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Differential and pseudo-differential operators.
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    Return kth derivative (or integral) of a periodic sequence x.

    If x_j and y_j are Fourier coefficients of periodic functions x
    and y, respectively, then::

      y_j = pow(sqrt(-1)*j*2*pi/period, order) * x_j
      y_0 = 0 if order is not 0.

    Parameters
    ----------
    x : array_like
        Input array.
    order : int, optional
        The order of differentiation. Default order is 1. If order is
        negative, then integration is carried out under the assumption
        that ``x_0 == 0``.
    period : float, optional
        The assumed period of the sequence. Default is ``2*pi``.

    Notes
    -----
    If ``sum(x, axis=0) = 0`` then ``diff(diff(x, k), -k) == x`` (within
    numerical accuracy).

    For odd order and even ``len(x)``, the Nyquist mode is taken zero.

    r��?N���?�c�0�|rt||z|��SdS�Nr)�pow)�k�order�cs   �=/usr/lib/python3/dist-packages/scipy/fftpack/_pseudo_diffs.py�kernelzdiff.<locals>.kernelAs!���
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!�!�1�	�	�	�	��0��6�E�>?�A�A�A��#���%��{���c�1�%�%�K���S��e�a�i�)4�6�6�6�6r%c��t|��}t|��r0t|j||��dt|j||��zzS|�|dzt
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    Return h-Tilbert transform of a periodic sequence x.

    If x_j and y_j are Fourier coefficients of periodic functions x
    and y, respectively, then::

        y_j = sqrt(-1)*coth(j*h*2*pi/period) * x_j
        y_0 = 0

    Parameters
    ----------
    x : array_like
        The input array to transform.
    h : float
        Defines the parameter of the Tilbert transform.
    period : float, optional
        The assumed period of the sequence. Default period is ``2*pi``.

    Returns
    -------
    tilbert : ndarray
        The result of the transform.

    Notes
    -----
    If ``sum(x, axis=0) == 0`` and ``n = len(x)`` is odd, then
    ``tilbert(itilbert(x)) == x``.

    If ``2 * pi * h / period`` is approximately 10 or larger, then
    numerically ``tilbert == hilbert``
    (theoretically oo-Tilbert == Hilbert).

    For even ``len(x)``, the Nyquist mode of ``x`` is taken zero.

    rNrrc�4�|rdt||z��zSdS)Nrr�r�r �hs  r#r$ztilbert.<locals>.kernel�s#���
%��4��!��9�9�}�$��1r%r�r'r))rrrr,r-r
r.r/r0rr1r�	r2r<r3r4r5r6r7r$r+s	         r#rrSs+��H�!�*�*�C��C���1��s�x��F�+�+��G�C�H�a��0�0�0�1�	1���
��E�B�J�����A���A��J�J��1�v���E��}��v�;�;�����
!���� � � ��
!��	�	�	�	��0��F�a�@�@�@�����!�u�
��c�1�%�%�K���S��a�K�P�P�P�Pr%c��t|��}t|��r0t|j||��dt|j||��zzS|�|dzt
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|��dkr|r|���|�|fd�}tj
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    Return inverse h-Tilbert transform of a periodic sequence x.

    If ``x_j`` and ``y_j`` are Fourier coefficients of periodic functions x
    and y, respectively, then::

      y_j = -sqrt(-1)*tanh(j*h*2*pi/period) * x_j
      y_0 = 0

    For more details, see `tilbert`.

    rNrrc�0�|rt||z��SdSrr:r;s  r#r$zitilbert.<locals>.kernel�s!���
"��Q�q�S�	�	�z�!��1r%rr=r))rrrr,r-r
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�a�C��F�6�M���A���A��J�J��!�u���E��}��v�;�;�����
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��c�1�%�%�K���S��a�K�P�P�P�Pr%c���t|��}t|��r,t|j��dt|j��zzSt|��}|�|��}|�Jt|��dkr|r|���|�d�}tj	||d���}|||<t||��}tj||d|���S)a�
    Return Hilbert transform of a periodic sequence x.

    If x_j and y_j are Fourier coefficients of periodic functions x
    and y, respectively, then::

      y_j = sqrt(-1)*sign(j) * x_j
      y_0 = 0

    Parameters
    ----------
    x : array_like
        The input array, should be periodic.
    _cache : dict, optional
        Dictionary that contains the kernel used to do a convolution with.

    Returns
    -------
    y : ndarray
        The transformed input.

    See Also
    --------
    scipy.signal.hilbert : Compute the analytic signal, using the Hilbert
                           transform.

    Notes
    -----
    If ``sum(x, axis=0) == 0`` then ``hilbert(ihilbert(x)) == x``.

    For even len(x), the Nyquist mode of x is taken zero.

