The word multiplicity is a general term meaning "the number of values for which a given condition holds." For example, the term is used to refer to the value of the totient valence function or the number of times a given polynomial equation has a root at a given point.

Let z_0 be a root of a function f, and let n be the least positive integer n such that f^((n))(z_0)!=0. Then the power series of f about z_0 begins with the nth term,


and f is said to have a root of multiplicity (or "order") n. If n=1, the root is called a simple root (Krantz 1999, p. 70).

See also

Degenerate, Module Multiplicity, Multiple Root, Noether's Fundamental Theorem, Root, Simple Root, Totient Valence Function

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Krantz, S. G. "Zero of Order n." §5.1.3 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 70, 1999.

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Cite this as:

Weisstein, Eric W. "Multiplicity." From MathWorld--A Wolfram Web Resource.

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