§
    JŠtjá>  ã                   óV  — d dl mZmZ d„ Zed„ ¦   «         Zed„ ¦   «         Zed„ ¦   «         Zed„ ¦   «         Zed„ ¦   «         Z	 ed„ ¦   «         Z	ed	„ ¦   «         Z
ed
„ ¦   «         Zed„ ¦   «         Zedd„¦   «         Zedd„¦   «         Zed„ ¦   «         Zed„ ¦   «         Zed„ ¦   «         ZdS )é   )ÚdefunÚdefun_wrappedc                 óB  — |                       |¦  «        \  }}|                      |¦  «        }| j         }|s7d| j        g|dgg ||dz
  z  gg g df}|r|d         dxx         ||z  z  cc<   |fS |                      | ¦  «        pJ|                      |¦  «        dk    p1|                      |¦  «        dk    o|                      |¦  «        dk    }| j        dz  dz   }|r\|                      |                      |||¬¦  «        dd	¬
¦  «        }	|                      ||  	                    d|¬¦  «        |¬¦  «        }
n|}
|                      |
|
|¬¦  «        }|  
                    d||¬¦  «        }|                      |d	¬
¦  «        }|                      |
d	¬
¦  «        }|rd|
g||gg g ||z  ||dz
  z  gg |f}|g}nBd|g||gg g ||z  ||dz
  z  gg |f}d| j        |g|dz   ddgg ||z  g||dz
  z  gd|z
  g|f}||g}|r•|                      |	¦  «        }t          t          |¦  «        ¦  «        D ]c}||         d         dxx         ||z  z  cc<   ||         d                              |¦  «         ||         d                              d¦  «         Œdt!          |¦  «        S )z”
    Combined calculation of the Hermite polynomial H_n(z) (and its
    generalization to complex n) and the parabolic cylinder
    function D.
    é   ç      à?r   é    é   é   )Úprecç      Ð¿T©Úexact)Ú_convert_paramÚconvertÚmpq_1_2ÚpiÚisnpintÚreÚimr   ÚfmulÚsqrtÚfdivÚfnegÚexpÚrangeÚlenÚappendÚtuple)ÚctxÚnÚzÚparabolic_cylinderÚntypÚqÚT1Úcan_use_2f0ÚexpprecÚuÚwÚw2Úrw2Únrw2ÚnwÚtermsÚT2ÚexpuÚis                      úY/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/mpmath/functions/orthogonal.pyÚ_hermite_paramr3      sô  € ð × Ò  Ñ#Ô#�G€A€tØ�Š�A‰Œ€AØ	Œˆ€Að  ð Ø�”ˆ[˜1˜c˜( B¨¨A¨a©C©¨	°2°r¸1Ð<ˆØð 	ØˆqŒE�!ˆHˆHŒH˜˜!™‰OˆHˆH‰HØˆsˆ
