Polylogarithmic function

Not to be confused with Polylogarithm.

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

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

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

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

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.

References

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