Bessel Function Jn(x) (Not in Base Package)

Calculates the Bessel function of the first kind of order n for any real number x. Details

x is any real number.
n is the nonnegative order (integer) of the Bessel function.
Jn(x) is the Bessel function of the first kind of order n, where both x and n are given.

Bessel Function Jn(x) Details

The Bessel function is defined by

Generally the power series for is not computationally useful. More useful are recurrence relations and rational approximations of and

.

The recurrences are

The treatment of the cases x n and x > n is completely different for the sake of accuracy.