P-adic exponential function

{{Short description|Mathematical function}} {{DISPLAYTITLE:''p''-adic exponential function}} In mathematics, particularly ''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'''_''p'', the completion of the algebraic closure of '''Q'''_''p'', by :<math>\exp_p(z)=\sum_{n=0}^\infty\frac{z^n}{n!}.</math> 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 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>\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 '''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 '''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 '''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 '''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''.

For ''z'' in the domain of exp_''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 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 p-adic exponential and p-adic logarithm]

Category:Exponentials Category:p-adic numbers