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zero of a function∗
mathcam†
2013-03-21 16:45:42
closed set
Suppose X is a set and f a 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.
The zero set of f is the set
Z(f ) := {x ∈ X | f (x) = 0}.
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
• For any z ∈ C, define ẑ : X → C by ẑ(x) = z. Then Z(0̂) = X and
Z(ẑ) = ∅ if z 6= 0.
• Suppose p is a polynomial p : C → C of degree n ≥ 1. Then p has at
most n zeroes. That is, |Z(p)| ≤ n.
• 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 7→ f (x)g(x).
• For any f : X → R, then
Z(f ) = Z(|f |) = Z(f n ),
where f n is the defined f n (x) = (f (x))n .
∗ hZeroOfAFunctioni created: h2013-03-21i by: hmathcami version: h34921i Privacy
setting: h1i hDefinitioni h26E99i
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• If f and g are both real-valued functions, then
Z(f ) ∩ Z(g) = Z(f 2 + g 2 ) = Z(|f | + |g|).
• If X is a topological space and f : X → C is a function, then the support
of f is given by:
supp f = Z(f ){
Further, if f is continuous, then Z(f ) is closed in X (assuming that C is
given the usual topology of the complex plane where {0} is a closed set).
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