Sec 35-36 The Power function, 37 Trig functions

Question 1

$z^c$ called

Answer 1

the multiple valued power function

$z^C$ for $z \not \in 0$ and $c \in \mathbb{C}$

Question 2

$z^c$ =

Answer 2

$e^{c \ logz}$

Question 3

$\frac{1}{z^c} = $

Answer 3

$z^{-c}$

Question 4

$\frac{d}{dz} (z^c_{\alpha}) = $

Answer 4

$c \ z^{c-1}_{\alpha} $

Question 5

$z^c_{\alpha} = $

Answer 5

$e^{c \ log_{\alpha}z}$

Question 6

Principle values of $z^c$, the multiple valued complex power function is

Answer 6

$e^{c\ Log z}$

Question 7

how is $\sin{z}$ defined for complex z

Answer 7

$\sin{z} = \frac{e^{iz} - e^{-iz}}{2i}$

Question 8

how is $\cos{z}$ defined for complex z

Answer 8

$\cos{z} = \frac{e^{iz} + e^{-iz}}{2}$

Question 9

$sin(z_1 + z_2)$

Answer 9

As real case

=$sin(z_1) cos(z_2) + cos(z_1)sin(z_2) $

Question 10

$cos(z_1 + z_2)$

Answer 10

As real case

=$cos(z_1) cos(z_2) - sin(z_1)sin(z_2) $

Question 11

how is $\sinh{z}$ defined for complex z

same as reals

Answer 11

$\sinh{z} = \frac{e^{z} - e^{-z}}{2}$

Question 12

how is $\cosh{z}$ defined for complex z

same as reals

Answer 12

$\cosh{z} = \frac{e^{z} + e^{-z}}{2}$