LegendreP

• Mathematical function, suitable for both symbolic and numerical manipulation.

• Explicit formulas are given for integers and .

• 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.