Fast Fourier Transform

[toshk]

since we don't time interval you are sampling i am assuming 1sec;
hence the frequency is 1; N=3; n=0..N-3; w*n=2*pi*f*i*n
e^(2*π*i*n/N)
your summing formula edited (Σ( x_n * e^(i * 2 * π * k * n / N) ) for n = 0 to N-1) in matrix form on Prime
X_K=vandermonde([1,e^(2*π*i/3),e^(4*π*i/3)])*[[0.54],[0.66],[0.52]]; and the inverse is true for X_n=inv(vandermonde([1,e^(2*π*i/3),e^(4*π*i/3)]))*[[1.72],[−0.05-0.12124355653*i],[−0.05+0.12124355653*i]]
or
simply fft([0.54,0.66,0.52])
Yes there are formulae for fft all depending how samples data are handle;(math, physics, statistics, signals...etc )

However Wolfram Alpha returns: normalize of Prime ans;
fft([0.54,0.66,0.52])/1.7211976093