o
    lh>                     @   s   d dl mZmZ dd Zedd Zedd Zedd	 Zed
d Zedd Z	 edd Z	edd Z
edd Zedd Zed"ddZed"ddZedd Zedd Zedd  Zd!S )#   )defundefun_wrappedc                 C   s8  |  |\}}| |}| j }|s6d| jg|dgg ||d  gg g df}|r3|d d  || 7  < |fS | | pP| |dkpP| |dkoP| |dk}| jd d }|rv| j| j|||ddd	d
}	| j|| j	d|d|d}
n|}
| j|
|
|d}| j
d||d}| j|d	d
}| j|
d	d
}|rd|
g||gg g || ||d  gg |f}|g}n6d|g||gg g || ||d  gg |f}d| j|g|d ddgg || g||d  gd| g|f}||g}|r| |	}tt|D ]"}|| d d  || 7  < || d | || d d qt|S )z
    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i r0   o/var/www/html/construction_image-detection-poc/venv/lib/python3.10/site-packages/mpmath/functions/orthogonal.py_hermite_param   sB   
&**:
r2   c                         j  fddg fi |S )Nc                         t  dS )Nr   r2   r0   r   r   r   r0   r1   <lambda>>       zhermite.<locals>.<lambda>	hypercombr   r   r   kwargsr0   r6   r1   hermite<   s    r=   c                    r3   )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                      r4   Nr   r5   r0   r6   r0   r1   r7   v   r8   zpcfd.<locals>.<lambda>r9   r;   r0   r6   r1   pcfd@   s    6r?   c                 K   s"   |  |\}}| | | j |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

    )r   r?   r   )r   ar   r<   r   _r0   r0   r1   pcfux   s   ,rB   c                    s     |\}  j j|dkrB d rB fdd} j|g fi |} r@ r@ |}|S  fdd} j|gfi |S )a  
    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

    Qr   c                     s    j ddd} t   d}t  | d}|D ]}|d d |d d |d   q    d j  }|D ]}|d | |d d qJ|| S )	Ny             Tr   r   r                 ?   r   )r   r2   r   expjpir   r   )jzT1termsT2termsTr&   r   r   r"   rr   r0   r1   h   s   "zpcfv.<locals>.hc           
         s     d}  d} |} j |g}| |   dg|   gg |   gg|f}|g  |   ddgd |   gg |  d  gd g|f} |   \}}|d | |d | ||fD ]}	|	d d |	d |   qy||fS )Nr
   r   r   r   rE   )square_exp_argr   r   cospi_sinpir   )
r   r'   r&   elY1Y2csY)r   r"   rL   r   r0   r1   rM      s   
8L)r   r   r   mpq_1_4isintr:   _is_real_type_re)r   r@   r   r<   ntyperM   vr0   rK   r1   pcfv   s    

r]   c                    sT     |\}  fdd} |} r( r( |}|S )aI  
    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

    c                  3   s
      d j  }  d j   d j   d }  jd d|   } d d j     j  } |d  d j   }| |   j  	d  V  | |    j   	d  V  d S )Nr   y               @   r   r   g      ?r
   )
arggammajloggammar   r   r   expjrB   rF   )phi2rhokCr6   r0   r1   r,     s   ,.",4zpcfw.<locals>.terms)r   r   sum_accuratelyrY   rZ   )r   r@   r   r<   rA   r,   r\   r0   r6   r1   pcfw   s   

	
ri   c                    s   |   rd  S |   d r/|  d rtdfdd}| j| gfi |S  fdd}| j|gfi |S )Nr   r   r   z#Gegenbauer function with two limitsc              	      sF   d|  }g g  | g d |g   | g| d gdd  f}|gS Nr   r   r   r0   )r@   a2rJ   r   r   r0   r1   rM   =     8zgegenbauer.<locals>.hc              	      sF   d  }g g | | g| d |g|  | | g d gdd  f}|gS rj   r0   )r   rk   rJ   )r@   r   r0   r1   rM   B  rm   )r   NotImplementedErrorr:   r   r   r@   r   r<   rM   r0   )r@   r   r   r1   
gegenbauer3  s   
rp   c                    s   |   s fdd}| j||gfi |S | s0fdd}| j|| gfi |S | |  || j| d|     d d d fi | S )Nc                    sJ   g g  |  d g| d  d g|    |  d g d gd d ffS Nr   r   r0   )r   r@   bxr0   r1   rM   K  s   Jzjacobi.<locals>.hc                    sF   g g   g| d   |  g|  |  |  d g d gd d ffS rq   r0   )r   r@   )rs   rt   r0   r1   rM   O  s   Fr   r   )r   r:   rX   binomialhyp2f1)r   r   r@   rs   rt   r<   rM   r0   rr   r1   jacobiH  s   

