p-adic analyticPAdicAnalytic2005-05-02 19:41:392005-05-02 19:41:39Definitioncomplex p-adic numbers$p$-adic analysisp-adic analysis% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Cl}{\operatorname{Cl}}\begin{defn}
Let $\Complex_p$ be the field of \PMlinkname{complex $p$-adic numbers}{ComplexPAdicNumbers}. Let $U$ be a domain in $\Complex_p$. A function $f: U \longrightarrow \Complex_p$ is {\em $p$-adic analytic} if $f$ has a Taylor series (with coefficients in $\Complex_p$) about each point $z \in U$ that converges to the function $f$ in an open neighborhood of $z$.
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For example, the \PMlinkname{$p$-adic exponential function}{PAdicExponentialAndPAdicLogarithm} is analytic on its domain of definition:
$$U=\{ z\in \Complex_p : |z|_p<\frac{1}{p^{1/(p-1)}}\}.$$
The study of $p$-adic analytic functions is usually called {\em $p$-adic analysis} and it is very similar to complex analysis in many respects, although there are important differences coming from the distinct topologies of $\Complex$ and $\Complex_p$.