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'''.

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]].

==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>

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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

One can also define a '''''p''-adic logarithm function''' by the power series

:<math>\log_p(1+zx)=\sum_{n=1}^\infty \frac{(-1)^{n+1}zx^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 Butfunction thislog_''p'' can be extended to all of '''C'''{{SubSup||''p''|×}} (the set of nonzero elements of '''C'''_''p'') by writingimposing anythat it continues to satisfy this last property and setting log_''zp''(''p'')&nbsp;=&nbsp;0. inSpecifically, every element ''w'' of '''C'''<sub>{{SubSup||''p''|×}} can be written as ''w''&nbsp;=&nbsp;''p<sup>r</subsup>''·ζ·''z'' aswith ''r'' a [[rational number]], ζ a [[root of unity]], and |''z''&nbsp;=&minus;&nbsp;1|<sub>''p''<sup/sub>&nbsp;&lt;&nbsp;1,<ref>{{harvnb|Cohen|2007|loc=Proposition 4.4.44}}</ref> in which case log<sub>''mp''</supsub>·(''uw''·)&nbsp;=&nbsp;log_''vp'',(''z'').{{efn|In wherefactoring ''mw'' as above, there is a rationalchoice number,of a root involved in writing ''up^r'' since ''r'' is rational; however, different choices differ only by multiplication by a root of unity, ofwhich ordergets coprimeabsorbed tointo the factor ζ.}} This function on '''C'''{{SubSup||''p'',|×}} andis sometimes called the ''v'Iwasawa logarithm''' liesto inemphasize the original ___domainchoice of convergence for log_''p''(''p'')&nbsp;=&nbsp;0. In fact, sothere is an extension of the logarithm from |''vz''&nbsp;&minus;&nbsp;1|_''p''&nbsp;&lt;&nbsp;1. to Weall thenof define'''C'''{{SubSup||''p''|×}} for each choice of log_''p''(''zp'') toin be log'''C'''_''p''(''v'').<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''.

And for suitableFor ''z'', soin thatthe everything___domain isof definedexp_''p'', we have exp_''p''(log_''p''(1+''z''))&nbsp;=&nbsp;1+''z'' and log_''p''(exp_''p''(''z''))&nbsp;=&nbsp;''z''.

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.

==Notes==

{{Notelist}}

==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

| 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

}}

*{{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]]

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

[[Category:p-adic numbers]]