# The logarithm of a complex number.

Definition 4.4.1

 ,
where is defined up to an additive multiple of .

Example 4.4.2

We denote Log the principal value of , i.e. the value corresponding to the principal value of (recall that ).

Example 4.4.3

 Log Log

Proposition 4.4.4

1. .
2. .
3. .

Proposition 4.4.5   The logarithmic function is analytic on its domain.

For a proof, use Cauchy-Riemann equations (v.s. 3).

Example 4.4.6

We resume our work from Example 3.5.

Example 4.4.7

Proposition 4.4.8

The proof is simple: let , where . Then we have:

and

Proposition 4.8 means that the function is not exactly the inverse of the complex exponential function in basis .

Noah Dana-Picard 2007-12-24