Relationship between logarithm (multivalued) and logarithm

Theorem

The following formula holds: $$\mathrm{Log}\left(re^{i\theta}\right) = \left\{ \log(re^{i\theta}+2k\pi i \colon k \in \mathbb{Z} \right\},$$ where $\mathrm{Log}$ denotes the logarithm (multivalued), $\log$ denotes the logarithm, $i$ denotes the imaginary number, and $\pi$ denotes pi.

Note

Sometimes this formula is written as $$\mathrm{Log}\left(re^{i\theta}\right) = \log(re^{i\theta})+2k\pi i, \quad k \in \mathbb{Z}.$$ Writing it this way emphasizes the multi-valued nature of $\mathrm{Log}$. For our purposes, this is not sufficient since we defined the codomain of the multivalued logarithm to be the power set of $\mathbb{C}$.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 4.1.4