y[k] = np.sum(x * np.exp(-2j * np.pi * k * np.arange(n)/n))

The frequency term ``f=k/n`` is found at ``y[k]``. At ``y[n/2]`` we reach
the Nyquist frequency and wrap around to the negative-frequency terms. So,
for an 8-point transform, the frequencies of the result are
[0, 1, 2, 3, -4, -3, -2, -1]. To rearrange the fft output so that the
zero-frequency component is centered, like [-4, -3, -2, -1, 0, 1, 2, 3],
use `fftshift`.

Transforms can be done in single, double, or extended precision (long
double) floating point. Half precision inputs will be converted to single
precision and non-floating-point inputs will be converted to double
precision.

If the data type of ``x`` is real, a "real FFT" algorithm is automatically
used, which roughly halves the computation time. To increase efficiency
a little further, use `rfft`, which does the same calculation, but only
outputs half of the symmetrical spectrum. If the data are both real and
symmetrical, the `dct` can again double the efficiency, by generating
half of the spectrum from half of the signal.

When ``overwrite_x=True`` is specified, the memory referenced by ``x`` may
be used by the implementation in any way. This may include reusing the
memory for the result, but this is in no way guaranteed. You should not
rely on the contents of ``x`` after the transform as this may change in
future without warning.

The ``workers`` argument specifies the maximum number of parallel jobs to
split the FFT computation into. This will execute independent 1-D
FFTs within ``x``. So, ``x`` must be at least 2-D and the
non-transformed axes must be large enough to split into chunks. If ``x`` is
too small, fewer jobs may be used than requested.

References
----------
.. [1] Cooley, James W., and John W. Tukey, 1965, "An algorithm for the
       machine calculation of complex Fourier series," *Math. Comput.*
       19: 297-301.
.. [2] Bluestein, L., 1970, "A linear filtering approach to the
       computation of discrete Fourier transform". *IEEE Transactions on
       Audio and Electroacoustics.* 18 (4): 451-455.

Examples
--------
>>> import scipy.fft
>>> import numpy as np
>>> scipy.fft.fft(np.exp(2j * np.pi * np.arange(8) / 8))
array([-2.33486982e-16+1.14423775e-17j,  8.00000000e+00-1.25557246e-15j,
        2.33486982e-16+2.33486982e-16j,  0.00000000e+00+1.22464680e-16j,
       -1.14423775e-17+2.33486982e-16j,  0.00000000e+00+5.20784380e-16j,
        1.14423775e-17+1.14423775e-17j,  0.00000000e+00+1.22464680e-16j])

In this example, real input has an FFT which is Hermitian, i.e., symmetric
in the real part and anti-symmetric in the imaginary part:

>>> from scipy.fft import fft, fftfreq, fftshift
>>> import matplotlib.pyplot as plt
>>> t = np.arange(256)
>>> sp = fftshift(fft(np.sin(t)))
>>> freq = fftshift(fftfreq(t.shape[-1]))
>>> plt.plot(freq, sp.real, freq, sp.imag)
[<matplotlib.lines.Line2D object at 0x...>,
 <matplotlib.lines.Line2D object at 0x...>]
>>> plt.show()

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