Brw   c                    s$    fdd}| j ||gfi |S )Nc                    s4   g g |   d g| d  d g  g| d gffS r>   r0   )r@   rl   r0   r1   rM   Z  s   4zlaguerre.<locals>.hr9   ro   r0   rl   r1   laguerreU  s   rx   c                 K   s   |  |r1t|}||dk  d@ r1|s|S | |}|d| j d k r%|S |dk r1|  j| 7  _| j| |d dd| d fi |S )Nr   r   
   r   )rX   intmagr	   rv   )r   r   rt   r<   r}   r0   r0   r1   legendre^  s   

&r~   r   c                    s   |  |}|  |}|s| j| fi |S |dkr, fdd}| j|||gfi |S |dkrB fdd}| j|||gfi |S td)Nr   c              	      sP   |d }d  d  g|| gg d| g|  | d gd| gdd   f}|fS Nr   r   r0   r   mgrJ   r   r0   r1   rM   w     Bzlegenp.<locals>.hrE   c              	      sP   |d } d  d g|| gg d| g|  | d gd| gdd   f}|fS r   r0   r   r   r0   r1   rM   }  r   requires type=2 or type=3)r   r~   r:   
ValueErrorr   r   r   r   typer<   rM   r0   r   r1   legenpm  s   

r   c                    s     |}  |}  dv r jS |dkr- fdd} j|||gfi |S |dkr]tdkrJ fdd} j|||gfi |S  fdd} j|||gfi |S td	)
N)r   r   c                    s     |\}}d|  j }|}d }d }|d }d d }	||||gdd|| gg d| g|  | d gd| g|	f}
| ||gd| |g| | d g| | d |d g|  | d g|d g|	f}|
|fS Nr   r   r   )rO   r   )r   r   cossinrU   rT   r@   rs   r&   r'   r#   r-   r   r   r0   r1   rM     s    2zlegenq.<locals>.hrE   r   c                    s     |d jd d gd|  d d|  | d d| d| g| | d g| d gdd|  |  dd|  |  g| d gd f}|gS )Nr   r   r   g      ?ry   )rF   r   )r   r   r#   r   r0   r1   rM     s   &,c           
         s   d  |  j } |}d }d }|d }d d }||||gdd|| gg d| g|  | d gd| g|f}| |||gdd| |g| | d g| | d |d g|  | d g|d g|f}	||	fS r   )sinpir   rF   )
r   r   rU   rT   r@   rs   r&   r'   r#   r-   r   r0   r1   rM     s   
 6r   )r   nanr:   absr   r   r0   r   r1   legenq  s   


r   c                 K   sN   |s|  |rt| |d dkr|d S | j| |dd| d fi |S )Nr   r   r   )r   r   rX   r|   rZ   rv   r   r   rt   r<   r0   r0   r1   chebyt  s   $"r   c                 K   sZ   |s|  |rt| |d dkr|d S |d | j| |d dd| d fi | S )Nr   r   r   )rE   r   r   r   r0   r0   r1   chebyu  s   $.r   c           
         s     |}  |}     |}|o|dk} |}|r;|dk r;|r; j|d  |fi |S dkrJ|rJ|dk rJ jd S |rb|rbt||krY jd S  fdd}	n fdd}	 j|	||gfi |S )Nr   r   rD   c              
      s   t |}d | d|  d  | |   j  | |   d  |dg}d|  |d  ddd| d| d g}||g g ||  | | d g|d g d d ffS )Nr   r   r   r   )r   rc   facr   r   sign)rQ   r   absmrg   Pr   phithetar0   r1   rM     s   ,
,"zspherharm.<locals>.hc                    s     | | d s  | | d s  d| r%dgdgg g g g dffS  d \}}d |  d|  d  j  | | d  | | d |d |d g}ddddd| d| g}||g d| g|  | d gd| g|d ffS )Nr   r   r   r   r   g      )r   cos_sinrc   r   r`   )rQ   r   r   r   rg   r   r   r0   r1   rM     s   2  .)r   rX   	spherharmzeror   r:   )
r   rQ   r   r   r   r<   l_isint	l_naturalm_isintrM   r0   r   r1   r     s"   

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

	r   N)r   )	functionsr   r   r2   r=   r?   rB   r]   ri   rp   rw   rx   r~   r   r   r   r   r   r0   r0   r0   r1   <module>   s>    9

7
.
H
%



5

