Infinite family of approximations of the Digamma function

The aim of this work is to find good approximations to the Digamma function Psi. We construct an infinite family of basic functions {Ia, a is an element of [0, 1]} covering the Digamma function. These functions are shown to approximate Psi locally and asymptotically, and it is shown that for any x is an element of R+, there exists an a such that Psi (x) = Ia (x). Local and global bounding error functions are found and, as a consequence, new inequalities for the Digamma function are introduced. The approximations are compared to another, well-known, approximation of the Digamma function and we show that an infinite number of members of the family are better. (c) 2005 Elsevier Ltd. All rights reserved.