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