Definition:Natural Logarithm/Complex/Definition 1

Definition
Let $z = r e^{i \theta}$ be a complex number expressed in exponential form such that $z \ne 0$.

The complex natural logarithm of $z \in \C_{\ne 0}$ is the multifunction defined as:


 * $\ln \left({z}\right) := \left\{{\ln \left({r}\right) + i \theta + 2 k \pi i: k \in \Z}\right\}$

where $\ln \left({r}\right)$ is the natural logarithm of the (strictly) positive real number $r$.

Also defined as
It can also be written:


 * $\ln \left({z}\right) := \ln \left|{z}\right| + i \arg \left({z}\right)$

where:
 * $\left|{z}\right|$ is the modulus of $z$
 * $\arg \left({z}\right)$ is the argument of $z$, which is a multifunction.

Also see

 * Equivalence of Definitions of Complex Natural Logarithm