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LegendreP

  • LegendreP[ n , x ] gives the Legendre polynomial .
  • LegendreP[ n , m , x ] gives the associated Legendre polynomial .
  • Mathematical function (see Section A.3.10).
  • Explicit formulas are given for integer 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.
  • See the Mathematica book: Section 3.2.9Section 3.2.10.
  • See also: SphericalHarmonicY.

    Further Examples

    Here are the first 4 Legendre polynomials.

    In[1]:=

    Out[1]//TableForm=

    Here are the first four associated Legendre polynomials.

    In[2]:=

    Out[2]//TableForm=

    This is the derivative.

    In[3]:=

    Out[3]=

    This is the integral.

    In[4]:=

    Out[4]=

    This is a series expansion around .

    In[5]:=

    Out[5]=