Sec 35-36 The Power function, 37 Trig functions Flashcards
$z^c$ called
the multiple valued power function
$z^C$ for $z \not \in 0$ and $c \in \mathbb{C}$
$z^c$ =
$e^{c \ logz}$
$\frac{1}{z^c} = $
$z^{-c}$
$\frac{d}{dz} (z^c_{\alpha}) = $
$c \ z^{c-1}_{\alpha} $
$z^c_{\alpha} = $
$e^{c \ log_{\alpha}z}$
Principle values of $z^c$, the multiple valued complex power function is
$e^{c\ Log z}$
how is $\sin{z}$ defined for complex z
$\sin{z} = \frac{e^{iz} - e^{-iz}}{2i}$
how is $\cos{z}$ defined for complex z
$\cos{z} = \frac{e^{iz} + e^{-iz}}{2}$
$sin(z_1 + z_2)$
As real case
=$sin(z_1) cos(z_2) + cos(z_1)sin(z_2) $
$cos(z_1 + z_2)$
As real case
=$cos(z_1) cos(z_2) - sin(z_1)sin(z_2) $
how is $\sinh{z}$ defined for complex z
same as reals
$\sinh{z} = \frac{e^{z} - e^{-z}}{2}$
how is $\cosh{z}$ defined for complex z
same as reals
$\cosh{z} = \frac{e^{z} + e^{-z}}{2}$