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{{DISPLAYTITLE:''p''-adic exponential function}}
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In [[mathematics]], particularly [[P-adic analysis|''p''-adic analysis]], the '''''p''-adic exponential function''' is a ''p''-adic analogue of the usual [[exponential function]] on the [[complex numbers]]. As in the complex case, it has an inverse function named the '''''p''-adic logarithm'''.
 
==Definition==
The usual exponential function on '''C''' is defined by the infinite series
:<math>\exp(z)=\sum_{n=0}^\infty \frac{z^n}{n!}.</math>
Entirely analogously, one defines the exponential function on '''C'''<sub>''p''</sub>, the completion of the algebraic closure of '''Q'''<sub>''p''</sub>, by
:<math>\exp_p(z)=\sum_{n=0}^\infty\frac{z^n}{n!}.</math>
However, unlike exp which converges on all of '''C''', exp<sub>''p''</sub> only converges on the disc
:<math>|z|_p<p^{-1/(p-1)}.</math>
This is because ''p''-adic series converge if and only if the summands tend to zero, and since the ''n''! in the denominator of each summand tends to make them very large ''p''-adically, rather a small value of ''z'' is needed in the numerator.
 
==''p''-adic logarithm function==
 
The power series
:<math>\log(1+x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}x^n}{n},</math>
converges for ''x'' in '''C'''<sub>''p''</sub> satisfying |''x''|<sub>''p''</sub>&nbsp;&lt;&nbsp;1 and so defines the '''''p''-adic logarithm function''' log<sub>''p''</sub>(''z'') for |''z''&nbsp;&minus;&nbsp;1|<sub>''p''</sub>&nbsp;&lt;&nbsp;1 satisfying the usual property log<sub>''p''</sub>(''zw'')&nbsp;=&nbsp;log<sub>''p''</sub>''z''&nbsp;+&nbsp;log<sub>''p''</sub>''w''. The function log<sub>''p''</sub> can be extended to all of {{SubSup|'''C'''|''p''|×}} (the set of nonzero elements of '''C'''<sub>''p''</sub>) by imposing that it continue to satisfy this last property and setting log<sub>''p''</sub>(''p'')&nbsp;=&nbsp;0. Specifically, every element ''w'' of {{SubSup|'''C'''|''p''|×}} can be written as ''w''&nbsp;=&nbsp;''p<sup>r''</sup>·ζ·''z'' with ''r'' a rational number, ζ a root of unity, and |''z''&nbsp;&minus;&nbsp;1|<sub>''p''</sub>&nbsp;&lt;&nbsp;1,<ref>{{harvnb|Cohen|2007|loc=Proposition 4.4.44}}</ref> in which case log<sub>''p''</sub>(''w'')&nbsp;=&nbsp;log<sub>''p''</sub>(''z'').<ref>In factoring ''w'' as above, there is a choice of a root involved in writing ''p<sup>r</sup>'' since ''r'' is rational; however, different choices differ only by multiplication by a root of unity, which gets absorbed into the factor ζ.</ref> This function on {{SubSup|'''C'''|''p''|×}} is sometimes called the '''Iwasawa logarithm''' to emphasize the choice of log<sub>''p''</sub>(''p'')&nbsp;=&nbsp;0. In fact, there is an extension of the logarithm from |''z''&nbsp;&minus;&nbsp;1|<sub>''p''</sub>&nbsp;&lt;&nbsp;1 to all of {{SubSup|'''C'''|''p''|×}} for each choice of log<sub>''p''</sub>(''p'') in '''C'''<sub>''p''</sub>.<ref>{{harvnb|Cohen|2007|loc=§4.4.11}}</ref>
 
==Properties==
 
If ''z'' and ''w'' are both in the radius of convergence for exp<sub>''p''</sub>, then their sum is too and we have the usual addition formula: exp<sub>''p''</sub>(''z''&nbsp;+&nbsp;''w'')&nbsp;=&nbsp;exp<sub>''p''</sub>(''z'')exp<sub>''p''</sub>(''w'').
 
Similarly if ''z'' and ''w'' are nonzero elements of '''C'''<sub>''p''</sub> then log<sub>''p''</sub>(''zw'')&nbsp;=&nbsp;log<sub>''p''</sub>''z''&nbsp;+&nbsp;log<sub>''p''</sub>''w''.
 
And for suitable ''z'', so that everything is defined, we have exp<sub>''p''</sub>(log<sub>''p''</sub>(''z''))&nbsp;=&nbsp;''z'' and log<sub>''p''</sub>(exp<sub>''p''</sub>(''z''))&nbsp;=&nbsp;''z''.
 
The roots of the Iwasawa logarithm log<sub>''p''</sub>(''z'') are exactly the elements of '''C'''<sub>''p''</sub> of the form ''p<sup>r''</sup>·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'''<sub>''p''</sub> of [[Euler's identity]], ''e''<sup>2''πi''</sup>&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<sub>''p''</sub> is much smaller than that of log<sub>''p''</sub>.  A modified exponential function &mdash; the [[Artin–Hasse exponential]] &mdash; can be used instead which converges on |''z''|<sub>''p''</sub>&nbsp;&lt;&nbsp;1.
 
==Notes==
 
{{reflist}}
 
==References==
* Chapter 12 of {{cite book | last=Cassels | first=J. W. S. | authorlink=J. W. S. Cassels | title=Local fields | series=[[London Mathematical Society|London Mathematical Society Student Texts]] | publisher=[[Cambridge University Press]] | year=1986 | isbn=0-521-31525-5 }}
*{{Citation
| last=Cohen
| first=Henri
| author-link=Henri Cohen (number theorist)
| title=Number theory, Volume I: Tools and Diophantine equations
| publisher=Springer
| location=New York
| series=[[Graduate Texts in Mathematics]]
| volume=239
| year=2007
| isbn=978-0-387-49922-2
| mr=2312337
| doi=10.1007/978-0-387-49923-9
}}
 
==External links==
* {{planetmath reference|id=7000|title=p-adic exponential and p-adic logarithm}}
 
[[Category:Exponentials]]

Latest revision as of 10:52, 18 June 2014



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