In Complex Analysis
Let z0 be a root of a holomorphic function ƒ, and let n be the least positive integer such that, the nth derivative of ƒ evaluated at z0 differs from zero. Then the power series of ƒ about z0 begins with the nth term, and ƒ 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).
We can also define the multiplicity of the zeroes and poles of a meromorphic function thus: If we have a meromorphic function ƒ = g/h, take the Taylor expansions of g and h about a point z0, and find the first non-zero term in each (denote the order of the terms m and n respectively). if m = n, then the point has non-zero value. If m > n, then the point is a zero of multiplicity m − n. If m < n, then the point has a pole of multiplicity n − m.
Read more about this topic: Multiplicity (mathematics)
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