Understanding the Exponential Property of Complex Numbers

[swoosh]

I'm currently learning complex analysis, and there's something I can't quite get...

Usually:

e^2 * e^3 = e^(2+3) = e^5

So why is that:

e^(i*PI*t) * e^(i*PI) = -e^(i*PI*t)

?

I was able to get the solution from:

e^(i*PI*t) * e^(i*PI) = 
  = (Cos(PI*t) + iSin(PI*t)) * (Cos(PI) + iSin(PI)) =
  = (Cos(PI*t) + iSin(PI*t)) * (-1) =
  = -e^(i*PI*t)

But, what am I missing?

Thanks!

 

[daudaudaudau]

I don't think you are missing anything, it is just another way to say the same thing.

$$e^{i\pi t}e^{i\pi}=e^{i\pi t+i\pi}=-e^{i\pi t}$$

Because as you have derived yourself
$$e^{i\pi}=-1$$

 

[swoosh]

Hm, I see.

The first time I came across this I just assumed:
$$e^{i\pi t}e^{i\pi}=e^{i\pi t+i\pi}=e^{i\pi (t+1)}$$
was the simplest way it could get.

Well, thanks again!