Ë
    3^(há>  ã                   ó   — 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y)é   )ÚdefunÚdefun_wrappedc                 óž  — | j                  |«      \  }}| j                  |«      }| j                   }|s6d| j                  g|dgg ||dz
  z  gg g df}|r|d   dxx   ||z  z  cc<   |fS | j	                  | «      xs@ | j                  |«      dkD  xs* | j                  |«      dk(  xr | j                  |«      dkD  }| j                  dz  dz   }|rL| j                  | j                  |||¬«      dd	¬
«      }	| j                  || j                  d|¬«      |¬«      }
n|}
| j                  |
|
|¬«      }| j                  d||¬«      }| j                  |d	¬
«      }| j                  |
d	¬
«      }|rd|
g||gg g ||z  ||dz
  z  gg |f}|g}nGd|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}|rn| j                  	«      }t        t        |«      «      D ]F  }||   d   dxx   ||z  z  cc<   ||   d   j                  |«       ||   d   j                  d«       ŒH t!        |«      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/skyplay_api_hub/venv/lib/python3.12/site-packages/mpmath/functions/orthogonal.pyÚ_hermite_paramr3      s”  € ð × Ñ  Ó#�G€A€tØ�‰�A‹€AØ	�‰ˆ€Añ  Ø�—‘ˆ[˜1˜c˜( B¨¨A¨a©C©¨	°2°r¸1Ð<ˆÙØˆq‰E�!‹H˜˜!™‰O‹HØˆsˆ
Ø—+‘+˜q˜b“/ò + S§V¡V¨A£Y°¡]ò +Ø	�‰�‹�a‰Ò	)˜CŸF™F 1›I¨™Mð à�h‰h�q‰j˜2‰o€GÙØ�H‰H�S—X‘X˜a  w�XÓ/°¸dˆHÓCˆØ�H‰H�Q˜Ÿ™ ¨'˜Ó2¸ˆHÓA‰àˆØ	�‰�!�Q˜WˆÓ	%€BØ
�(‰(�1�b˜wˆ(Ó
'€CØ�8‰8�C˜tˆ8Ó$€DØ	�‰�!˜4ˆÓ	 €BÙØ�ˆV�a˜�V˜R  a¨¡c¨1¨a°©c©7 ^°R¸Ð=ˆØ�‰à�ˆW�q˜!�f˜b " q¨¡s¨A¨q°©s©G n°b¸$Ð>ˆØ�—‘˜ˆ_˜q ™s C¨˜m¨R°!°A±#°¸¸A¸a¹C¹¸	ÀAÀaÁCÀ5È"ÐLˆØ�B�ˆáØ�w‰w�q‹zˆÜ”s˜5“zÓ"ò 	"ˆAØ�!‰H�Q‰K˜‹N˜a ™cÑ!‹NØ�!‰H�Q‰K×Ñ˜tÔ$Ø�!‰H�Q‰K×Ñ˜qÕ!ð	"ô �‹<Ðó    c                 ó:   ‡ ‡‡—  ‰ 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¸¸1Ó!=€ r4   ©Ú	hypercomb©r   r    r!   Úkwargss   ``` r2   Úhermiter?   <   s   ú€ àˆ3�=‰=Õ=¸rÑLÀVÑLÐLr4   c                 ó:   ‡ ‡‡—  ‰ 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   @   s   ú€ ðl ˆ3�=‰=Õ=¸rÑLÀVÑLÐLr4   c                 ój   — | j                  |«      \  }}| j                  | | 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   s4   € ðX ×Ñ˜aÓ �D€A€qØ�8‰8�Q�B�s—{‘{‘N AÓ&Ð&r4   c                 ó–  ‡ ‡‡‡‡	— ‰ j                  |«      \  Š}‰ j                  ‰«      Š‰ j                  Š‰ j                  Š	|dk(  rf‰ j	                  ‰dz  «      rRˆ ˆˆˆ	ˆfd„} ‰ j
                  |g fi |¤Ž}‰ j                  ‰«      r"‰ j                  ‰«      r‰ j                  |«      }|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                  óÒ  •— ‰j                  ‰	dd¬«      } t        ‰‰ ‰z
  ‰	d«      }t        ‰‰‰z
  | d«      }|D ]A  }|d   j                  d«       |d   j                  d«       |d   j                  ‰‰z
  «       ŒC ‰j                  ‰‰z  ‰z
