Need this derivation to solidify understanding

[HasuChObe]

Can someone please give a concise and complete derivation of the following Fourier Transform:

x(t) = e^(j*w0*t) (w0 is a constant)

Please explain all parts. Thank you very much!

 

[Bob S]

Consider the relation

z(t) = A[cos(w₀t) + i sin(w₀t)] in the complex plane

Where z(t) is a complex variable z(t) = x(t) + i y(t)

For the differential equation d²x/dt² + w₀² x = 0
the solution is x = A e^{i w₀ t}

where e^{i w₀ t} as a rotating unit-amplitude vector in the complex plane and i = sqrt(-1):

z(t) = A e^{i w₀ t}