Ø—+’+˜q˜b‘/”/ð + S§V¢V¨A¡Y¤Y°¢]ð +Ø	�Š�‰Œ�aŠÐ	)˜CŸFšF 1™IœI¨šMð àŒh�q‰j˜2‰o€GØð Ø�HŠH�S—X’X˜a  w�XÑ/Ô/°¸dˆHÑCÔCˆØ�HŠH�Q˜Ÿš ¨'˜Ñ2Ô2¸ˆHÑAÔAˆˆàˆØ	�Š�!�Q˜WˆÑ	%Ô	%€BØ
�(Š(�1�b˜wˆ(Ñ
'Ô
'€CØ�8Š8�C˜tˆ8Ñ$Ô$€DØ	�Š�!˜4ˆÑ	 Ô	 €BØð Ø�ˆV�a˜�V˜R  a¨¡c¨1¨a°©c©7 ^°R¸Ð=ˆØ�ˆˆà�ˆW�q˜!�f˜b " q¨¡s¨A¨q°©s©G n°b¸$Ð>ˆØ�”˜ˆ_˜q ™s C¨˜m¨R°!°A±#°¸¸A¸a¹C¹¸	ÀAÀaÁCÀ5È"ÐLˆØ�B�ˆàð "Ø�wŠw�q‰zŒzˆÝ•s˜5‘z”zÑ"Ô"ð 	"ð 	"ˆAØ�!ŒH�QŒK˜ˆNˆNŒN˜a ™cÑ!ˆNˆN‰NØ�!ŒH�QŒK×Ò˜tÑ$Ô$Ð$Ø�!ŒH�QŒK×Ò˜qÑ!Ô!Ð!Ð!Ý�‰<Œ<Ðó    c                 ó0   ‡ ‡‡—  ‰ j         ˆ ˆˆfd„g fi |¤ŽS )Nc                  ó(   •— t          ‰ ‰‰d¦  «        S )Nr   ©r3   ©r   r    r!   s   €€€r2   ú<lambda>zhermite.<locals>.<lambda>>   ó   ø€ ¥°°Q¸¸1Ñ!=Ô!=€ r4   ©Ú	hypercomb©r   r    r!   Úkwargss   ``` r2   Úhermiter?   <   s1   øøø€ àˆ3Œ=Ð=Ð=Ð=Ð=Ð=Ð=¸rÐLÐLÀVÐLÐLÐLr4   c                 ó0   ‡ ‡‡—  ‰ j         ˆ ˆˆfd„g fi |¤Ž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                  ó(   •— t          ‰ ‰‰d¦  «        S ©Nr   r7   r8   s   €€€r2   r9   zpcfd.<locals>.<lambda>v   r:   r4   r;   r=   s   ``` r2   ÚpcfdrC   @   s4   øøø€ ðl ˆ3Œ=Ð=Ð=Ð=Ð=Ð=Ð=¸rÐLÐLÀVÐLÐLÐLr4   c                 óp   — |                       |¦  «        \  }}|                      | | j        z
  |¦  «        S )aå  
    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   rC   r   )r   Úar!   r>   r    Ú_s         r2   ÚpcfurG   x   s8   € ðX ×Ò˜aÑ Ô �D€A€qØ�8Š8�Q�B�s”{‘N AÑ&Ô&Ð&r4   c                 ó   ‡ ‡‡‡‡	— ‰                       |¦  «        \  Š}‰                      ‰¦  «        Š‰ j        Š‰ j        Š	|dk    rq‰                      ‰dz  ¦  «        rYˆ ˆˆˆ	ˆfd„} ‰ j        |g fi |¤Ž}‰                      ‰¦  «        r*‰                      ‰¦  «        r‰                      |¦  «        }|S ˆ ˆˆ	ˆfd„} ‰ j        |‰gfi |¤ŽS )aÞ  
    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

    ÚQr   c                  ó.  •— ‰                      ‰	dd¬¦  «        } t          ‰‰ ‰z
  ‰	d¦  «        }t          ‰‰‰z
  | d¦  «        }|D ]V}|d                              d¦  «         |d                              d¦  «         |d                              ‰‰z
  ¦  «         ŒW‰                     ‰‰z  ‰z
