 P-adic exponential function

The p-adic exponential function is $\exp (x)={\sum }_{n\ge 0}{x}^n/n!$ in a complete extension of ${\mathbb{Q}}_p$. It converges when $|x{|}_p<p^{-1/(p-1)}$, as the p-adic valuations of $n!$ determine the coefficient growth.

On ${\mathbb{Q}}_p$, this includes $p{\mathbb{Z}}_p$ for odd $p$, and $4{\mathbb{Z}}_2$ for $p=2$. In a sufficiently small neighborhood it satisfies $\exp (x+y)=\exp (x)\exp (y)$ and is inverse to the p-adic logarithm. Unlike the real exponential, its defining series is not convergent on the whole field, so its domain is a substantive part of its definition.

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.