|
|
|
Viewing Version
9
of
'properties of the exponential'
|
[ view 'properties of the exponential'
|
back to history
]
| Title of object: |
properties of the exponential |
| Canonical Name: |
PropertiesOfTheExponential |
| Type: |
Theorem |
| Created on: |
2002-02-27 09:59:07 |
| Modified on: |
2003-09-11 08:32:26 |
| Classification: |
msc:26A03 |
Revision comment (for changes between this and next version):
| Changes for correction #2971 ('order'). |
Preamble:
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\newcommand{\reals}{\mathbb{R}}
\newcommand{\natnums}{\mathbb{N}}
\newcommand{\cnums}{\mathbb{C}}
\newcommand{\znums}{\mathbb{Z}}
\newcommand{\lp}{\left(}
\newcommand{\rp}{\right)}
\newcommand{\lb}{\left[}
\newcommand{\rb}{\right]}
\newcommand{\supth}{^{\text{th}}}
\newtheorem{proposition}{Proposition} |
Content:
The exponential operation possesses the following properties.
\begin{itemize}
\item \PMlinkescapetext{{\bf Homogeneity.}} For $x,y\in\reals^+, p\in \reals$ we have
$$(xy)^p = x^p y^p$$
\item \PMlinkescapetext{{\bf Exponent additivity.}} For $x\in\reals^+$ we have
$$x^0=1,\qquad x^1 = x.$$
Furthermore
$$x^{p+q} = x^p x^q,\qquad p,q\in\reals.$$
\item \PMlinkname{\bf Order.}{TotalOrder} For $x,y\in\reals^+$ with $x<y$
and $p\in \reals^+$ we have
$$x^p < y^p,\qquad x^{-p} > y^{-p}.$$
\item {\bf Continuity.}
The exponential operation is continuous with respect to its
arguments. To be more precise, the following function is continuous:
$$P:\reals^+\times\reals\rightarrow \reals,\qquad P(x,y)=x^y.$$
\end{itemize}
Let us also note that the exponential operation is characterized (in
the sense of existence and uniqueness) by the {\em additivity} and
{\em continuity} properties. [{\bf Author's note}: One can probably get away with
substantially less, but I haven't given this enough thought.] |
|
|
|
|
|
