P-adic exponential function: Difference between revisions

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However, unlike exp which converges on all of '''C''', exp_''p'' 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 large ''p''-adically, a small value of ''z'' is needed in the numerator. It follows from [[Legendre's formula]] that if <math>|z|_p < p^{-1/(p-1)}</math> then <math>\frac{z^n}{n!}</math> tends to <math>0</math>, ''p''-adically.

Although the ''p''-adic exponential is sometimes denoted ''e''^''x'', the [[e (mathematical constant)|number ''e'']] itself has no ''p''-adic analogue. This is because the power series exp_''p''(''x'') does not converge at {{nowrap|''x'' {{=}} 1}}. It is possible to choose a number ''e'' to be a ''p''-th root of exp_''p''(''p'') for {{nowrap|''p'' ≠ 2}},{{efn|or a 4th root of exp₂(4), for {{nowrap|''p'' {{=}} 2}}}} but there are multiple such roots and there is no canonical choice among them.<ref>{{harvnb|Robert|2000|p=252}}</ref>

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The power series

:<math>\log_p(1+x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}x^n}{n},</math>

converges for ''x'' in '''C'''_''p'' satisfying |''x''|_''p''&nbsp;&lt;&nbsp;1 and so defines the '''''p''-adic logarithm function''' log_''p''(''z'') for |''z''&nbsp;&minus;&nbsp;1|_''p''&nbsp;&lt;&nbsp;1 satisfying the usual property log_''p''(''zw'')&nbsp;=&nbsp;log_''p''''z''&nbsp;+&nbsp;log_''p''''w''. The function log_''p'' can be extended to all of {{SubSup|'''C'''{{SubSup||''p''|×}} (the set of nonzero elements of '''C'''_''p'') by imposing that it continues to satisfy this last property and setting log_''p''(''p'')&nbsp;=&nbsp;0. Specifically, every element ''w'' of {{SubSup|'''C'''{{SubSup||''p''|×}} can be written as ''w''&nbsp;=&nbsp;''p^r''·ζ·''z'' with ''r'' a [[rational number]], ζ a [[root of unity]], and |''z''&nbsp;&minus;&nbsp;1|_''p''&nbsp;&lt;&nbsp;1,<ref>{{harvnb|Cohen|2007|loc=Proposition 4.4.44}}</ref> in which case log_''p''(''w'')&nbsp;=&nbsp;log_''p''(''z'').{{efn|In factoring ''w'' as above, there is a choice of a root involved in writing ''p^r'' since ''r'' is rational; however, different choices differ only by multiplication by a root of unity, which gets absorbed into the factor ζ.}} This function on {{SubSup|'''C'''{{SubSup||''p''|×}} is sometimes called the '''Iwasawa logarithm''' to emphasize the choice of log_''p''(''p'')&nbsp;=&nbsp;0. In fact, there is an extension of the logarithm from |''z''&nbsp;&minus;&nbsp;1|_''p''&nbsp;&lt;&nbsp;1 to all of {{SubSup|'''C'''{{SubSup||''p''|×}} for each choice of log_''p''(''p'') in '''C'''_''p''.<ref>{{harvnb|Cohen|2007|loc=§4.4.11}}</ref>

==Properties==

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==References==

=== Citations ===

{{reflist}}

=== List of 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