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P-adic exponential function

Mathematical function

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Source updated July 19, 2026

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.

01Definition

The usual exponential function on $\mathbb {C}$ is defined by the infinite series

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

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

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

However, unlike exp which converges on all of $\mathbb {C}$ , $\exp _{p}$ 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 $\exp _{p}(x)$ does not converge at $x=1$ . It is possible to choose a number $e$ to be a p-th root of $\exp _{p}(p)$ for $p\neq 2$ , but there are multiple such roots and there is no canonical choice among them.

02p-adic logarithm function

The power series

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

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

03Properties

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: $\exp _{p}(z+w)=\exp _{p}(z)\exp _{p}(w)$ .

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

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

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

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

Another major difference to the situation in $\mathbb {C}$ 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 $|z|_{p}<1$ .

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Sources and credits

This article is adapted from the Wikipedia article “P-adic exponential function”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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