# Difference between revisions of "Artin–Hasse exponential"

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2010 Mathematics Subject Classification: Primary: 11S80 [MSN][ZBL]

A modification of the exponential function in the p-adic number domain. In classical analysis we have $$\exp(z) = \prod_{n=1}^\infty \left({ 1-z^n }\right)^{-\mu(n)/n} \ ,$$ which is an identity in formal power series over the rational numbers.

Over the field of p-adic numbers we define $$E_p(z) = \prod_{n=1; p\not\mid n}^\infty \left({ 1-z^n }\right)^{-\mu(n)/n} \ ,$$ removing the factors for which $n$ is divisible by $p$. This has radius of convergence $1$ and defines an analytic function with the property that $$E_p(z) = \exp\left({ z + \frac{z^p}{p} + \frac{z^{p^2}}{p^2} + \cdots }\right)$$ and is given by a power series with rational $p$-integral coefficients.

How to Cite This Entry:
Artin–Hasse exponential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Artin%E2%80%93Hasse_exponential&oldid=50966