Logarithm (multivalued) of product is a sum of logarithms (multivalued)

Theorem

The following formula holds for any $z_1,z_2 \in \mathbb{C}$: $$\mathrm{Log}\left( z_1z_2 \right) \subset \mathrm{Log}(z_1) + \mathrm{Log}(z_2),$$ where $\mathrm{Log}$ denotes the logarithm (multivalued) and $\mathrm{Log}(z_1) + \mathrm{Log}(z_2)$ denotes the sumset of $\mathrm{Log}(z_1)$ and $\mathrm{Log}(z_2)$.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $4.1.6$