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

$\blacksquare$