2-adic exponential
= 2-adic exponential
{title2=$\exp(x)$}
The <2-adic exponential> is the <power series> $\exp(x)=\sum_{n\ge0}x^n/n!$ interpreted with the <2-adic absolute value>. Over $\mathbb Q_2$, it converges for $v_2(x)>1$, equivalently $x\in4\mathbb Z_2$. The general <p-adic exponential function> has convergence condition $|x|_p<p^{-1/(p-1)}$.
On $4\mathbb Z_2$, the <2-adic exponential> and <2-adic logarithm> give inverse <group homomorphisms> between the additive <group> $4\mathbb Z_2$ and the multiplicative <group> $1+4\mathbb Z_2$. The stronger domain restriction compared with odd primes comes from the powers of two dividing the factorial denominators.