zero of a function (original) (raw)

Suppose X is a set and f a complex (http://planetmath.org/Complex)-valued function f:X→ℂ. Then a zero of f is an element x∈X such that f⁢(x)=0. It is also said that f vanishes at x.

Remark. When X is a “simple” space, such as ℝ or ℂ a zero is also called a root. However, in pure mathematics and especially if Z⁢(f) is infinite, it seems to be customary to talk of zeroes and the zero set instead of roots.

Examples

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  For any z∈ℂ, define z^:X→ℂ by z^⁢(x)=z. Then Z⁢(0^)=X and Z⁢(z^)=∅ if z≠0.
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  Suppose p is a polynomial (http://planetmath.org/Polynomial) p:ℂ→ℂ of degree n≥1. Then p has at most n zeroes. That is, |Z⁢(p)|≤n.
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  If f and g are functions f:X→ℂ and g:X→ℂ, then

  Z⁢(f⁢g)	=	Z⁢(f)∪Z⁢(g),
  Z⁢(f⁢g)	⊇	Z⁢(f),
  where f⁢g is the function x↦f⁢(x)⁢g⁢(x).

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  For any f:X→ℝ, then
  where fn is the defined fn⁢(x)=(f⁢(x))n.
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  If f and g are both real-valued functions, then
  | Z⁢(f)∩Z⁢(g)=Z⁢(f2+g2)=Z⁢(|f|+|g|). |
  | ------------------------------------- |
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