Trigonométrie complexe

Extension des fonctions circulaires

$\sin z = \frac {e^{iz} - e^{-iz}} {2i} = \frac {\sinh iz} {i} = \sum _{k=0}^{\infty}{\frac {(-1)^k z^{2k+1}} {(2k+1)!}}$

$\cos z = \frac {e^{iz} + e^{-iz}} {2} = {\cosh iz} = \sum _{k=0}^{\infty}{\frac {(-1)^k z^{2k}} {(2k)!}}$

$\tan z = \frac {\sin z} {\cos z} = -i \frac {\sinh iz} {\cosh iz} = -i \tanh iz = -i \frac {e^{iz} - e^{-iz}} {e^{iz} + e^{-iz}}$

fonctions inverses:

$\arcsin z = -i \ln \left( i z + \sqrt { 1-z^2} \right)$

$\arccos z = -i \ln \left( z + \sqrt {z^2-1} \right)$

rappel: $e^{a+ ib} = e^a (\cos b + i \sin b)\,\!$