Multiplicity

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  be a root of a function , and let  be the least positive integer  such that . Then the power seriesof  about  begins with the th term,

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

REFERENCES:

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