Theorem: The natural logarithm of a complex variable:
Proof: Let , then take natural logs of both sides to yield, . Then we have .
- This is an infinitely valued function. To recover the single valued function familiar for real numbers, we designate as a branch point and the negative real axis as a branch cut. This allows us to define the principal branch of this multivalued function as . If is real, then and .
- We have used the fact that the complex exponential function has a period of . Thus