P-adic exponential function: Difference between revisions

Revision as of 22:29, 1 May 2013

Revision as of 22:53, 1 May 2013

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The roots of the Iwasawa logarithm log_''p''(''z'') are exactly the elements of '''C'''_''p'' of the form ''p^r''·ζ where ''r'' is a rational number and ζ is a root of unity.<ref>{{harvnb|Cohen|2007|loc=Proposition 4.4.45}}</ref>

Note that there is no analogue in '''C'''_''p'' of [[Euler's identity]], ''e''^2''πi''&nbsp;=&nbsp;1. This is a corollary of [[Strassmann's theorem]].

Another major difference to the situation in '''C''' is that the ___domain of convergence of exp_''p'' is much smaller than that of log_''p''. A modified exponential function &mdash; the [[Artin–Hasse exponential]] &mdash; can be used instead which converges on |''z''|_''p''&nbsp;&lt;&nbsp;1.

==See also==

* [[Double exponential function]]

==Notes==

{{reflist}}

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==External links==

* {{planetmath reference|id=7000|title=p-adic exponential and p-adic logarithm}}

* [http://homes.esat.kuleuven.be/~fvercaut/talks/pAdic.pdf Efficient p-adic arithmetic] (slides)

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[[Category:Exponentials]]

[[Category:p-adic algorithms]]

[[Category:p-adic analysis]]

[[Category:p-adic arithmetic]]

[[Category:p-adic numbers]]

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