for type 3. - For certain special arguments, LegendreP automatically evaluates to exact values.
- LegendreP can be evaluated to arbitrary numerical precision.
- LegendreP automatically threads over lists.
Legendre polynomial:
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