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zero of a function (Definition)

Suppose $ X$ is a set and $ f$ a complex-valued function $ f\colon X\to \mathbb{C}$. Then a zero of $ f$ is an element $ x\in X$ such that $ f(x) = 0$. It is also said that $ f$ vanishes at $ x$.

The zero set of $ f$ is the set

$\displaystyle Z(f) := \{ x\in X \mid f(x)=0\}.$

Remark. When $ X$ is a “simple” space, such as $ \mathbb{R}$ or $ \mathbb{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\in \mathbb{C}$, define $ \hat{z}:X\to \mathbb{C}$ by $ \hat{z}(x)=z$. Then $ Z(\hat{0})=X$ and $ Z(\hat{z})=\varnothing$ if $ z\ne 0$.
  • Suppose $ p$ is a polynomial $ p\colon\mathbb{C}\to\mathbb{C}$ of degree $ n\ge 1$. Then $ p$ has at most $ n$ zeroes. That is, $ \vert Z(p)\vert\le n$.
  • If $ f$ and $ g$ are functions $ f\colon X\to\mathbb{C}$ and $ g\colon X\to\mathbb{C}$, then
    $\displaystyle Z(fg)$ $\displaystyle =$ $\displaystyle Z(f)\cup Z(g),$  
    $\displaystyle Z(fg)$ $\displaystyle \supseteq$ $\displaystyle Z(f),$  

    where $ fg$ is the function $ x\mapsto f(x) g(x)$.
  • For any $ f\colon X\to \mathbb{R}$, then
    $\displaystyle Z(f)=Z(\vert f\vert)=Z(f^n),$
    where $ f^n$ is the defined $ f^n(x)=(f(x))^n$.
  • If $ f$ and $ g$ are both real-valued functions, then
    $\displaystyle Z(f)\cap Z(g)=Z(f^2+g^2)=Z(\vert f\vert+\vert g\vert).$
  • If $ X$ is a topological space and $ f:X\to \mathbb{C}$ is a function, then the support of $ f$ is given by:
    $\displaystyle \operatorname{supp} f = \overline{Z(f)^\complement}$
    Further, if $ f$ is continuous, then $ Z(f)$ is closed in $ X$ (assuming that $ \mathbb{C}$ is given the usual topology of the complex plane where $ \{0\}$ is a closed set).



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"zero of a function" is owned by mathcam. [ full author list (7) | owner history (2) ]
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See Also: support of function

Other names:  zero, vanish, vanishes
Also defines:  zero set

Attachments:
least and greatest zero (Theorem) by pahio
vanish at infinity (Definition) by asteroid
order of vanishing (Definition) by pahio
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Cross-references: complex plane, usual topology, continuous, topological space, degree, roots, infinite, function
There are 113 references to this entry.

This is version 27 of zero of a function, born on 2003-10-15, modified 2007-04-11.
Object id is 4921, canonical name is ZeroOfAFunction.
Accessed 6128 times total.

Classification:
AMS MSC: 26E99 (Real functions :: Miscellaneous topics :: Miscellaneous)
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