  ¦  «        ‰                     d‰j        z  ¦  «        z  }|D ]8}|d                              |¦  «         |d                              d¦  «         Œ9||z   S )	Ny       €      ð¿Tr   r   r   ù              ð?é   r   )r   r3   r   Úexpjpir   r   )
ÚjzÚT1termsÚT2termsÚTr(   r   r    r$   Úrr!   s
        €€€€€r2   Úhzpcfv.<locals>.hÍ   s  ø€ Ø—’˜!˜S¨�Ñ-Ô-ˆBÝ$ S¨1¨"¨Q©$°°1Ñ5Ô5ˆGÝ$ S¨!¨A©#¨r°1Ñ5Ô5ˆGØð !ð !�Ø�!”—’˜B‘”�Ø�!”—’˜A‘”�Ø�!”—’˜A˜a™CÑ Ô Ð Ð Ø—
’
˜A˜a™C ™EÑ#Ô# c§h¢h¨q°´©xÑ&8Ô&8Ñ8ˆAØð ð �Ø�!”—’˜A‘”�Ø�!”—’˜A‘”��Ø˜WÑ$Ð$r4   c                 ó”  •— ‰
                      ‰d¦  «        }‰
                      ‰d¦  «        }‰
                     |¦  «        }‰
j        ‰‰
                     |¦  «        g}|‰ | ‰z  ‰z   dg‰‰| z  z
  gg ‰| z  ‰z   g‰g|f}|‰gz   ‰ | ‰z  ‰z
  ddgd‰z
  ‰| z  z
  gg ‰| z  dz   ‰z
  gd‰z   g|f}‰
                     ‰‰| z  z   ¦  «        \  }}|d                              |¦  «         |d                              |¦  «         ||fD ];}	|	d                              d¦  «         |	d                              ‰| z
  ¦  «         Œ<||fS )Nr   r   r   r   rL   )Úsquare_exp_argr   r   Úcospi_sinpir   )r    r)   r(   ÚeÚlÚY1ÚY2ÚcÚsÚYr   r$   rR   r!   s             €€€€r2   rS   zpcfv.<locals>.hß   sl  ø€ Ø×"Ò" 1 eÑ,Ô,ˆAØ×"Ò" 1 cÑ*Ô*ˆAØ—’˜‘
”
ˆAØ”˜˜CŸGšG A™JœJÐ'ˆAØ�a�R˜˜1™˜Q™ �N Q q¨¡s¡U G¨R°!°A±#°a±%°¸1¸#¸qÐ@ˆBØ�a�S‘˜A˜2˜q ™s 1™u a¨Ð+¨a°©c°!°A±#©g¨Y¸¸Q¸q¹SÀ¹UÀ1¹W¸IÈÈ!ÉÀuÈaÐOˆBØ—?’? 1 Q q¡S¡5Ñ)Ô)‰DˆAˆqØˆqŒE�LŠL˜‰OŒOˆOØˆqŒE�LŠL˜‰OŒOˆOØ˜"�Xð !ð !�Ø�!”—’˜A‘”�Ø�!”—’˜A˜a™CÑ Ô Ð Ð Ø�r�6ˆMr4   )r   r   r   Úmpq_1_4Úisintr<   Ú_is_real_typeÚ_re)
r   rE   r!   r>   ÚntyperS   Úvr    r$   rR   s
   ` `    @@@r2   Úpcfvrd   §   s*  øøøøø€ ð@ ×!Ò! !Ñ$Ô$�H€A€uØ�Š�A‰Œ€AØŒ€AØŒ€AØ�‚|€|˜Ÿ	š	 ! A¡#™œ€|ð	%ð 	%ð 	%ð 	%ð 	%ð 	%ð 	%ð 	%ð 	%ð ˆCŒM˜!˜RÐ*Ð* 6Ð*Ð*ˆØ×Ò˜QÑÔð 	 C×$5Ò$5°aÑ$8Ô$8ð 	Ø—’˜‘
”
ˆAØˆð	ð 	ð 	ð 	ð 	ð 	ð 	ð 	ð ˆsŒ}˜Q  Ð.Ð. vÐ.Ð.Ð.r4   c                 ó  ‡ ‡‡— ‰                       |¦  «        \  Š}‰                      ‰¦  «        Šˆ ˆˆf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   ót  •K  — ‰                      ‰                     d‰j        ‰z  z   ¦  «        ¦  «        } ‰                     d‰j        ‰z  z   ¦  «        ‰                     d‰j        ‰z  z
