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Adumbrativus (talk | contribs) Add note on the lack of an analogue for the number e; separate Notes and References footnotes |
→Definition: +"It can be seen to follow from Legendre's formula." taken from Legendre's formula#Applications |
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However, unlike exp which converges on all of '''C''', exp<sub>''p''</sub> 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 follow from [[Legendre's formula]].
Although the ''p''-adic exponential is sometimes denoted ''e''<sup>''x''</sup>, the [[e (mathematical constant)|number ''e'']] itself has no ''p''-adic analogue. This is because the power series exp<sub>''p''</sub>(''x'') does not converge at {{nowrap|''x'' {{=}} 1}}. It is possible to choose a number ''e'' to be a ''p''-th root of exp<sub>''p''</sub>(''p'') for {{nowrap|''p'' ≠ 2}},{{efn|or a 4th root of exp<sub>2</sub>(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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