Complex Natural Logarithm/Examples/-1

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Examples of Complex Natural Logarithm

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

for all $k \in \Z$.


Proof

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

$\blacksquare$


Sources