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
\(\ds -2\) | \(=\) | \(\ds 2 e^{i \pi}\) | Euler's Identity | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \ln \paren {-2}\) | \(=\) | \(\ds \ln \paren {2 e^{i \pi + 2 k \pi i} }\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \ln 2 + i \pi + 2 k \pi i\) | Definition of Complex Natural Logarithm | |||||||||||
\(\ds \) | \(=\) | \(\ds \ln 2 + \paren {2 k + 1} \pi i\) |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.6$. The Logarithm: Examples: $\text {(i)}$