Simplification with exponentials and imaginary numbers

[JieXian]

Hi, I'm using euler's identity : exp(i∏) = exp (-i∏) = -1 to simplify the equation after integrating it.

[PLAIN]http://img443.imageshack.us/img443/5504/captureikm.jpg
Note: the equation to be integrated is exp(0.5it) + exp(-0.5it) and they have simplified it, it was actually a cos(0.5t) function and the 1/2 from cos(t) = 0.5[ exp(it) + exp(-it) ] isn't included.

However, after trying I got the numerator to be 0, because of the exp(i∏) = exp (-i∏) problem.

Letting x and y be any variable,

exp(i∏ x) - exp(i∏ y) - [ exp(i(-∏) x) - exp(i(-∏) y) ]

= exp(i∏ x) - exp(i(-∏) x) - exp(i∏ y) + exp(i(-∏) y) --------- (rearranging)

= exp(i∏ x) - exp(i(∏) x) - exp(i∏ y) + exp(i(∏) y) ----------- (since exp(i∏) = exp (-i∏))

= 0

Please help me, thank you very much.

 

[lanedance]

whilst it is true
<br /> e^{i \pi} = -1 = e^{-i \pi}<br />

unless x=2n+1 for some integer n, then
<br /> e^{i \pi x} \neq e^{-i \pi x}<br />

this makes your last step invalid, though I have not checked the rest