scipy.special.jacobi¶
- scipy.special.jacobi(n, alpha, beta, monic=False)[source]¶
Jacobi polynomial.
Defined to be the solution of
\[(1 - x^2)\frac{d^2}{dx^2}P_n^{(\alpha, \beta)} + (\beta - \alpha - (\alpha + \beta + 2)x) \frac{d}{dx}P_n^{(\alpha, \beta)} + n(n + \alpha + \beta + 1)P_n^{(\alpha, \beta)} = 0\]for \(\alpha, \beta > -1\); \(P_n^{(\alpha, \beta)}\) is a polynomial of degree \(n\).
- Parameters
- nint
Degree of the polynomial.
- alphafloat
Parameter, must be greater than -1.
- betafloat
Parameter, must be greater than -1.
- monicbool, optional
If True, scale the leading coefficient to be 1. Default is False.
- Returns
- Porthopoly1d
Jacobi polynomial.
Notes
For fixed \(\alpha, \beta\), the polynomials \(P_n^{(\alpha, \beta)}\) are orthogonal over \([-1, 1]\) with weight function \((1 - x)^\alpha(1 + x)^\beta\).