Complex Logarithm of 1
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Theorem
- $\ln 1 = \set {2 k \pi i: k \in \Z}$
where $\ln 1$ denotes the complex natural logarithm of $1$.
Proof
Denote as $\ln_\R$ the real natural logarithm.
\(\ds \map \ln {r e^{i \theta} }\) | \(=\) | \(\ds \set {\map {\ln_\R} r + i \theta + 2 k \pi i: k \in \Z}\) | Definition 1 of Complex Natural Logarithm | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \ln {1 e^{i \times 0} }\) | \(=\) | \(\ds \set {\map {\ln_\R} 1 + i \times 0 + 2 k \pi i: k \in \Z}\) | substituting $r = 1$ and $\theta = 0$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \ln {1 e^{i \times \paren {0 + 0 i} } }\) | \(=\) | \(\ds \set {\map {\ln_\R} 1 + i \times 0 + 2 k \pi i: k \in \Z}\) | Definition of Wholly Real | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \ln {1 e^0}\) | \(=\) | \(\ds \set {\map {\ln_\R} 1 + i \times 0 + 2 k \pi i: k \in \Z}\) | Definition of Complex Multiplication | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \ln {1 \times \paren {1 + 0 i} }\) | \(=\) | \(\ds \set {\map {\ln_\R} 1 + i \times 0 + 2 k \pi i: k \in \Z}\) | Definition of Complex Exponential Function | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \ln {1 \times 1}\) | \(=\) | \(\ds \set {\map {\ln_\R} 1 + i \times 0 + 2 k \pi i: k \in \Z}\) | Definition of Wholly Real | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \ln 1\) | \(=\) | \(\ds \set {\map {\ln_\R} 1 + i \times 0 + 2 k \pi i: k \in \Z}\) | Real Multiplication Identity is One | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \ln 1\) | \(=\) | \(\ds \set {0 + i \times 0 + 2 k \pi i: k \in \Z}\) | Real Logarithm of $1$ is $0$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \ln 1\) | \(=\) | \(\ds \set {2 k \pi i: k \in \Z}\) | Complex Addition Identity is Zero |
$\blacksquare$