Logarithm (multivalued) of a quotient is a difference of logarithms (multivalued)

Theorem

Let $z_1,z_2 \in \mathbb{C}$ with $z_2 \neq 0$. The following formula holds: $$\mathrm{Log} \left( \dfrac{z_1}{z_2} \right) \subset \mathrm{Log}(z_1) - \mathrm{Log}(z_2),$$ where $\mathrm{Log}$ denotes the logarithm (multivalued) and $\mathrm{Log}(z_1) - \mathrm{Log}(z_2)$ denotes the difference set of $\mathrm{Log}(z_1)$ and $\mathrm{Log}(z_2)$.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $4.1.8$