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# p-adic exponential and p-adic logarithm

Let $p$ be a prime number and let $\mathbb{C}_{p}$ be the field of complex $p$-adic numbers.

###### Definition 1.

The $p$-adic exponential is a function $\exp_{p}\colon R\to\mathbb{C}_{p}$ defined by

$\exp_{p}(s)=\sum_{{n=0}}^{\infty}\frac{s^{n}}{n!}$ |

where

$R=\{s\in\mathbb{C}_{p}:|s|_{p}<\frac{1}{p^{{1/(p-1)}}}\}.$ |

The domain of $\exp_{p}$ is restricted because the radius of convergence of the series $\sum_{{n=0}}^{\infty}z^{n}/n!$ over $\mathbb{C}_{p}$ is precisely $r=p^{{-1/(p-1)}}$. Recall that, for $z\in\mathbb{Q}_{p}$, we define

$|z|_{p}=\frac{1}{p^{{\nu_{p}(z)}}}$ |

where $\nu_{p}(z)$ is the largest exponent $\nu$ such that $p^{\nu}$ divides $z$. For example, if $p\geq 3$, then $\exp_{p}$ is defined over $p\mathbb{Z}_{p}$. However, $e=\exp_{p}(1)$ is never defined, but $\exp_{p}(p)$ is well-defined over $\mathbb{C}_{p}$ (when $p=2$, the number $e^{4}\in\mathbb{C}_{2}$ because $|4|_{2}=0.25<0.5=r$).

###### Definition 2.

The $p$-adic logarithm is a function $\log_{p}\colon S\to\mathbb{C}_{p}$ defined by

$\log_{p}(1+s)=\sum_{{n=1}}^{\infty}(-1)^{{n+1}}\frac{s^{n}}{n}$ |

where

$S=\{s\in\mathbb{C}_{p}:|s|_{p}<1\}.$ |

We extend the $p$-adic logarithm to the entire $p$-adic complex field $\mathbb{C}_{p}$ as follows. One can show that:

$\mathbb{C}_{p}=\{p^{t}\cdot w\cdot u:t\in\mathbb{Q},\ w\in W,\ u\in U\}=p^{{% \mathbb{Q}}}\times W\times U$ |

where $W$ is the group of all roots of unity of order prime to $p$ in $\mathbb{C}_{p}^{\times}$ and $U$ is the open circle of radius centered at $z=1$:

$U=\{s\in\mathbb{C}_{p}:|s-1|_{p}<1\}.$ |

We define $\log_{p}\colon\mathbb{C}_{p}\to\mathbb{C}_{p}$ by:

$\log_{p}(s)=log_{p}(u)$ |

where $s=p^{r}\cdot w\cdot u$, with $w\in W$ and $u\in U$.

###### Proposition (Properties of $\exp_{p}$ and $\log_{p}$).

With $\exp_{p}$ and $\log_{p}$ defined as above:

1. If $\exp_{p}(s)$ and $\exp_{p}(t)$ are defined then $\exp_{p}(s+t)=\exp_{p}(s)\exp_{p}(t)$.

2. $\log_{p}(s)=0$ if and only if $s$ is a rational power of $p$ times a root of unity.

3. $\log_{p}(xy)=\log_{p}(x)+\log_{p}(y)$, for all $x$ and $y$.

4. If $|s|_{p}<p^{{-1/(p-1)}}$ then

$\exp_{p}(\log_{p}(1+s))=1+s,\quad\log_{p}(\exp_{p}(s))=s.$

## Mathematics Subject Classification

12J12*no label found*11S99

*no label found*11S80

*no label found*

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