p-adic exponential and p-adic logarithm

Let p be a prime numberMathworldPlanetmath and let p be the field of complex p-adic numbers (http://planetmath.org/ComplexPAdicNumbers5).

Definition 1.

The p-adic exponential is a functionMathworldPlanetmath expp:RCp defined by




The domain of expp is restricted because the radius of convergenceMathworldPlanetmath of the series n=0zn/n! over p is precisely r=p-1/(p-1). Recall that, for zp, we define


where νp(z) is the largest exponentMathworldPlanetmath ν such that pν divides z. For example, if p3, then expp is defined over pp. However, e=expp(1) is never defined, but expp(p) is well-defined over p (when p=2, the number e42 because |4|2=0.25<0.5=r).

Definition 2.

The p-adic logarithm is a function logp:SCp defined by




We extend the p-adic logarithm to the entire p-adic complex field Cp as follows. One can show that:


where W is the group of all roots of unityMathworldPlanetmath of order prime to p in Cp× and U is the open circle of radius centered at z=1:


We define logp:CpCp by:


where s=prwu, with wW and uU.

Proposition (Properties of expp and logp).

With expp and logp defined as above:

  1. 1.

    If expp(s) and expp(t) are defined then expp(s+t)=expp(s)expp(t).

  2. 2.

    logp(s)=0 if and only if s is a rational power of p times a root of unity.

  3. 3.

    logp(xy)=logp(x)+logp(y), for all x and y.

  4. 4.

    If |s|p<p-1/(p-1) then


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


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