Polylogarithmic function

In mathematics, a polylogarithmic function in $n$ is a polynomial in the logarithm of $n$,


 * $$a_k (\log n)^k + a_{k-1} (\log n)^{k-1} + \cdots + a_1(\log n) + a_0. $$

The notation $logkn$ is often used as a shorthand for $(log n)k$, analogous to $sin2θ$ for $(sin θ)2$.

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 $n$ are $o(nε)$ 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 $Õ(n)$.