Legendre Polynomial Calculator
Calculate Legendre polynomials P_n(x) and associated Legendre functions.
Polynomial Parameters
Recurrence Relation
(n+1)P_(n+1) = (2n+1)xP_n - nP_(n-1)
P_3(x) Formula
-1.5x +2.5x^3
P_3(0.5)
-0.4375000000
Derivative P'_3(0.5)
0.37500000
Roots of P_3(x)
-0.7745970.0000000.774597
Used in Gauss-Legendre quadrature
Values by Order
P_0(0.5)1.000000
P_1(0.5)0.500000
P_2(0.5)-0.125000
P_3(0.5)-0.437500
P_4(0.5)-0.289063
P_5(0.5)0.089844
P_6(0.5)0.323242
Associated P^m_3(0.5)
P^0_3-0.437500
P^1_3-0.324760
P^2_35.625000
P^3_3-9.742786