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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.

Mathematics Encyclopedia

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