View source for Artin–Hasse exponential

{{MSC|11S80}}

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.  

====References====
* Cassels, J.W.S. ''Local fields'' London Mathematical Society Student Texts '''3''' Cambridge University Press (1986) ISBN 0-521-31525-5 {{ZBL|0595.12006}}
* Robert, Alain M. ''A course in ''p''-adic analysis'' Graduate Texts in Mathematics '''198''' Springer (2000) {{ZBL|0947.11035}}
* Schikhof, W.H. ''Ultrametric calculus. An introduction to p-adic analysis'' Cambridge Studies in Advanced Mathematics '''4''' Cambridge University Press (1984) {{ZBL|0553.26006}}

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