p-adic analytic

Definition.

Let ${\mathrm{C}}_p$ be the field of complex $p$-adic numbers (http://planetmath.org/ComplexPAdicNumbers). Let $U$ be a domain in ${\mathrm{C}}_p$. A function $f\mathrm{:}U\mathrm{⟶}{\mathrm{C}}_p$ is $p$-adic analytic if $f$ has a Taylor series (with coefficients in ${\mathrm{C}}_p$) about each point $z\mathrm{\in }U$ 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:

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 $ℂ$ and ${ℂ}_p$.