The Key Question Is... a different frequency will give a Zero result due to the orthogonality (of the Fourier Transform)...

By assuming a Complex Exponential without breaking it into the Real and Imaginary comiponents then different frequencies just equal ZERO when in fact they are noncommutative nonlocality!!

 Schrodinger Equation & Fourier Analysis: Quantum Superposition of Momentum States

of x if you want to find out how much of

frequency k

is present in that function then all you

do is multiply the function f of x

by e to the minus i k x you have to

multiply it by the complex

conjugate of the complex exponential and

so all you do is just take the negative

for the argument of the exponential so

multiplied by e to the minus ikx

the reason for that is we're trying to

pick out

e to the ikx that's present in f of x if

you want to pick that out

then it's going to have to latch on to

the case of e to the minus ikx

multiplied by e to the plus ikx

which is equal to e to the 0 which is

equal to 1 and will give us a finite

result

whereas when this is multiplied by any

other complex exponential of a different

frequency it will give a zero result due 

frequency it will give a zero result due

to orthogonality of these functions

so that's a very rapid overview of how

the fourier transform works

and that will give us the amount of each

frequency k that is present in f of x

and then we've got the amount of each

frequency k we can then use that

to re-synthesize the original waveform

just by saying well we've got the amount

air for frequency k

let's use that as the coefficient for

frequency k which is the complex

exponential