# Artin–Hasse exponential

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.

#### 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

**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