Infinite family of approximations of the Digamma function

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