Complex Natural Logarithm/Examples/-2

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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
\(\displaystyle \) \(=\) \(\displaystyle \ln 2 + \paren {2 k + 1} \pi i\)

$\blacksquare$


Sources