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    Combined calculation of the Hermite polynomial H_n(z) (and its
    generalization to complex n) and the parabolic cylinder
    function D.
    ���?r���)�prec�пT��exact)�_convert_param�convert�mpq_1_2�pi�isnpint�re�imr�fmul�sqrt�fdiv�fneg�exp�range�len�append�tuple)�ctx�n�z�parabolic_cylinder�ntyp�q�T1�can_use_2f0�expprec�u�w�w2�rw2�nrw2�nw�terms�T2�expu�is                   �=/usr/lib/python3/dist-packages/mpmath/functions/orthogonal.py�_hermite_paramr3s���� � ��#�G�A�t����A��A�	����A� 
�����[�1�c�(�B��A�a�C��	�2�r�1�
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��+�+�q�b�/�+�S�V�V�A�Y��]�+�	�����a��	)�C�F�F�1�I��M���h�h�q�j�2�o�G���H�H�S�X�X�a��w�X�/��d�H�C���H�H�Q�����'��2��H�A��
��	���!�Q�W��	%�B�

�(�(�1�b�w�(�
'�C��8�8�C�t�8�$�D�	���!�4��	 �B����V�a��V�R��a��c�1�a��c�7�^�R��
=�������W�q�!�f�b�"�q��s�A�q��s�G�n�b�$�
>�������_�q��s�C��m�R�!�A�#���A�a�C��	�A�a�C�5�"�
L���B�����w�w�q�z���s�5�z�"�	"�A��!�H�Q�K��N�a��c�!�N��!�H�Q�K���t�$��!�H�Q�K���q�!�	"���<��c�:�����j���fd�gfi|��S)Nc� ��t���d�S)Nr�r3�rr r!s���r2�<lambda>zhermite.<locals>.<lambda>>������Q��1�!=�r4��	hypercomb�rr r!�kwargss``` r2�hermiter?<s����3�=�=�=�r�L�V�L�Lr4c�:�����j���fd�gfi|��S)a8
    Gives the parabolic cylinder function in Whittaker's notation
    `D_n(z) = U(-n-1/2, z)` (see :func:`~mpmath.pcfu`).
    It solves the differential equation

    .. math ::

        y'' + \left(n + \frac{1}{2} - \frac{1}{4} z^2\right) y = 0.

    and can be represented in terms of Hermite polynomials
    (see :func:`~mpmath.hermite`) as

    .. math ::

        D_n(z) = 2^{-n/2} e^{-z^2/4} H_n\left(\frac{z}{\sqrt{2}}\right).

    **Plots**

    .. literalinclude :: /plots/pcfd.py
    .. image :: /plots/pcfd.png

    **Examples**

        >>> from mpmath import *
        >>> mp.dps = 25; mp.pretty = True
        >>> pcfd(0,0); pcfd(1,0); pcfd(2,0); pcfd(3,0)
        1.0
        0.0
        -1.0
        0.0
        >>> pcfd(4,0); pcfd(-3,0)
        3.0
        0.6266570686577501256039413
        >>> pcfd('1/2', 2+3j)
        (-5.363331161232920734849056 - 3.858877821790010714163487j)
        >>> pcfd(2, -10)
        1.374906442631438038871515e-9

    Verifying the differential equation::

        >>> n = mpf(2.5)
        >>> y = lambda z: pcfd(n,z)
        >>> z = 1.75
        >>> chop(diff(y,z,2) + (n+0.5-0.25*z**2)*y(z))
        0.0

    Rational Taylor series expansion when `n` is an integer::

        >>> taylor(lambda z: pcfd(5,z), 0, 7)
        [0.0, 15.0, 0.0, -13.75, 0.0, 3.96875, 0.0, -0.6015625]

    c� ��t���d�S�Nrr7r8s���r2r9zpcfd.<locals>.<lambda>vr:r4r;r=s``` r2�pcfdrC@s���l�3�=�=�=�r�L�V�L�Lr4c�j�|j|�\}}|j||jz
|�S)a�
    Gives the parabolic cylinder function `U(a,z)`, which may be
    defined for `\Re(z) > 0` in terms of the confluent
    U-function (see :func:`~mpmath.hyperu`) by

    .. math ::