  ¦  «        z
  dz  } ‰j        dz  d| z  z   }‰                     d‰                     d‰j        z  ‰z  ¦  «        z   ¦  «        ‰                     ‰j        ‰z  ¦  «        z
  }‰                     |dz  ¦  «        ‰                     d‰j        z  ‰z  ¦  «        z  }|‰                     |¦  «        z  ‰                     ‰j        ‰z  ‰‰ 	                    d¦  «        z  ¦  «        z  V — |‰                     | ¦  «        z  ‰                     ‰j         ‰z  ‰‰ 	                    d¦  «        z  ¦  «        z  V — d S )Nr   y               @é   r   r   g      Ð?r   )
ÚargÚgammaÚjÚloggammar   r   r   ÚexpjrG   rM   )Úphi2ÚrhoÚkÚCr   r    r!   s       €€€r2   r.   zpcfw.<locals>.terms  s‚  øè è € Ø�wŠw�s—y’y  s¤u¨Q¡w¡Ñ/Ô/Ñ0Ô0ˆØ—’˜S ¤ q¡™[Ñ)Ô)¨C¯LªL¸¸S¼UÀ1¹W¹Ñ,EÔ,EÑEÀrÑIˆØŒf�Q‰h˜˜T™Ñ!ˆà�HŠH�Q˜Ÿš  3¤6¡¨!¡Ñ,Ô,Ñ,Ñ-Ô-°·²¸¼¸q¹Ñ0AÔ0AÑAˆØ�HŠH�Q�q‘S‰MŒM˜CŸGšG D¨¬¡K°¡MÑ2Ô2Ñ2ˆØ�#—(’(˜3‘-”-Ñ #§(¢(¨3¬5°©7°A°c·j²jÀÑ6GÔ6GÑ4GÑ"HÔ"HÑHÐHÐHÐHØ�#—(’(˜C˜4‘.”.Ñ  3§8¢8¨S¬U¨F°1©H°a¸¿
º
À4Ñ8HÔ8HÑ6HÑ#IÔ#IÑIÐIÐIÐIÐIÐIr4   )r   r   Úsum_accuratelyr`   ra   )r   rE   r!   r>   rF   r.   rc   r    s   ` `    @r2   Úpcfwrr   ð   s«   øøø€ ð, ×Ò˜aÑ Ô �D€A€qØ�Š�A‰Œ€AðJð Jð Jð Jð Jð Jð Jð 	×Ò˜5Ñ!Ô!€AØ
×Ò˜ÑÔð  × 1Ò 1°!Ñ 4Ô 4ð Ø�GŠG�A‰JŒJˆØ€Hr4   c                 ó  ‡‡‡— |                       ‰¦  «        rd‰‰z   z  S |                       ‰dz   ¦  «        r=|                       ‰dz   ¦  «        rt          d¦  «        ‚ˆˆfd„} | j        |‰gfi |¤ŽS ˆˆfd„} | j        |‰gfi |¤ŽS )Nr   r   r   z#Gegenbauer function with two limitsc           	      óX   •— d| z  }g g ‰|z   g‰dz   |g‰ ‰|z   g| dz   gdd‰z
  z  f}|gS ©Nr   r   r   © )rE   Úa2rQ   r    r!   s      €€r2   rS   zgegenbauer.<locals>.h=  sM   ø€ Ø�1‘ˆBØ�B˜˜2™˜  1¡ b 	¨Q¨B°°"±¨:¸¸#¹°wÀÀQÀqÁSÁ	ÐIˆAØ�3ˆJr4   c           	      óX   •— d‰z  }g g | |z   g| dz   |g|  | |z   g‰dz   gdd‰z
  z  f}|gS ru   rv   )r    rw   rQ   rE   r!   s      €€r2   rS   zgegenbauer.<locals>.hB  sM   ø€ Øˆq‰SˆØ��Q�r‘T�F˜Q˜q™S "˜I¨¨¨A¨b©D z°A°c±E°7¸CÀÀ1Á¹IÐEˆØˆsˆ
r4   )r   ÚNotImplementedErrorr<   ©r   r    rE   r!   r>   rS   s    ```  r2   Ú
gegenbauerr{   3  sä   øøø€ ð ‡{‚{�1�~„~ð Ø�!�A‘#‰wˆØ
