Complex Natural Logarithm/Examples/-2

Examples of Complex Natural Logarithm

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

for all $k \in \Z$.

Proof

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

$\blacksquare$