p-adic exponential and p-adic logarithm

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

→

C

p

defined by

exp

p

⁡

(

s

)

=

∑

n

=

0

∞

s

n

n

!

where

R

=

{

s

∈

ℂ

p

:

|

s

|

p

<

1

p

1

/

(

p

-

1

)

}

.

The domain of

exp

p

is restricted because the

radius of convergence

of the series

∑

n

=

0

∞

z

n

/

n

!

over

ℂ

p

is precisely

r

=

p

-

1

/

(

p

-

1

)

. Recall that, for

z

∈

ℚ

p

, we define

|

z

|

p

=

1

p

ν

p

⁢

(

z

)

where

ν

p

⁢

(

z

)

is the largest

exponent

ν

such that

p

ν

divides

z

. For example, if

p

≥

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

∈

ℂ

2

because

|

4

|

2

=

0.25

<

0.5

=

r

).

Definition 2

.

The

p

-adic

logarithm

is a function

log

p

:

S

→

C

p

defined by

log

p

⁡

(

1

+

s

)

=

∑

n

=

1

∞

(

-

1

)

n

+

1

⁢

s

n

n

where

S

=

{

s

∈

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

⋅

w

⋅

u

:

t

∈

ℚ

,

w

∈

W

,

u

∈

U

}

=

p

ℚ

×

W

×

U

where

W

is the group of all

roots of unity

of order prime to

p

in

C

p

×

and

U

is the open circle of radius centered at

z

=

1

:

U

=

{

s

∈

ℂ

p

:

|

s

-

1

|

p

<

1

}

.

We define

log

p

:

C

p

→

C

p

by:

log

p

⁡

(

s

)

=

l

⁢

o

⁢

g

p

⁢

(

u

)

where

s

=

p

r

⋅

w

⋅

u

, with

w

∈

W

and

u

∈

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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