PlanetMath: p-adic analytic

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p-adic analytic

(Definition)

Definition 1 Let $\mathbb{C}_p$ be the field of complex -adic numbers. Let be a domain in $\mathbb{C}_p$ . A function $f: U \longrightarrow \mathbb{C}_p$ is -adic analytic if has a Taylor series (with coefficients in $\mathbb{C}_p$ ) about each point $z \in U$ that converges to the function in an open neighborhood of .

For example, the -adic exponential function is analytic on its domain of definition:

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

The study of -adic analytic functions is usually called -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 $\mathbb{C}_p$ .

"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, neighborhood, open, converges, point, coefficients, Taylor series, analytic, function, domain, field
There is 1 reference to this entry.

This is version 1 of p-adic analytic, born on 2005-05-02.
Object id is 7001, canonical name is PAdicAnalytic.
Accessed 2409 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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