Complex Analysis: Theory and Applications · Problem 5.2.10

Elementary Functions — Howell & Matthews

Mathematics — Complex Analysis · complex-logarithm-validity

Problem

For what values of

z

is it true that ((a))

\mathrm{Log}\left(\frac{z_1}{z_2}\right) = \mathrm{Log}(z_1) - \mathrm{Log}(z_2)

? Why? ((b))

\frac{d}{dz}\mathrm{Log}(z) = \frac{1}{z}

? Why? ((c))

\mathrm{Log}\left(\frac{1}{z}\right) = -\mathrm{Log}(z)

? Why?