= p-adic exponential function
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{title2=$\exp_p$}
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= p-adic exponential
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The <p-adic exponential function> is $\exp(x)=\sum_{n\ge0}x^n/n!$ 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.
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