Complex Natural Logarithm/Examples/1 - i tan alpha

Examples of Complex Natural Logarithm

$\ln \paren {1 - i \tan \alpha} = \ln \sec \alpha + i \paren {-\alpha + 2 k \pi}$

for all $k \in \Z$.

Proof

\(\ds 1 - i \tan \alpha\)

\(=\)

\(\ds 1 - i \dfrac {\sin \alpha} {\cos \alpha}\)

Definition of Tangent Function

\(\ds \)

\(=\)

\(\ds \dfrac {\cos \alpha - i \sin \alpha} {\cos \alpha}\)

\(\ds \)

\(=\)

\(\ds \dfrac 1 {\cos \alpha} \exp \paren {-i \alpha}\)

Euler's Formula: Corollary

\(\ds \)

\(=\)

\(\ds \ln \paren {\dfrac 1 {\cos \alpha} } -i \paren {\alpha + 2 k \pi}\)

Definition of Complex Natural Logarithm

\(\ds \)

\(=\)

\(\ds \ln \sec \alpha -i \paren {\alpha + 2 k \pi}\)

Definition of Complex Secant Function

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4$. Elementary Functions of a Complex Variable: Exercise $6 \ \text{(ii)}$