Difference between revisions of "Artin–Hasse exponential"
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| − | A modification of the [[exponential function]] in the [[p-adic number]] domain. In classical analysis we have | + | A modification of the [[exponential function]] in the [[p-adic number|$p$-adic number]] domain. In classical analysis we have |
$$ | $$ | ||
\exp(z) = \prod_{n=1}^\infty \left({ 1-z^n }\right)^{-\mu(n)/n} \ , | \exp(z) = \prod_{n=1}^\infty \left({ 1-z^n }\right)^{-\mu(n)/n} \ , | ||
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which is an identity in [[formal power series]] over the rational numbers. | which is an identity in [[formal power series]] over the rational numbers. | ||
| − | Over the field of p-adic numbers we define | + | 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} \ , | E_p(z) = \prod_{n=1; p\not\mid n}^\infty \left({ 1-z^n }\right)^{-\mu(n)/n} \ , | ||
Revision as of 17:46, 20 January 2022
2020 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
Artin–Hasse exponential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Artin%E2%80%93Hasse_exponential&oldid=51935