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p-adic exponential and p-adic logarithm
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(Definition)
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Let $p$ be a prime number and let $\Complex_p$ be the field of complex $p$ adic numbers.
Definition 1 The $p$ adic exponential is a function $\exp_p\colon R \to \Complex_p$ defined by $$\exp_p(s)=\sum_{n=0}^\infty \frac{s^n}{n!}$$ where $$R=\{ s\in \Complex_p : |s|_p<\frac{1}{p^{1/(p-1)}}\}.$$
The domain of $\exp_p$ is restricted because the radius of convergence of the series $\sum_{n=0}^\infty z^n/n!$ over $\Complex_p$ is precisely $r=p^{-1/(p-1)}$ Recall that, for $z\in \Rats_p$ 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 $p\Ints_p$ However, $e=\exp_p(1)$ is never defined, but $\exp_p(p)$ is well-defined over $\Complex_p$ (when $p=2$ the number $e^4\in \Complex_2$ because $|4|_2=0.25<0.5=r$ .
Definition 2 The $p$ adic logarithm is a function $\log_p\colon S\to \Complex_p$ defined by $$\log_p(1+s)=\sum_{n=1}^\infty (-1)^{n+1}\frac{s^n}{n}$$ where $$S=\{ s\in \Complex_p : |s|_p<1\}.$$ We extend the $p$ adic logarithm to the entire $p$ adic complex field $\Complex_p$ as follows. One can show that: $$\Complex_p=\{ p^t\cdot w\cdot u: t\in \Rats,\ w\in W,\ u\in U\}=p^{\Rats}\times W \times U$$ where $W$ is the group of all roots of unity of order prime to $p$ in $\Complex_p^\times$ and $U$ is the open circle of radius centered at $z=1$ $$U=\{ s\in \Complex_p : |s-1|_p < 1\}.$$ We define
$\log_p\colon \Complex_p \to \Complex_p$ by: $$\log_p(s)=log_p(u)$$ where $s=p^r\cdot w \cdot u$ with $w\in W$ and $u\in U$
Proposition 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 $s$ is a rational power of $p$ times a root of unity.
- $\log_p(xy)=\log_p(x)+\log_p(y)$ for all $x$ and $y$
- If $|s|_p<p^{-1/(p-1)}$ 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.
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"p-adic exponential and p-adic logarithm" is owned by alozano.
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Cross-references: similar, rational, radius, circle, open, prime, order, roots of unity, group, complex, entire, logarithm, number, 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 4880 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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Pending Errata and Addenda
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