PlanetMath: p-adic analytic

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p-adic analytic

(Definition)

Definition 1 Let $\Complex_p$ be the field of complex $p$ adic numbers. Let $U$ be a domain in $\Complex_p$ A function $f: U \longrightarrow \Complex_p$ is $p$ 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 $z$

For example, the $p$ adic exponential function 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 $p$ 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 $\Complex_p$

"p-adic analytic" is owned by alozano.

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Other names:

-adic analytic

Also defines:

-adic analysis, p-adic analysis

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Cross-references: topologies, differences, complex analysis, similar, analysis, analytic functions, domain of definition, neighborhood, open, converges, point, coefficients, Taylor series, analytic, function, domain, field
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This is version 1 of p-adic analytic, born on 2005-05-02.
Object id is 7001, canonical name is PAdicAnalytic.
Accessed 3562 times total.

Classification:

AMS MSC:	11S99 (Number theory :: Algebraic number theory: local and $p$-adic fields :: Miscellaneous)
	12J12 (Field theory and polynomials :: Topological fields :: Formally $p$-adic fields)
	11S80 (Number theory :: Algebraic number theory: local and $p$-adic fields :: Other analytic theory )

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