# Complex Natural Logarithm/Examples/-1

## Examples of Complex Natural Logarithm

$\map \ln {-1} = \paren {2 k + 1} \pi i$

for all $k \in \Z$.

## Proof

 $\ds -1$ $=$ $\ds e^{i \pi}$ Euler's Identity $\ds \leadsto \ \$ $\ds \map \ln {-1}$ $=$ $\ds \map \ln {e^{i \pi + 2 k \pi i} }$ $\ds$ $=$ $\ds \ln 1 + i \pi + 2 k \pi i$ Definition of Complex Natural Logarithm $\ds$ $=$ $\ds 0 + \paren {2 k + 1} \pi i$

$\blacksquare$