Complex logarithm

$\log z = \ln|z| + i\arg z$ -- multivalued through $\arg z$ , unless restricted to a branch.

The complex logarithm of a nonzero $z$ is $\log z = \ln|z| + i\arg z$ , where $\ln|z|$ is the ordinary real natural logarithm of the modulus and $\arg z$ is any valid argument of $z$ . Because $\arg z$ is multivalued, so is $\log z$ ; restricting to the principal argument gives the principal branch, written $\operatorname{Log} z = \ln|z|+i\operatorname{Arg} z$ , which is what most calculations default to unless another branch is specified.

Examples

$\operatorname{Log}(1) = \ln 1 + i\cdot 0 = 0$ , but every $2\pi i k$ for integer $k$ is also a valid value of $\log(1)$ .
$\operatorname{Log}(-e) = 1+i\pi$ , since $|-e|=e$ and $\operatorname{Arg}(-e)=\pi$ .