|
|
|
Viewing Version
20
of
'zero of a function'
|
[ view 'zero of a function'
|
back to history
]
| Title of object: |
zero of a function |
| Canonical Name: |
ZeroOfAFunction |
| Type: |
Definition |
| Created on: |
2003-10-15 01:25:30 |
| Modified on: |
2004-01-02 15:42:13 |
| Classification: |
msc:26E99 |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\def\C{\mathbb{C}}
\def\R{\mathbb{R}} |
Content:
\PMlinkescapeword{root}
\PMlinkescapeword{simple}
{\bf Definition}
Suppose $X$ is a set, and suppose $f$ is a complex-valued function $f\colon X\to \C$.
Then a {\em zero} of $f$ is an element $x\in X$ such that $f(x)=0$.
The {\em zero set} of $f$ is the set
$$ Z(f) = \{ x\in X \mid f(x)=0\}.$$
{\bf 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.
{\bf Examples}
\begin{itemize}
\item Suppose $p$ is a \PMlinkname{polynomial}{Polynomial} $p\colon\C\to\C$ of degree $n\ge 1$. Then $p$ has at most $n$ zeroes. That is, $|Z(p)|\le n$.
\item If $f$ and $g$ are functions $f\colon X\to\C$ and $g\colon X\to\C$, then
$$ Z(fg)=Z(f)\cup Z(g)$$
where $fg$ is the function $x\mapsto f(x) g(x)$.
\item If $X$ is a topological space and $f:X\to \C$ is a function, then
$$\operatorname{supp} f = \overline{Z(f)^\complement}.$$
Further, if $f$ is continuous, then $Z(f)$ is a closed in $X$ (assuming that
$\C$ is given the usual topology of the complex plane where
$\{0\}$ is a closed set).
\end{itemize} |
|
|
|
|
|
