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| 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 10:54:20 |
| Classification: |
msc:26E99 |
Revision comment (for changes between this and next version):
| page images mode is broken again now - trying to fix |
Preamble:
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\usepackage{amssymb}
\usepackage{amsmath}
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%\usepackage{psfrag}
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%\usepackage{graphicx}
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Content:
{\bf Definition}
Suppose $X$ is a set, and suppose $f$ is a complexvalued function $f:X\to \mathbb{C}$.
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
$$ Z(f) &=& \{ x\in X \mid f(x)=0\}.$$
\subsubsection{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.
\subsubsection{Examples}
\begin{itemize}
\item Suppose $p$ is a polynomial $p:\mathbb{C}\to \mathb{R}$ 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 $f:X\to \mathbb{C}$ and $g:X\to \mathbb{C}$, then
$$ Z(fg)=Z(f)\cup Z(g)$$
where $fg$ is the function $x\mapsto f(x) g(x)$.
\end{itemize} |
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