Mathematics - Complex Exponentiation

Deranged · Posted: 18 Sep 2010 06:06 am Post subject: Mathematics - Complex Exponentiation

Please help me with complex exponentiation.
I'm trying to work out how the imaginary unit 'i' actually works when used as an indice.
I've started with euler's formula:
e^(i*pi)+1=0
e^(i*pi)=-1
(e^(i*pi))^2=(-1)^2
e^(2*pi*i)=1
So how would I expand e^(2*pi*i), slowly and one step at a time?
This is the only way I will truly understand complex exponentiation...

Timothy_N · Posted: 18 Oct 2010 05:49 pm Post subject:

I don't really understand what you want help with.
If you can be more precise I will try to help you.

Deranged · Posted: 19 Oct 2010 02:50 am Post subject:

Timothy_N · Posted: 19 Oct 2010 06:50 am Post subject:

I'm still not sure what you want me to do. You can't really expand e^(2*pi*i).

I'm just going to write down some stuff that I think might help you and you can
tell me if I'm on the right track or not.

e^(i*x) = cos (x) + i*sin (x)
e^(i*2*pi) = cos (2*pi) + i*sin (2*pi) = 1 + i*0 = 1
e^(i*pi) = cos (pi) + i*sin (pi) = -1 + i*0 = -1
e^(i*pi/2) = cos (pi/2) + i*sin (pi/2) = 0 + i*1 = i
e^(i*3*pi/2)= cos (3*pi/2) + i*sin (3*pi/2) = 0 + i*(-1) = -i

e^(i*x) = e^(i*(x+2*pi*n)) , where n is a integer
e^(i*(-4*pi)) = e^(i*(-2*pi)) = e^(i*0) = e^(i*2*pi) = e^(i*4*pi)

Deranged · Posted: 30 Oct 2010 08:07 am Post subject:

What I don't get is that e^(pi) = 23 and e^(-pi )= 0.04 but e^(i*pi) = -1
I know that when you put something to the power of a positive number you just times it by itself that many times.
When you put something to the power of a negative number you put the negative reciprocal of it (swap numerator and denominator).
When you put something to the power of a fraction you put the numerator as an exponentiator and the denominator as a root.
Eg. 27^(-2/3) = 1/cbrt(27^2) = 1/cbrt(729) = 1/9
But how can you possibly use an imaginary number as an exponentiator?!
How does it work?

Kolba

Deranged

EternalSilence