Zero (complex analysis)

In complex analysis, a zero (sometimes called a root) of a holomorphic function f is a complex number a such that f(a) = 0.

Multiplicity of a zero

A complex number a is a simple zero of f, or a zero of multiplicity 1 of f, if f can be written as

$f(z)=(z-a)g(z)\,$

where g is a holomorphic function g such that g(a) is not zero.

Generally, the multiplicity of the zero of f at a is the positive integer n for which there is a holomorphic function g such that

$f(z)=(z-a)^ng(z)\ \mbox{and}\ g(a)\neq 0.\,$

The multiplicity of a zero a is also known as the order of vanishing of the function at a.

Existence of zeros

The fundamental theorem of algebra says that every nonconstant polynomial with complex coefficients has at least one zero in the complex plane. This is in contrast to the situation with real zeros: some polynomial functions with real coefficients have no real zeros. An example is f(x) = x² + 1.

Properties

An important property of the set of zeros of a holomorphic function of one variable (that is not identically zero) is that the zeros are isolated. In other words, for any zero of a holomorphic function there is a small disc around the zero which contains no other zeros. There are also some theorems in complex analysis which show the connections between the zeros of a holomorphic (or meromorphic) function and other properties of the function. In particular Jensen's formula and Weierstrass factorization theorem are results for complex functions which have no counterpart for functions of a real variable.

See also

• Root of a function
• Pole (complex analysis)
• Hurwitz's theorem (complex analysis)
• Rouché's theorem
• Filter design
• Nyquist stability criterion in control theory
• Marden's theorem
• Sendov's conjecture
• Gauss–Lucas theorem

References

• Conway, John (1986). Functions of One Complex Variable I. Springer. ISBN 0-387-90328-3.
• Conway, John (1995). Functions of One Complex Variable II. Springer. ISBN 0-387-94460-5.

External links

• Weisstein, Eric W. "Root". MathWorld.

• Module for Zeros and Poles by John H. Mathews
• Free C++ solver for zeros of an analytic function (with C and Fortran bindings) by Ivan B. Ivanov