The logarithm of a complex number $z$ is defined as every complex number $w$ which satisfies the equation

(1)

The solution of (1) is obtained by using the form $e^w = re^{i\varphi}$ , where $r = |z|$ and $\varphi = \arg{z}$ ; the result is $$w = \log{z} = \ln{|z|}+i\arg{z}.$$ Here, the $\ln|z|$ means the usual Napierian or natural logarithm (`logarithmus naturalis') of the real number $|z|$ . If we fix the phase angle $\varphi$ of $|z|$ so that $0 \leqq \varphi < 2\pi$ , we can write $$\log{z} = \ln{r}+i\varphi+n\cdot 2\pi i\quad(n = 0,\,\pm1,\,\pm2,\,...).$$

The complex logarithm $\log{z}$ is defined for all $z \neq 0$ and it is infinitely multivalued $-$ e.g. $\log{(-1)} = (2n+1)\pi i$ where $n$ is an arbitrary integer. The values with $n = 0$ are called the principal values of the logarithm; if $z$ is real, the principal value of $\log{z}$ coincides with $\ln{z}$ .