P-adic exponential function: Difference between revisions

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{{Short description|Mathematical function}}

{{DISPLAYTITLE:''p''-adic exponential function}}

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'''.

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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 very large ''p''-adically, rather 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>

==''p''-adic logarithm function==

The power series

:<math>\loglog_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'').<ref>{{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.</ref>}} 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==

If ''z'' and ''w'' are both in the [[radius of convergence]] for exp_''p'', then their sum is too and we have the usual addition formula: exp_''p''(''z''&nbsp;+&nbsp;''w'')&nbsp;=&nbsp;exp_''p''(''z'')exp_''p''(''w'').

Similarly if ''z'' and ''w'' are nonzero elements of '''C'''_''p'' then log_''p''(''zw'')&nbsp;=&nbsp;log_''p''''z''&nbsp;+&nbsp;log_''p''''w''.

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==Notes==

{{Notelist}}

==References==

=== Citations ===

{{reflist}}

=== List of references ===

==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

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| doi=10.1007/978-0-387-49923-9

}}

*{{Citation |last=Robert |first=Alain M. |year=2000 |title=A Course in ''p''-adic Analysis |publisher=Springer |isbn=0-387-98669-3}}

==External links==

* {{[https://planetmath.org/PadicExponentialAndPadicLogarithm reference|id=7000|title=p-adic exponential and p-adic logarithm}}]

[[Category:Exponentials]]