The -adic exponential converges near zero.
On , the power series has radius of convergenceIndeed, Legendre formula gives , and the Cauchy-Hadamard theorem gives the stated radius.
The -adic logarithm is the convergent series on a sufficiently small group of principal units.
It is locally inverse to the p-adic exponential function.
It is locally inverse to the p-adic exponential function.
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The \( p \)-adic exponential function is an important concept in \( p \)-adic analysis, which is a branch of mathematics that deals with \( p \)-adic numbers. The \( p \)-adic numbers are a system of numbers that extend the rational numbers and provide a different perspective on number theory and algebra.