        U(a,z) = 2^{-\frac{1}{4}-\frac{a}{2}} e^{-\frac{1}{4} z^2}
            U\left(\frac{a}{2}+\frac{1}{4},
            \frac{1}{2}, \frac{1}{2}z^2\right)

    or, for arbitrary `z`,

    .. math ::

        e^{-\frac{1}{4}z^2} U(a,z) =
            U(a,0) \,_1F_1\left(-\tfrac{a}{2}+\tfrac{1}{4};
            \tfrac{1}{2}; -\tfrac{1}{2}z^2\right) +
            U'(a,0) z \,_1F_1\left(-\tfrac{a}{2}+\tfrac{3}{4};
            \tfrac{3}{2}; -\tfrac{1}{2}z^2\right).

    **Examples**

    Connection to other functions::

        >>> from mpmath import *
        >>> mp.dps = 25; mp.pretty = True
        >>> z = mpf(3)
        >>> pcfu(0.5,z)
        0.03210358129311151450551963
        >>> sqrt(pi/2)*exp(z**2/4)*erfc(z/sqrt(2))
        0.03210358129311151450551963
        >>> pcfu(0.5,-z)
        23.75012332835297233711255
        >>> sqrt(pi/2)*exp(z**2/4)*erfc(-z/sqrt(2))
        23.75012332835297233711255
        >>> pcfu(0.5,-z)
        23.75012332835297233711255
        >>> sqrt(pi/2)*exp(z**2/4)*erfc(-z/sqrt(2))
        23.75012332835297233711255

    )rrCr)r�ar!r>r �_s      r2�pcfurGxs4��X���a� �D�A�q��8�8�Q�B�s�{�{�N�A�&�&r4c������	��j|�\�}�j����j��j�	|dk(rf�j	�dz�rR����	�fd�}�j
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��r"�j
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    Gives the parabolic cylinder function `V(a,z)`, which can be
    represented in terms of :func:`~mpmath.pcfu` as

    .. math ::

        V(a,z) = \frac{\Gamma(a+\tfrac{1}{2}) (U(a,-z)-\sin(\pi a) U(a,z)}{\pi}.

    **Examples**

    Wronskian relation between `U` and `V`::

        >>> from mpmath import *
        >>> mp.dps = 25; mp.pretty = True
        >>> a, z = 2, 3
        >>> pcfu(a,z)*diff(pcfv,(a,z),(0,1))-diff(pcfu,(a,z),(0,1))*pcfv(a,z)
        0.7978845608028653558798921
        >>> sqrt(2/pi)
        0.7978845608028653558798921
        >>> a, z = 2.5, 3
        >>> pcfu(a,z)*diff(pcfv,(a,z),(0,1))-diff(pcfu,(a,z),(0,1))*pcfv(a,z)
        0.7978845608028653558798921
        >>> a, z = 0.25, -1
        >>> pcfu(a,z)*diff(pcfv,(a,z),(0,1))-diff(pcfu,(a,z),(0,1))*pcfv(a,z)
        0.7978845608028653558798921
        >>> a, z = 2+1j, 2+3j
        >>> chop(pcfu(a,z)*diff(pcfv,(a,z),(0,1))-diff(pcfu,(a,z),(0,1))*pcfv(a,z))
        0.7978845608028653558798921

    �Qrc�����j�	dd��}t���z
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�C�M�M�!�R�*�6�*�����Q��C�$5�$5�a�$8�����
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    Gives the parabolic cylinder function `W(a,z)` defined in (DLMF 12.14).

    **Examples**

    Value at the origin::

        >>> from mpmath import *
        >>> mp.dps = 25; mp.pretty = True
        >>> a = mpf(0.25)
        >>> pcfw(a,0)
        0.9722833245718180765617104
        >>> power(2,-0.75)*sqrt(abs(gamma(0.25+0.5j*a)/gamma(0.75+0.5j*a)))
        0.9722833245718180765617104
        >>> diff(pcfw,(a,0),(0,1))
        -0.5142533944210078966003624
        >>> -power(2,-0.25)*sqrt(abs(gamma(0.75+0.5j*a)/gamma(0.25+0.5j*a)))
        -0.5142533944210078966003624

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