In mathematics, a polylogarithmic function in is a polynomial in the logarithm of ,
: <math>a_k (\log n)^k + a_{k-1} (\log n)^{k-1} + \cdots + a_1(\log n) + a_0. </math>
The notation is often used as a shorthand for , analogous to for .
In computer science, polylogarithmic functions occur as the order of time for some data structure operations. Additionally, the exponential function of a polylogarithmic function produces a function with quasi-polynomial growth, and algorithms with this as their time complexity are said to take quasi-polynomial time.
All polylogarithmic functions of are for every exponent (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 .