Polylogarithmic function

Not to be confused with Polylogarithm.

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

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

The notation $\log ^{k}n$ is often used as a shorthand for $(\log n)^{k}$, analogous to $\sin ^{2}\theta$ for $(\sin \theta )^{2}$.

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

All polylogarithmic functions of $n$ are $o(n^{\varepsilon })$ 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 Õ(n).

References

• Black, Paul E. (2004-12-17). "polylogarithmic". Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. Retrieved 2010-01-10.