legendre_Q( n , z )

The Legendre function of the second kind of z in SageMath. A solution of the differential equation

\[ ( 1-z^2 ) \frac{ d^2 f }{ dz^2 } -2z \frac{ d f }{ dz } + n(n+1) f = 0 \]

The second linearly independent solution of this equation is legendre_P.

Equivalent to gen_legendre_Q( n , 0 , z ).

Explicit form:

Plot on the real axis:

Series expansion about the origin:

Special values:

Related functions:   legendre_P   gen_legendre_Q

Function category: orthogonal polynomials sagemath-docs