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 ${\displaystyle \mathbb {C} }$ is defined by the infinite series

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

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

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

However, unlike exp which converges on all of ${\displaystyle \mathbb {C} }$, ${\displaystyle \exp _{p}}$ only converges on the disc

${\displaystyle |z|_{p}<p^{-1/(p-1)}.}$

This is because p-adic series converge if and only if the summands tend to zero, and since the ${\displaystyle n!}$ 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 ${\displaystyle |z|_{p}<p^{-1/(p-1)}}$ then ${\displaystyle {\frac {z^{n}}{n!}}}$ tends to ${\displaystyle 0}$, p-adically.

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

The power series

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

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

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

Similarly if ${\displaystyle z}$ and ${\displaystyle w}$ are nonzero elements of ${\displaystyle \mathbb {C} _{p}}$ then ${\displaystyle \log _{p}(zw)=\log _{p}(z)+\log _{p}(w)}$.

For ${\displaystyle z}$ in the domain of ${\displaystyle \exp _{p}}$, we have ${\displaystyle \exp _{p}(\log _{p}(1+z))=1+z}$ and ${\displaystyle \log _{p}(\exp _{p}(z))=z}$.

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

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

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

• Chapter 12 of Cassels, J. W. S. (1986). Local fields. London Mathematical Society Student Texts. Cambridge University Press. ISBN 0-521-31525-5.
• Cohen, Henri (2007), Number theory, Volume I: Tools and Diophantine equations, Graduate Texts in Mathematics, vol. 239, New York: Springer, doi:10.1007/978-0-387-49923-9, ISBN 978-0-387-49922-2, MR 2312337
• Robert, Alain M. (2000), A Course in p-adic Analysis, Springer, ISBN 0-387-98669-3

• p-adic exponential and p-adic logarithm