p-adic analytic

Definition.

Let $\mathbb{C}_{p}$ be the field of complex $p$ -adic numbers (http://planetmath.org/ComplexPAdicNumbers). Let $U$ be a domain in $\mathbb{C}_{p}$ . A function $f:U\longrightarrow\mathbb{C}_{p}$ is $p$ -adic analytic if $f$ has a Taylor series (with coefficients in $\mathbb{C}_{p}$ ) about each point $z\in U$ that converges to the function $f$ in an open neighborhood of $z$ .

For example, the $p$ -adic exponential function (http://planetmath.org/PAdicExponentialAndPAdicLogarithm) is analytic on its domain of definition:

$U=\{z\in\mathbb{C}_{p}:|z|_{p}<\frac{1}{p^{1/(p-1)}}\}.$

The study of $p$ -adic analytic functions is usually called $p$ -adic analysis and it is very similar to complex analysis in many respects, although there are important differences coming from the distinct topologies of $\mathbb{C}$ and $\mathbb{C}_{p}$ .

Title	p-adic analytic
Canonical name	PadicAnalytic
Date of creation	2013-03-22 15:13:53
Last modified on	2013-03-22 15:13:53
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	4
Author	alozano (2414)
Entry type	Definition
Classification	msc 11S99
Classification	msc 12J12
Classification	msc 11S80
Synonym	$p$ -adic analytic
Related topic	Analytic
Related topic	PAdicExponentialAndPAdicLogarithm
Defines	$p$ -adic analysis
Defines	p-adic analysis