###### 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 PadicAnalytic 2013-03-22 15:13:53 2013-03-22 15:13:53 alozano (2414) alozano (2414) 4 alozano (2414) Definition msc 11S99 msc 12J12 msc 11S80 $p$-adic analytic Analytic PAdicExponentialAndPAdicLogarithm $p$-adic analysis p-adic analysis