‡{‚{�1�S‘5ÑÔð 	/ð �;Š;�q˜‘sÑÔð 	MÝ%Ð&KÑLÔLÐLð	ð 	ð 	ð 	ð 	ð 	ð ˆsŒ}˜Q  Ð.Ð. vÐ.Ð.Ð.ðð ð ð ð ð ð ˆ3Œ=˜˜Q˜CÐ*Ð* 6Ð*Ð*Ð*r4   c                 ó4  ‡‡‡— |                       ‰¦  «        sˆˆˆfd„} | j        ||gfi |¤ŽS |                      ‰¦  «        sˆˆfd„} | j        ||‰gfi |¤ŽS |                      |‰z   |¦  «         | j        | d|z   ‰z   ‰z   ‰dz   d‰z
  dz  fi |¤Žz  S )Nc                 ób   •— g g ‰| z   dz   g| dz   ‰dz   g|  ‰‰z   | z   dz   g‰dz   gd‰z
  dz  ffS ©Nr   r   rv   ©r    rE   ÚbÚxs    €€€r2   rS   zjacobi.<locals>.hK  sQ   ø€ Ø˜˜a ™c !™e˜W q¨¡s¨A¨a©C j°A°2°q¸±s¸1±u¸Q±w°-À!ÀAÁ#ÀÈÈ1ÉÈcÉ	ÐRÐTÐTr4   c                 óZ   •— g g ‰ g| dz   ‰ | z
  g|  |‰z   | z   dz   g‰dz   g‰dz   dz  ffS r~   rv   r   s     €€r2   rS   zjacobi.<locals>.hO  sM   ø€ Ø˜˜q˜b˜T A a¡C¨!¨¨A© ;°!°°Q°q±S¸±U¸1±W°ÀÀ!Á¸uÀqÈÁsÈCÁiÐPÐRÐRr4   r   r   )r   r<   r_   ÚbinomialÚhyp2f1)r   r    rE   r€   r�   r>   rS   s     ```  r2   Újacobir…   H  s  øøø€ à�;Š;�q‰>Œ>ð /ð	Uð 	Uð 	Uð 	Uð 	Uð 	Uð 	UàˆsŒ}˜Q  Ð.Ð. vÐ.Ð.Ð.Ø�9Š9�Q‰<Œ<ð 2ð	Sð 	Sð 	Sð 	Sð 	Sð 	SàˆsŒ}˜Q  A Ð1Ð1¨&Ð1Ð1Ð1à�<Š<˜˜!™˜AÑÔ  ¤¨Q¨B¨q°©s°1©u°Q©w°q¸±s¸A¸a¹CÀ¹7Ð!MÐ!MÀfÐ!MÐ!MÑMÐMr4   c                 ó2   ‡‡— ˆˆfd„} | j         ||gfi |¤ŽS )Nc                 óB   •— g g | ‰z   dz   g| dz   ‰dz   g‰ g| dz   g‰ffS rB   rv   )rE   r    r!   s    €€r2   rS   zlaguerre.<locals>.hZ  s;   ø€ Ø�R˜!˜A™#˜a™%˜ 1 Q¡3¨¨!© *°¨r¨d°Q°q±S°E¸1Ð=Ð?Ð?r4   r;   rz   s    ` `  r2   Úlaguerrerˆ   U  sF   øø€ ð
@ð @ð @ð @ð @ð @àˆ3Œ=˜˜Q˜CÐ*Ð* 6Ð*Ð*Ð*r4   c                 ó  — |                       |¦  «        r^t          |¦  «        }||dk     z   dz  rC|s|S |                      |¦  «        }|d| j        z  dz
  k     r|S |dk     r| xj        | z  c_         | j        | |dz   dd|z
  dz  fi |¤ŽS )Nr   r   éþÿÿÿé
   éûÿÿÿr   )r_   ÚintÚmagr   r„   )r   r    r�   r>   rŽ   s        r2   Úlegendrer�   ^  s³   € à
‡y‚y��|„|ð 
!Ý�‰FŒFˆà��Q’‰K˜1Ñð 	!Øð Ø�Ø—'’'˜!‘*”*ˆCØ�R˜œ‘[ ‘^Ò#Ð#Ø�Ø�RŠxˆxØ�”˜S˜DÑ �”Øˆ3Œ:�q�b˜˜1™˜Q  !¡ Q™wÐ1Ð1¨&Ð1Ð1Ð1r4   r   c                 ó  ‡— |                       |¦  «        }|                       |¦  «        }|s | j        |‰fi |¤ŽS |dk    rˆfd„} | j        |||gfi |¤ŽS |dk    rˆfd„} | j        |||gfi |¤ŽS t          d¦  «        ‚)Nr   c           	      ód   •— |dz  }d‰z   d‰z
