# 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

$\blacksquare$