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LegendreP[n, x]
gives the Legendre polynomial P_n(x).
LegendreP[n, m, x]
gives the associated Legendre polynomial .
  • 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 (1-x^2)(d^2​y/d​x^2)-2x(d​y/d​x)+n​(n+1)​y=0.
  • 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 (1-z^2)^(m/2), of type 2 involves (1+z)^(m/2)/(1-z)^(m/2) and of type 3 involves (1+z)^(m/2)/(-1+z)m/2.
  • Type 1 is defined only for z 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 -1 and from +1 to +∞ in the complex z plane.
  • Type 3 functions have a single branch cut from -∞ to +1.
  • For certain special arguments, LegendreP automatically evaluates to exact values.
  • LegendreP can be evaluated to arbitrary numerical precision.
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