# Boost C++ Libraries

...one of the most highly regarded and expertly designed C++ library projects in the world.

This is the documentation for a snapshot of the master branch, built from commit 2d55b6d6f1.

### Jacobi Polynomials

##### Synopsis
```#include <boost/math/special_functions/jacobi.hpp>
```
```namespace boost{ namespace math{

template<typename Real>
Real jacobi(unsigned n, Real alpha, Real beta, Real x);

template<typename Real>
Real jacobi_derivative(unsigned n, Real alpha, Real beta, Real x, unsigned k);

template<typename Real>
Real jacobi_prime(unsigned n, Real alpha, Real beta, Real x);

template<typename Real>
Real jacobi_double_prime(unsigned n, Real alpha, Real beta, Real x);

}} // namespaces
```

Jacobi polynomials are a family of orthogonal polynomials.

A basic usage is as follows:

```using boost::math::jacobi;
double x = 0.5;
double alpha = 0.3;
double beta = 7.2;
unsigned n = 3;
double y = jacobi(n, alpha, beta, x);
```

All derivatives of the Jacobi polynomials are available. The k-th derivative of the n-th Gegenbauer polynomial is given by

```using boost::math::jacobi_derivative;
double x = 0.5;
double alpha = 0.3;
double beta = 7.2;
unsigned n = 3;
double y = jacobi_derivative(n, alpha, beta, x, k);
```

For consistency with the rest of the library, `jacobi_prime` is provided which simply returns ```jacobi_derivative(n, lambda, x,1)```.

#### Implementation

The implementation uses the 3-term recurrence for the Jacobi polynomials, rising.