# Thread: Complex Analysis - Mapping of a straight line and Principal value

1. ## Complex Analysis - Mapping of a straight line and Principal value

I've done part a) I just simply cubed z and expanded, then seperated real and imaginary. I assume this is all that's needed?

For part b) I have no idea how to do it.

2. ## Re: Complex Analysis - Mapping of a straight line and Principal value

Originally Posted by fourierT

I've done part a) I just simply cubed z and expanded, then seperated real and imaginary. I assume this is all that's needed?

For part b) I have no idea how to do it.
I doubt they are talking about the Cauchy Principal value so
what I'm going to assume they mean in (b) is simply the value of $\small 2^{i+1} \text { for } (0 \leq \theta <2\pi)$

$\small 2^{i+1}=e^{(i+1)ln(2)}$

$\text{You can probably come up with } z=r e^{i\theta}\text{ for that.}$

3. ## Re: Complex Analysis - Mapping of a straight line and Principal value

$\small 2^{i+1}=e^{(i+1)ln(2)}$

$\text{You can probably come up with } z=r e^{i\theta}\text{ for that.}$

Is it cos(ln2) + (i+1)sin(ln2)

?

4. ## Re: Complex Analysis - Mapping of a straight line and Principal value

Originally Posted by fourierT
$\small 2^{i+1}=e^{(i+1)ln(2)}$

$\text{You can probably come up with } z=r e^{i\theta}\text{ for that.}$

Is it cos(ln2) + (i+1)sin(ln2)

?
no...

$e^{(i+1)ln(2)} = e^{ln(2) + ln(2) i} = 2 * e^{ln(2) i} = 2(cos(ln(2))+i sin(ln(2)))$

you could have started with

$2^{i+1}=2*2^{i}=2e^{ln(2)i}$

I guess that's a bit quicker

5. ## Re: Complex Analysis - Mapping of a straight line and Principal value

Originally Posted by romsek
no...

$e^{(i+1)ln(2)} = e^{ln(2) + ln(2) i} = 2 * e^{ln(2) i} = 2(cos(ln(2))+i sin(ln(2)))$

you could have started with

$2^{i+1}=2*2^{i}=2e^{ln(2)i}$

I guess that's a bit quicker
How can we evaluate that in algebraic form..wouldn't it just be numbers.

6. ## Re: Complex Analysis - Mapping of a straight line and Principal value

Originally Posted by fourierT
How can we evaluate that in algebraic form..wouldn't it just be numbers.
you see any variables in $2^{i+1}$ ?

7. ## Re: Complex Analysis - Mapping of a straight line and Principal value

Sure, i is algebraic, but is that all it wants, the real part and the imaginary part?

8. ## Re: Complex Analysis - Mapping of a straight line and Principal value

the algebraic form of a complex number is z = x + i y

that's all it is.