  g|| gg d|z
  g|  | dz   gd|z
  gdd‰z
  z  f}|fS ©Nr   r   rv   ©r    ÚmÚgrQ   r!   s       €r2   rS   zlegenp.<locals>.hw  óW   ø€ Ø�#‘ˆAØ�1‘�a˜‘c�
˜Q  ˜G R¨!¨A©#¨°!°°Q°q±S°	¸A¸a¹C¸5À#ÀqÈÁsÁ)ÐKˆAØ�4ˆKr4   rL   c           	      ód   •— |dz  }‰dz   ‰dz
  g|| gg d|z
  g|  | dz   gd|z
  gdd‰z
  z  f}|fS r’   rv   r“   s       €r2   rS   zlegenp.<locals>.h}  r–   r4   úrequires type=2 or type=3)r   r�   r<   Ú
ValueError©r   r    r”   r!   Útyper>   rS   s      `   r2   Úlegenprœ   m  sÞ   ø€ ð 	�Š�A‰Œ€AØ�Š�A‰Œ€Aàð ,ØˆsŒ|˜A˜qÐ+Ð+ FÐ+Ð+Ð+àˆq‚y€yð	ð 	ð 	ð 	ð 	ð ˆsŒ}˜Q  1 Ð0Ð0¨Ð0Ð0Ð0Øˆq‚y€yð	ð 	ð 	ð 	ð 	ð ˆsŒ}˜Q  1 Ð0Ð0¨Ð0Ð0Ð0Ý
Ð0Ñ
1Ô
1Ð1r4   c                 ó€  ‡ ‡— ‰                       |¦  «        }‰                       |¦  «        }‰                       ‰¦  «        Š‰dv r‰ j        S |dk    rˆ ˆfd„} ‰ j        |||gfi |¤ŽS |dk    rAt          ‰¦  «        dk    rˆ ˆfd„} ‰ j        |||gfi |¤ŽS ˆ ˆfd„} ‰ j        |||gfi |¤ŽS t	          d¦  «        ‚)	N)r   éÿÿÿÿr   c                 ó"  •— ‰                      |¦  «        \  }}d|z  ‰j        z  }|}d‰z   }d‰z
  }|dz  }d‰z
  dz  }	||||gdd|| gg d|z
  g|  | dz   gd|z
  g|	f}
| ||gd| |g| |z   dz   g| |z
  dz   |dz   g|  | dz   g|dz   g|	f}|
|fS ©Nr   r   rž   )rV   r   )r    r”   ÚcosÚsinr\   r[   rE   r€   r(   r)   r%   r/   r   r!   s               €€r2   rS   zlegenq.<locals>.h�  sî   ø€ Ø—’ qÑ)Ô)‰HˆC�Ø�C‘˜#œ&Ñ ˆAØˆAØ�!‘ˆAØ�!‘ˆAØ�!‘ˆAØ�1‘�a‘ˆAØ�Q˜˜1�  A q¨1¨"˜~¨r°A°a±C°5Ø��Q�q‘S�	˜A˜a™C˜5 !ð$ˆBà�"�a˜�˜b 1 " a˜[¨1¨Q©3¨q©5¨'°A°a±C¸±E¸1¸Q¹3°<Ø��Q�q‘S�	˜A˜a™C˜5 !ð$ˆBà�r�6ˆMr4   rL   r   c                 óÞ   •— ‰                      |¦  «        d‰j        ‰‰dz
  ‰dz   gd|  dz
  d|  |z
  dz
  d|z  d|z  g| |z   dz   g| dz   gdd| z   |z   z  dd| z   |z   z  g| dz   g‰dz  f}|gS )Nr   r   r   g      ø?rŠ   )rM   r   )r    r”   r%   r   r!   s      €€r2   rS   zlegenq.<locals>.h¢  s£   ø€ Ø—j’j ‘m”m Q¨¬°°1°Q±3¸¸!¹Ð<Ø˜!˜˜A™˜s Q B q¡D¨¡F¨C°©E°3°q±5Ð9Ø˜‘c˜!‘e�W˜q ™u˜gØ˜1˜Q™3˜q™5‘k 3¨¨!©¨A©¡;Ð/°!°C±%°¸!¸b¹'ðB�ð �t�r4   c                 óB  •— d‰
                      |¦  «        z  ‰
