Complex logarithm
Properties

Not all identities satisfied by ln {\displaystyle \ln } extend to complex numbers. It is true that e Log z = z {\displaystyle e^{\operatorname {Log} z}=z} for all z ≠ 0 {\displaystyle z\not =0} (this is what it means for Log z {\displaystyle \operatorname {Log} z} to be a logarithm of z {\displaystyle z}), but the identity Log (e z) = z {\displaystyle \operatorname {Log} (e^{z})=z} fails for z {\displaystyle z} outside the strip S {\displaystyle S}. For this reason, one cannot always apply Log {\displaystyle {\text{Log}}} to both sides of an identity e z = e w {\displaystyle e^{z}=e^{w}} to deduce z = w {\displaystyle z=w}. Also, the identity Log (z 1 z 2) = Log z 1 + Log z 2 {\displaystyle \operatorname {Log} (z_{1}z_{2})=\operatorname {Log} z_{1}+\operatorname {Log} z_{2}} can fail: the two sides can differ by an integer multiple of 2 π i {\displaystyle 2\pi i}; for instance,

but

The function Log z {\displaystyle \operatorname {Log} z} is discontinuous at each negative real number, but continuous everywhere else in C ∗ {\displaystyle \mathbb {C} ^{*}}. To explain the discontinuity, consider what happens to arg z {\displaystyle \arg z} as z {\displaystyle z} approaches a negative real number a {\displaystyle a}. If z {\displaystyle z} approaches a {\displaystyle a} from above, then arg z {\displaystyle \arg z} approaches π, {\displaystyle \pi,} which is also the value of arg a {\displaystyle \arg a} itself. But if z {\displaystyle z} approaches a {\displaystyle a} from below, then arg z {\displaystyle \arg z} approaches − π. {\displaystyle -\pi.} So arg z {\displaystyle \arg z} "jumps" by 2 π {\displaystyle 2\pi } as z {\displaystyle z} crosses the negative real axis, and similarly Log z {\displaystyle \operatorname {Log} z} jumps by 2 π i. {\displaystyle 2\pi i.}
change_case:english_lowercase