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
\(\ds -1\) | \(=\) | \(\ds e^{i \pi}\) | 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 | |||||||||||
\(\ds \) | \(=\) | \(\ds 0 + \paren {2 k + 1} \pi i\) |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4$. Elementary Functions of a Complex Variable: Exercise $6 \ \text{(i)}$