j        z  }‰
                     |¦  «        }d‰z   }‰dz
  }|dz  }d‰z
  dz  }||||gdd|| gg d|z
  g|  | dz   gd|z
  g|f}| |||gdd| |g| |z   dz   g| |z
  dz   |dz   g|  | dz   g|dz   g|f}	||	fS r    )Úsinpir   rM   )r    r”   r\   r[   rE   r€   r(   r)   r%   r/   r   r!   s             €€r2   rS   zlegenq.<locals>.h«  sò   ø€ Ø˜Ÿ	š	 !™œÑ$ s¤vÑ-�Ø—J’J˜q‘M”M�Ø�a‘C�Ø�a‘C�Ø�a‘C�Ø�q‘S˜!‘G�Ø˜˜A˜q�\ B¨¨1¨q¨b >°2¸¸!¹°uØ�R˜˜1™�I  !¡˜u að(�à�b˜!˜Q �] R¨¨Q¨B° N°Q°q±S¸±U°G¸aÀ¹cÀ!¹eÀQÀqÁS¸\Ø�R˜˜1™�I  !¡˜u að(�à˜2�v�r4   r˜   )r   Únanr<   Úabsr™   rš   s   `  `   r2   Úlegenqr¨   „  s7  øø€ ð 	�Š�A‰Œ€AØ�Š�A‰Œ€AØ�Š�A‰Œ€AØˆG€|€|ð ŒwˆØˆq‚y€yð	ð 	ð 	ð 	ð 	ð 	ð ˆsŒ}˜Q  A Ð1Ð1¨&Ð1Ð1Ð1Øˆq‚y€yõ ˆq‰6Œ6�AŠ:ˆ:ðð ð ð ð ð ð !�3”=  Q¨ FÐ5Ð5¨fÐ5Ð5Ð5ðð ð ð ð ð ð !�3”=  Q¨ FÐ5Ð5¨fÐ5Ð5Ð5Ý
Ð0Ñ
1Ô
1Ð1r4   c                 ó¼   — |sC|                       |¦  «        r.t          |                      |¦  «        ¦  «        dz  dk    r|dz  S  | j        | |dd|z
  dz  fi |¤ŽS )Nr   r   r   )r   r   ©r_   r�   ra   r„   ©r   r    r�   r>   s       r2   Úchebytr¬   º  sm   € àð �3—9’9˜Q‘<”<ð ¥C¨¯ª°©
¬
¡O¤O°aÑ$7¸1Ò$<Ð$<Ø�1‰uˆØˆ3Œ:�q�b˜˜5 ! A¡# q¡Ð3Ð3¨FÐ3Ð3Ð3r4   c                 óÎ   — |sC|                       |¦  «        r.t          |                      |¦  «        ¦  «        dz  dk    r|dz  S |dz    | j        | |dz   dd|z
  dz  fi |¤Žz  S )Nr   r   r   )rL   r   rª   r«   s       r2   Úchebyur®   À  sz   € àð �3—9’9˜Q‘<”<ð ¥C¨¯ª°©
¬
¡O¤O°aÑ$7¸1Ò$<Ð$<Ø�1‰uˆØˆa‰C�:�3”:˜q˜b ! A¡# u¨q°©s°A©gÐ@Ð@¸Ð@Ð@Ñ@Ð@r4   c                 ó  ‡ ‡‡— ‰                       |¦  «        }‰                       |¦  «        }‰                       ‰¦  «        Š‰                       ‰¦  «        Š‰                      |¦  «        }|o|dk    }‰                      |¦  «        }|r|dk     r|r ‰ j        |dz    |‰‰fi |¤ŽS ‰dk    r|r|dk     r
‰ j        dz  S |r'|r%t	          |¦  «        |k    r
‰ j        dz  S ˆ ˆˆfd„}	nˆ ˆˆfd„}	 ‰ j        |	||gfi |¤ŽS )Nr   r   rK   c           
      óê  •— t          |¦  «        }d‰                     |‰z  ¦  «        d| z  dz   ‰                     | |z   ¦  «        z  ‰j        z  ‰                     | |z
