1[/ 2 Copyright 2019, Nick Thompson 3 Distributed under the Boost Software License, Version 1.0. 4 (See accompanying file LICENSE_1_0.txt or copy at 5 http://www.boost.org/LICENSE_1_0.txt). 6] 7 8[section:jacobi Jacobi Polynomials] 9 10[h4 Synopsis] 11 12`` 13#include <boost/math/special_functions/jacobi.hpp> 14`` 15 16 namespace boost{ namespace math{ 17 18 template<typename Real> 19 Real jacobi(unsigned n, Real alpha, Real beta, Real x); 20 21 template<typename Real> 22 Real jacobi_derivative(unsigned n, Real alpha, Real beta, Real x, unsigned k); 23 24 template<typename Real> 25 Real jacobi_prime(unsigned n, Real alpha, Real beta, Real x); 26 27 template<typename Real> 28 Real jacobi_double_prime(unsigned n, Real alpha, Real beta, Real x); 29 30 }} // namespaces 31 32Jacobi polynomials are a family of orthogonal polynomials. 33 34A basic usage is as follows: 35 36 using boost::math::jacobi; 37 double x = 0.5; 38 double alpha = 0.3; 39 double beta = 7.2; 40 unsigned n = 3; 41 double y = jacobi(n, alpha, beta, x); 42 43All derivatives of the Jacobi polynomials are available. 44The /k/-th derivative of the /n/-th Gegenbauer polynomial is given by 45 46 using boost::math::jacobi_derivative; 47 double x = 0.5; 48 double alpha = 0.3; 49 double beta = 7.2; 50 unsigned n = 3; 51 double y = jacobi_derivative(n, alpha, beta, x, k); 52 53For consistency with the rest of the library, `jacobi_prime` is provided which simply returns `jacobi_derivative(n, lambda, x,1)`. 54 55[$../graphs/jacobi.svg] 56 57[h3 Implementation] 58 59The implementation uses the 3-term recurrence for the Jacobi polynomials, rising. 60 61 62[endsect] 63
