Let $p$ be a prime number and let $\mathbb{C}_{p}$ be the field of complex $p$-adic numbers (http://planetmath.org/ComplexPAdicNumbers5).

###### 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. 1.

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

2. 2.

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

3. 3.

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

4. 4.

If $|s|_{p} then

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

In a similar way one defines the general $p$-adic power by:

 $s^{z}=\exp_{p}(z\log_{p}(s))$

where it makes sense.

Title p-adic exponential and p-adic logarithm PadicExponentialAndPadicLogarithm 2013-03-22 15:13:50 2013-03-22 15:13:50 alozano (2414) alozano (2414) 6 alozano (2414) Definition msc 12J12 msc 11S99 msc 11S80 $p$-adic exponential $p$-adic logarithm PAdicRegulator PAdicAnalytic GeneralPower general $p$-adic power