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p-adic exponential and p-adic logarithm

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

Definition 1.

The $p$ -adic exponential is a function $\exp_{p} : R \to C_{p}$ defined by

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

where

R = {s \in 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 $ℂ_{p}$ is precisely $r = p^{- 1 / (p - 1)}$ . Recall that, for $z \in ℚ_{p}$ , we define

{| z |}_{p} = \frac{1}{p^{ν_{p} (z)}}

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

Definition 2.

The $p$ -adic logarithm is a function $\log_{p} : S \to C_{p}$ defined by

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

where

S = {s \in ℂ_{p} : {| s |}_{p} < 1} .

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

ℂ_{p} = {p^{t} \cdot w \cdot u : t \in ℚ, w \in W, u \in U} = p^{ℚ} \times W \times U

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

U = {s \in ℂ_{p} : {| s - 1 |}_{p} < 1} .

We define $\log_{p} : C_{p} \to C_{p}$ by:

\log_{p} (s) = l o g_{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.

$\log_{p} (x y) = \log_{p} (x) + \log_{p} (y)$ , for all $x$ and $y$ .
4.

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

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

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
Canonical name	PadicExponentialAndPadicLogarithm
Date of creation	2013-03-22 15:13:50
Last modified on	2013-03-22 15:13:50
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	6
Author	alozano (2414)
Entry type	Definition
Classification	msc 12J12
Classification	msc 11S99
Classification	msc 11S80
Synonym	$p$ -adic exponential
Synonym	$p$ -adic logarithm
Related topic	PAdicRegulator
Related topic	PAdicAnalytic
Related topic	GeneralPower
Defines	general $p$ -adic power

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p-adic exponential and p-adic logarithm

Definition 1.

Definition 2.

Proposition (Properties of expp and logp).

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