PlanetMath: p-adic exponential and p-adic logarithm

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p-adic exponential and p-adic logarithm

(Definition)

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

Definition 1 The -adic exponential is a function $\exp_p\colon R \to \mathbb{C}_p$ defined by

$\displaystyle \exp_p(s)=\sum_{n=0}^\infty \frac{s^n}{n!}$

where

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

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

$\displaystyle \vert z\vert _p=\frac{1}{p^{\nu_p(z)}}$

where $\nu_p(z)$ is the largest exponent $\nu$ such that $p^\nu$ divides

. 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

, the number $e^4\in \mathbb{C}_2$ because $\vert 4\vert _2=0.25<0.5=r$ ).

Definition 2 The -adic logarithm is a function $\log_p\colon S\to \mathbb{C}_p$ defined by

$\displaystyle \log_p(1+s)=\sum_{n=1}^\infty (-1)^{n+1}\frac{s^n}{n}$

where

$\displaystyle S=\{ s\in \mathbb{C}_p : \vert s\vert _p<1\}.$

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

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

where is the group of all roots of unity of order prime to in $\mathbb{C}_p^\times$ and is the open circle of radius centered at :

$\displaystyle U=\{ s\in \mathbb{C}_p : \vert s-1\vert _p < 1\}.$

We define $\log_p\colon \mathbb{C}_p \to \mathbb{C}_p$ by:

$\displaystyle \log_p(s)=log_p(u)$

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

Proposition 1 (Properties of $\exp_p$ and $\log_p$ ) With $\exp_p$ and $\log_p$ defined as above:

If $\exp_p(s)$ and $\exp_p(t)$ are defined then $\exp_p(s+t)=\exp_p(s)\exp_p(t)$ .
$\log_p(s)=0$ if and only if is a rational power of times a root of unity.
$\log_p(xy)=\log_p(x)+\log_p(y)$ , for all and .
If $\vert s\vert _p<p^{-1/(p-1)}$ then
$\displaystyle \exp_p(\log_p(1+s))=1+s,\quad \log_p(\exp_p(s))=s.$

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

$\displaystyle s^z=\exp_p(z\log_p(s))$

where it makes sense.

"p-adic exponential and p-adic logarithm" is owned by alozano.

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See Also:

-adic regulator, p-adic analytic, general power

Other names:

-adic exponential, $p$

-adic logarithm

Also defines:

general

-adic power

This object's parent.

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Cross-references: similar, rational, radius, circle, open, prime, order, roots of unity, group, complex, entire, logarithm, well-defined, divides, exponent, series, radius of convergence, restricted, domain, function, exponential, field, prime number
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This is version 3 of p-adic exponential and p-adic logarithm, born on 2005-05-02, modified 2005-05-03.
Object id is 7000, canonical name is PAdicExponentialAndPAdicLogarithm.
Accessed 3007 times total.

Classification:

AMS MSC:	11S99 (Number theory :: Algebraic number theory: local and $p$-adic fields :: Miscellaneous)
	12J12 (Field theory and polynomials :: Topological fields :: Formally $p$-adic fields)
	11S80 (Number theory :: Algebraic number theory: local and $p$-adic fields :: Other analytic theory )

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