# Polylogarithmic function

Template:Distinguish A polylogarithmic function in n is a polynomial in the logarithm of n,

${\displaystyle a_{k}\log ^{k}(n)+\cdots +a_{1}\log(n)+a_{0}.\,}$

In computer science, polylogarithmic functions occur as the order of memory used by some algorithms (e.g., "it has polylogarithmic order").

All polylogarithmic functions are

${\displaystyle P_{\ell }(x)=o(x^{\varepsilon })\,}$

for every exponent ε > 0 (for the meaning of this symbol, see small o notation), that is, a polylogarithmic function grows more slowly than any positive exponent. This observation is the basis for the soft O notation.