    The sign of the returned transform does not have a factor -1 that is more
    often than not found in the definition of the Hilbert transform. Note also
    that `scipy.signal.hilbert` does have an extra -1 factor compared to this
    function.

    rNrc�&�|dkrdS|dkrdSdS)Nrrg�g�)r s r#r$zhilbert.<locals>.kernel�s#���1�u�u��s��Q����t��3r%rr=r))rrrr,r-r.r/r0rr1r)r2r4r5r6r7r$r+s       r#rr�s���N�!�*�*�C��C���6��s�x� � ��G�C�H�$5�$5�!5�5�5��A���A��J�J�q�M�M�E��}��v�;�;�����
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!�	�	�	��0��6�A�>�>�>����q�	��c�1�%�%�K���S��a�K�P�P�P�Pr%c�"�t|��S)z�
    Return inverse Hilbert transform of a periodic sequence x.

    If ``x_j`` and ``y_j`` are Fourier coefficients of periodic functions x
    and y, respectively, then::

      y_j = -sqrt(-1)*sign(j) * x_j
      y_0 = 0

    )r)r2s r#rrs��
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    Return (a,b)-cosh/sinh pseudo-derivative of a periodic sequence.

    If ``x_j`` and ``y_j`` are Fourier coefficients of periodic functions x
    and y, respectively, then::

      y_j = -sqrt(-1)*cosh(j*a*2*pi/period)/sinh(j*b*2*pi/period) * x_j
      y_0 = 0

    Parameters
    ----------
    x : array_like
        The array to take the pseudo-derivative from.
    a, b : float
        Defines the parameters of the cosh/sinh pseudo-differential
        operator.
    period : float, optional
        The period of the sequence. Default period is ``2*pi``.

    Returns
    -------
    cs_diff : ndarray
        Pseudo-derivative of periodic sequence `x`.

    Notes
    -----
    For even len(`x`), the Nyquist mode of `x` is taken as zero.

    rNrrc�V�|r&t||z��t||z��zSdSr)rr�r �a�bs   r#r$zcs_diff.<locals>.kernel@s0���
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    Return (a,b)-sinh/cosh pseudo-derivative of a periodic sequence x.

    If x_j and y_j are Fourier coefficients of periodic functions x
    and y, respectively, then::

      y_j = sqrt(-1)*sinh(j*a*2*pi/period)/cosh(j*b*2*pi/period) * x_j
      y_0 = 0

    Parameters
    ----------
    x : array_like
        Input array.
    a,b : float
        Defines the parameters of the sinh/cosh pseudo-differential
        operator.
    period : float, optional
        The period of the sequence x. Default is 2*pi.

    Notes
    -----
    ``sc_diff(cs_diff(x,a,b),b,a) == x``
    For even ``len(x)``, the Nyquist mode of x is taken as zero.

    rNrrc�T�|r%t||z��t||z��zSdSr)rrrGs   r#r$zsc_diff.<locals>.kernelxs.���
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    Return (a,b)-sinh/sinh pseudo-derivative of a periodic sequence x.

    If x_j and y_j are Fourier coefficients of periodic functions x
    and y, respectively, then::

      y_j = sinh(j*a*2*pi/period)/sinh(j*b*2*pi/period) * x_j
      y_0 = a/b * x_0

    Parameters
    ----------
    x : array_like
        The array to take the pseudo-derivative from.
    a,b
        Defines the parameters of the sinh/sinh pseudo-differential
        operator.
    period : float, optional
        The period of the sequence x. Default is ``2*pi``.

    Notes
    -----
    ``ss_diff(ss_diff(x,a,b),b,a) == x``

    rNrrc�t�|r%t||z��t||z��zSt|��|zS�N)r�floatrGs   r#r$zss_diff.<locals>.kernel�s9���
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    Return (a,b)-cosh/cosh pseudo-derivative of a periodic sequence.

    If x_j and y_j are Fourier coefficients of periodic functions x
    and y, respectively, then::

      y_j = cosh(j*a*2*pi/period)/cosh(j*b*2*pi/period) * x_j

    Parameters
    ----------
    x : array_like
        The array to take the pseudo-derivative from.
    a,b : float
        Defines the parameters of the sinh/sinh pseudo-differential
        operator.
    period : float, optional
        The period of the sequence x. Default is ``2*pi``.

    Returns
    -------
    cc_diff : ndarray
        Pseudo-derivative of periodic sequence `x`.

    Notes
    -----
    ``cc_diff(cc_diff(x,a,b),b,a) == x``

    rNrrc�L�t||z��t||z��zSrO)rrGs   r#r$zcc_diff.<locals>.kernel�s!����!��9�9�T�!�A�#�Y�Y�&�&r%rQ)rrrr,r-r
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    Shift periodic sequence x by a: y(u) = x(u+a).

    If x_j and y_j are Fourier coefficients of periodic functions x
    and y, respectively, then::

          y_j = exp(j*a*2*pi/period*sqrt(-1)) * x_f

    Parameters
    ----------
    x : array_like
        The array to take the pseudo-derivative from.
    a : float
        Defines the parameters of the sinh/sinh pseudo-differential
    period : float, optional
        The period of the sequences x and y. Default period is ``2*pi``.
    rNrrc�&�t||z��SrO)r�r rHs  r#�kernel_realzshift.<locals>.kernel_real����q��s�8�8�Or%c�&�t||z��SrO)rrVs  r#�kernel_imagzshift.<locals>.kernel_imagrXr%rr&rrQ)
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