Section 31- 32, The Logarithmic function

Question 1

let $0 \not = z \in C$

If $e^w = z$ then

Answer 1

$w = ln|z| + i arg(z) = ln|z| + i(Arg(z) + 2 \pi i) , n \in Z$

Question 2

Multiple valued logarithmic function log$z$ is defined as

Answer 2

log $z = ln |z| + i arg(z) = ln|z| + i(Arg(z) + 2 \pi i) , n \in Z$

Question 3

Clearly $e^{log(z)}=$

Answer 3

z

for $z \not = 0, z \in C)

Question 4

log($e^z$) = ?

Answer 4

$z + 2 n \pi i, n \in Z$

Question 5

Log is a single valued function onwhat domain

How is Log z into log z

Answer 5

domain of Log z is $\mathbb{C}${0}

and

$log z = Log z + 2 n \pi i, n\in Z$

Question 6

When and how does the complex logarithm turn into the real case

Answer 6

If $z = x + i 0, x>0$ then $|z| = x$ and Arg z = 0

so that Log z = ln x reduces to teh real logarithmic function

Question 7

log$(z_1 z_2) = $

Answer 7

$log z_1 + log z_2$

Question 8

log $\frac{z_1}{z_2} = $

Answer 8

$log z_1 - log z_2$

Question 9

$z^n =$

Answer 9

$e^{nlogz} \qquad n \in Z$

Question 10

$z^{1/n}=$

Answer 10

$e^{(1/n)logz} \qquad n\in Z$

Question 11

log$(z_1 z_2) = $

Answer 11

NOT
$Log z_1 + Log z_2$

UNLESS
Re$(z_1) > 0$ and Re$(z_2)>0$