P-adic exponential function: Difference between revisions

Revision as of 21:53, 5 March 2023

Revision as of 21:00, 9 March 2023

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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 can be seen to followfollows 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>