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In mathematics, particularly p-adic analysis, the p-adic exponential function is a p-adic analogue of the usual exponential function on the complex numbers. As in the complex case, it has an inverse function, named the p-adic logarithm.

The usual exponential function on $ℂ$ is defined by the infinite series

$\exp \left(z\right)=\sum _{n=0}^{\infty }\frac{z^n}{n!}.$

Entirely analogously, one defines the exponential function on ${ℂ}_p$, the completion of the algebraic closure of ${\mathbb{Q}}_p$, by

${\exp }_p\left(z\right)=\sum _{n=0}^{\infty }\frac{z^n}{n!}.$

However, unlike exp which converges on all of $ℂ$, ${\exp }_p$ only converges on the disc

$|z{|}_p<p^{-1/(p-1)}.$

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 large p-adically, a small value of z is needed in the numerator. It follows from Legendre's formula that if $|z{|}_p<p^{-1/(p-1)}$ then $\frac{z^n}{n!}$ tends to $0$, p-adically.

Although the p-adic exponential is sometimes denoted $e^x$, the number e itself has no p-adic analogue. This is because the power series ${\exp }_p\left(x\right)$ does not converge at $x=1$. It is possible to choose a number $e$ to be a p-th root of ${\exp }_p\left(p\right)$ for $p\ne 2$,^[a] but there are multiple such roots and there is no canonical choice among them.^[1]

The power series

${\log }_p(1+x)=\sum _{n=1}^{\infty }\frac{(-1{)}^{n+1}{x}^n}{n},$

converges for $x$ in ${ℂ}_p$ satisfying $|x{|}_p<1$ and so defines the p-adic logarithm function ${\log }_p\left(z\right)$ for $|z-1{|}_p<1$ satisfying the usual property ${\log }_p\left(zw\right)={\log }_p\left(z\right)+{\log }_p\left(w\right)$. The function ${\log }_p$ can be extended to all of $ℂ$×
p  (the set of nonzero elements of ${ℂ}_p$) by imposing that it continues to satisfy this last property and setting ${\log }_p\left(p\right)=0$. Specifically, every element $w$ of $ℂ$×
p  can be written as $w=p^r\cdot \zeta \cdot z$ with $r$ a rational number, $\zeta$ a root of unity, and $|z-1{|}_p<1$,^[2] in which case ${\log }_p\left(w\right)={\log }_p\left(z\right)$.^[b] This function on $ℂ$×
p  is sometimes called the Iwasawa logarithm to emphasize the choice of ${\log }_p\left(p\right)=0$. In fact, there is an extension of the logarithm from $|z-1{|}_p<1$ to all of $ℂ$×
p  for each choice of ${\log }_p\left(p\right)$ in ${ℂ}_p$.^[3]

If $z$ and $w$ are both in the radius of convergence for ${\exp }_p$, then their sum is too and we have the usual addition formula: ${\exp }_p(z+w)={\exp }_p\left(z\right){\exp }_p\left(w\right)$.

Similarly if $z$ and $w$ are nonzero elements of ${ℂ}_p$ then ${\log }_p\left(zw\right)={\log }_p\left(z\right)+{\log }_p\left(w\right)$.

For $z$ in the domain of ${\exp }_p$, we have ${\exp }_p\left({\log }_p\right(1+z\left)\right)=1+z$ and ${\log }_p\left({\exp }_p\right(z\left)\right)=z$.

The roots of the Iwasawa logarithm ${\log }_p\left(z\right)$ are exactly the elements of ${ℂ}_p$ of the form $p^r\cdot \zeta$ where $r$ is a rational number and $\zeta$ is a root of unity.^[4]

Note that there is no analogue in ${ℂ}_p$ of Euler's identity, $e^{2\pi i}=1$. This is a corollary of Strassmann's theorem.

Another major difference to the situation in $ℂ$ is that the domain of convergence of ${\exp }_p$ is much smaller than that of ${\log }_p$. A modified exponential function — the Artin–Hasse exponential — can be used instead which converges on $|z{|}_p<1$.

• p-adic exponential and p-adic logarithm