Chap 3: Elementary functions, Sec 30: The Exponential Function

Question 1

$e^z$ is defined as

What happens in the real case

Answer 1

$e^{x}e^{iy} = e^{x}(\cos{y}+i\sin{y})$
in the real case where y = 0 this reduces to $e^z = e^x $
as in the real case

Question 2

$|e^z|$

Answer 2

$e^x$

Question 3

arg($e^z$)

Answer 3

$y+2 n \pi \quad n \ in Z$

Question 4

Properties of $e^z$ similar to real case

Answer 4

$e^z$ is an entire function
derivitives, product quotient rules all same
$e^z \ \not = 0 \ \forall \ z \ \in C$

Question 5

Periodicity of complex exponential function

Answer 5

$e^z$ is periodic with period $2\pi i$
so that
$e^{z+2 \pi i} = e^z$

Question 6

Negativity of complex exponential function

Answer 6

$e^z$ CAN be negative for all odd powers in exp

$e^{i(2n+1)\pi} = -1 \ \forall \ n \ \in \ Z$

Question 7

Getting any $z \in C$ from exponential function, given that z is not zero

Answer 7

Given any $0 \not = z \in C, \exists \ w \ \in C$ st $e^w = z$