Infinite family of approximations of the Digamma function

Abstract

The aim of this work is to find “good” approximations to the Digamma function Ψ . We construct an infinite family of “basic” functions { I a , a ∈ [ 0 , 1 ] } covering the Digamma function. These functions are shown to approximate Ψ locally and asymptotically, and it is shown that for any x ∈ R + , there exists an a such that Ψ ( x ) = I a ( 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.