Associated Legendre functions

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The associated Legendre functions P^m_n(x) are polynomials which are most conveniently defined in terms of derivatives of the Legendre polynomials:

 P^m_n(x)= (1-x^2)^{m/2} \frac{d^m}{dx^m} P_n(x)


The first associated Legendre polynomials are:


P_0^0 (x) =1


P_1^0 (x) =x


P_1^1 (x) =-(1-x^2)^{1/2}


P_2^0 (x) =\frac{1}{2}(3x^2-1)


P_2^1 (x) =-3x(1-x^2)^{1/2}


P_2^2 (x) =3(1-x^2)

etc.