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:


 * $\ln \paren z := \set {\ln \paren r + i \theta + 2 k \pi i: k \in \Z}$

where $\ln \paren r$ is the natural logarithm of the (strictly) positive real number $r$.

Also defined as
It can also be written:


 * $\ln \paren z := \ln \cmod z + i \arg z$

where:
 * $\cmod z$ is the modulus of $z$
 * $\arg z$ is the argument of $z$, which is a multifunction.

Also see

 * Equivalence of Definitions of Complex Natural Logarithm