  ¦  «        z  ‰                     ‰¦  «        dz  ‰                     |¦  «        dg}d|z  ‰                     |¦  «        dz   z  ddd|z  d| dz
  g}||g g || z
  | |z   dz   g|dz   g‰                     d‰z  ¦  «        dz  ffS )Nrž   r   r   r   )r§   rl   Úfacr   r¢   Úsign)rX   r”   Úabsmrp   ÚPr   ÚphiÚthetas        €€€r2   rS   zspherharm.<locals>.hØ  s  ø€ Ý�q‘6”6ˆDØ�S—X’X˜a ™e‘_”_Ø�A‘#�a‘%˜Ÿš  4¡™œÑ(¨¬Ñ/°·²¸¸$¹±´Ñ?Ø—’˜‘” Ñ"Ø—’˜‘” ð#ˆAð �Q‘˜Ÿš ™œ A™Ñ&¨¨3°°D±¸"¸t¸eÀA¹gÐFˆAØ˜˜2˜r D¨¡F¨A¨d©F°1©HÐ#5¸¸Q¹°xØ—’˜˜E™	Ñ"Ô" AÑ%ð'ð )ð )r4   c                 ó  •— ‰                      | |z
  dz   ¦  «        s3‰                      | |z   dz   ¦  «        s‰                      d|z
  ¦  «        rdgdgg g g g dffS ‰                     d‰z  ¦  «        \  }}d‰                     |‰z  ¦  «        z  d| z  dz   ‰j        z  ‰                     | |z
  dz   ¦  «        ‰                     | |z   dz   ¦  «        |dz  |dz  g}ddddd|z  d|z  g}||g d|z
  g|  | dz   gd|z
  g|dz  ffS )Nr   r   rž   r   r   g      à¿)r   Úcos_sinrl   r   ri   )	rX   r”   r¡   r¢   rp   r´   r   rµ   r¶   s	         €€€r2   rS   zspherharm.<locals>.hä  s;  ø€ Ø�{Š{˜1˜Q™3˜q™5Ñ!Ô!ð 9 S§[¢[°°1±°Q±Ñ%7Ô%7ð 9¸3¿;º;ÀqÈÁsÑ;KÔ;Kð 9Ø˜˜r˜d B¨¨B°°AÐ6Ð8Ð8Ø—{’{ 3 u¡9Ñ-Ô-‰HˆC�Ø�S—X’X˜a ™e‘_”_Ñ$ q¨¡s¨1¡u¨c¬f¡nØ—’˜1˜Q™3˜q™5Ñ!Ô! 3§9¢9¨Q¨q©S°©UÑ#3Ô#3Ø�a‘˜˜a™ð!ˆAð �C˜˜d C¨¡E¨4°©6Ð2ˆAØ˜˜2  !¡˜u¨ r¨!¨A©# h°°1±°°s¸A±vÐ>Ð@Ð@r4   )r   r_   Ú	spherharmÚzeror§   r<   )
r   rX   r”   r¶   rµ   r>   Úl_isintÚ	l_naturalÚm_isintrS   s
   `  ``     r2   r¹   r¹   Æ  sl  øøø€ à�Š�A‰Œ€AØ�Š�A‰Œ€AØ�KŠK˜ÑÔ€EØ
�+Š+�cÑ
Ô
€CØ�iŠi˜‰lŒl€GØÐ"˜A šF€IØ�iŠi˜‰lŒl€GØð >�1�q’5�5˜W�5ØˆsŒ}˜q ™s˜V Q¨¨sÐ=Ð=°fÐ=Ð=Ð=Ø�‚z€z�g€z ! a¢% %ØŒx˜"‰}ÐØð A�Wð AÝˆq‰6Œ6�AŠ:ˆ:Ø”8˜b‘=Ð ð	)ð 	)ð 	)ð 	)ð 	)ð 	)ð 	)ð 	)ð	Að 	Að 	Að 	Að 	Að 	Að 	Að ˆ3Œ=˜˜Q˜q˜EÐ,Ð, VÐ,Ð,Ð,r4   N)r   )Ú	functionsr   r   r3   r?   rC   rG   rd   rr   r{   r…   rˆ   r�   rœ   r¨   r¬   r®   r¹   rv   r4   r2   ú<module>r¿      sÔ  ðØ +Ð +Ð +Ð +Ð +Ð +Ð +Ð +ð7ð 7ð 7ðr ðMð Mñ „ðMð ð5Mð 5Mñ „ð5Mðn ð,'ð ,'ñ „ð,'ð\ ðE/ð E/ñ „ðE/ðP ð#ð #ñ „ð#ðJð: ð+ð +ñ „ð+ð( ð
Nð 
Nñ „ð
Nð ð+ð +ñ „ð+ð ð2ð 2ñ „ð2ð ð2ð 2ð 2ñ „ð2ð, ð32ð 32ð 32ñ „ð32ðj ð4ð 4ñ „ð4ð
 ðAð Añ „ðAð
 ð&-ð &-ñ „ð&-ð &-ð &-r4   