
        The "backward", "forward" values were added.

out : complex ndarray, optional
    If provided, the result will be placed in this array. It should be
    of the appropriate shape and dtype.

    .. versionadded:: 2.0.0

Returns
-------
out : complex ndarray
    The truncated or zero-padded input, transformed along the axis
    indicated by `axis`, or the last one if `axis` is not specified.
    If `n` is even, the length of the transformed axis is ``(n/2)+1``.
    If `n` is odd, the length is ``(n+1)/2``.

Raises
------
IndexError
    If `axis` is not a valid axis of `a`.

See Also
--------
numpy.fft : For definition of the DFT and conventions used.
irfft : The inverse of `rfft`.
fft : The one-dimensional FFT of general (complex) input.
fftn : The *n*-dimensional FFT.
rfftn : The *n*-dimensional FFT of real input.

Notes
-----
When the DFT is computed for purely real input, the output is
Hermitian-symmetric, i.e. the negative frequency terms are just the complex
conjugates of the corresponding positive-frequency terms, and the
negative-frequency terms are therefore redundant.  This function does not
compute the negative frequency terms, and the length of the transformed
axis of the output is therefore ``n//2 + 1``.

When ``A = rfft(a)`` and fs is the sampling frequency, ``A[0]`` contains
the zero-frequency term 0*fs, which is real due to Hermitian symmetry.

If `n` is even, ``A[-1]`` contains the term representing both positive
and negative Nyquist frequency (+fs/2 and -fs/2), and must also be purely
real. If `n` is odd, there is no term at fs/2; ``A[-1]`` contains
the largest positive frequency (fs/2*(n-1)/n), and is complex in the
general case.

If the input `a` contains an imaginary part, it is silently discarded.

Examples
--------
>>> import numpy as np
>>> np.fft.fft([0, 1, 0, 0])
array([ 1.+0.j,  0.-1.j, -1.+0.j,  0.+1.j]) # may vary
>>> np.fft.rfft([0, 1, 0, 0])
array([ 1.+0.j,  0.-1.j, -1.+0.j]) # may vary

Notice how the final element of the `fft` output is the complex conjugate
of the second element, for real input. For `rfft`, this symmetry is
exploited to compute only the non-negative frequency terms.

TrK