Computations of the Hill functions of higher order

Abstract:

In this note, we express the hill function ${\phi _n}(x)$ of an order n as a Fourier cosine series which is of simple form that allows proving the function’s basic properties. For the hill functions of higher order $(15 < n < 50)$ the form of the coefficients makes the series "essentially" self-truncating. For such high order hill functions, this truncated series (with thirty terms) computes the hill function with the same accuracy as the method of Legendre polynomials with local coordinates, but without the latter required ${n^2}$ coefficients which are to be computed in advance. The preliminary time analysis indicates that the time for the two methods starts to be the same at $n \sim 15$, changes slightly for the cosine series for $n > 15$ and varies roughly as ${n^3}$ for the localized Legendre polynomial method. In comparison with the most recent efficient methods which require a storage of order n, this note’s method required a storage of the order 25-40 for $n < 60$, executed with almost the same speed and accuracy and stayed stable as long as the above methods did.