We call the point c a zero of the function f (z) if f (c) = 0. For example, the zeros of sin z are at z = nπ for n = 0,±1,±2, . . . . We define the order ormultiplicity of a zero as follows. Suppose f (z) has Taylor

expansion

f (z) = ∞∑ 0 an(z− c)n =

f (n)(c)

n! (z− c) n

at z = c. We say c is a zero of order n if a0 = a1 = · · · = an−1 = 0, but an = 0. Equivalently, if f (c) = f ′(c) = · · · = f (n−1)(c) = 0, but f (n)(c) = 0. A zero of order 1 is called a simple zero, a zero of order 2 is called a double zero, etc. For example, the zeros of f (z) = sin z are all simple since f ′(z) = cos z = ±1 at z = nπ . However, for example, g(z) = z sin z has a double zero at z = 0 since the Maclaurin expansion is

z sin z = z ( z− z

3! + · · · ) = z2 − z

3! + · · ·

Theorem 1 (Fundamental theorem of algebra) Every polynomial of degree n with complex coefficients has n zeros in the complex plane taking account of multiplicity.