  «      ‰j	                  d‰j
                  z  «      z  }|D ]*  }|d   j                  |«       |d   j                  d«       Œ, ||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$   Úrr!   s
        €€€€€r2   Úhzpcfv.<locals>.hÍ   sê   ø€ Ø—‘˜!˜S¨�Ó-ˆBÜ$ S¨1¨"¨Q©$°°1Ó5ˆGÜ$ S¨!¨A©#¨r°1Ó5ˆGØò !�Ø�!‘—‘˜B”Ø�!‘—‘˜A”Ø�!‘—‘˜A˜a™CÕ ð!ð —
‘
˜A˜a™C ™EÓ# c§h¡h¨q°·±©xÓ&8Ñ8ˆAØò �Ø�!‘—‘˜A”Ø�!‘—‘˜A•ðð ˜WÑ$Ð$r4   c                 óB  •— ‰
j                  ‰d«      }‰
j                  ‰d«      }‰
j                  |«      }‰
j                  ‰‰
j                  |«      g}|‰ | ‰z  ‰z   dg‰‰| z  z
  gg ‰| z  ‰z   g‰g|f}|‰gz   ‰ | ‰z  ‰z
  ddgd‰z
  ‰| z  z
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  gd‰z   g|f}‰
j                  ‰‰| z  z   «      \  }}|d   j	                  |«       |d   j	                  |«       ||fD ]-  }	|	d   j	                  d«       |	d   j	                  ‰| z
  «       Œ/ ||fS )Nr   r   r   r   rL   )Úsquare_exp_argr   r   Úcospi_sinpir   )r    r)   r(   ÚeÚlÚY1ÚY2ÚcÚsÚYr   r$   rR   r!   s             €€€€r2   rS   zpcfv.<locals>.hß   sN  ø€ Ø×"Ñ" 1 eÓ,ˆAØ×"Ñ" 1 cÓ*ˆAØ—‘˜“
ˆAØ—‘˜˜CŸG™G A›JÐ'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¡5Ó)‰DˆAˆqØˆq‰E�L‰L˜ŒOØˆq‰E�L‰L˜ŒOØ˜"�Xò !�Ø�!‘—‘˜A”Ø�!‘—‘˜A˜a™CÕ ð!ð �r�6ˆMr4   )r   r   r   Úmpq_1_4Úisintr<   Ú_is_real_typeÚ_re)
r   rE   r!   r>   ÚntyperS   Úvr    r$   rR   s
   ` `    @@@r2   Úpcfvrd   §   s¿   ü€ ð@ ×!Ñ! !Ó$�H€A€uØ�‰�A‹€AØ�‰€AØ�‰€AØ�‚|˜Ÿ	™	 ! A¡#œ÷	%ð 	%ð ˆC�M‰M˜!˜RÑ* 6Ñ*ˆØ×Ñ˜QÔ C×$5Ñ$5°aÔ$8Ø—‘˜“
ˆAØˆ÷	ð ˆs�}‰}˜Q  Ñ. vÑ.Ð.r4   c                 óì   ‡ ‡‡— ‰ j                  |«      \  Š}‰ j                  ‰«      Šˆ ˆˆfd„}‰ j                  |«      }‰ j                  ‰«      r"‰ j                  ‰«      r‰ j	                  |«      }|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   óX  •K  — ‰j                  ‰j                  d‰j                  ‰z  z   «      «      } ‰j                  d‰j                  ‰z  z   «      ‰j                  d‰j                  ‰z  z
  «      z
  dz  } ‰j                  dz  d| z  z   }‰j                  d‰j                  d‰j                  z  ‰z  «      z   «      ‰j                  ‰j                  ‰z  «      z
  }‰j                  |dz  «      ‰j                  d‰j                  z  ‰z  «      z  }|‰j                  |«      z  ‰j                  ‰j                  ‰z  ‰‰j                  d«      z  «      z  –— |‰j                  | «      z  ‰j                  ‰j                   ‰z  ‰‰j                  d«      z  «      z  –— y ­w)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  sX  øè ø€ Ø�w‰w�s—y‘y  s§u¡u¨Q¡w¡Ó/Ó0ˆØ—‘˜S §¡ q¡™[Ó)¨C¯L©L¸¸S¿U¹UÀ1¹W¹Ó,EÑEÀrÑIˆØ�f‰f�Q‰h˜˜T™Ñ!ˆà�H‰H�Q˜Ÿ™  3§6¡6¡¨!¡Ó,Ñ,Ó-°·±¸¿¹¸q¹Ó0AÑAˆØ�H‰H�Q�q‘S‹M˜CŸG™G D¨¯©¡K°¡MÓ2Ñ2ˆØ�#—(‘(˜3“-Ñ #§(¡(¨3¯5©5°©7°A°c·j±jÀÓ6GÑ4GÓ"HÑHÒHØ�#—(‘(˜C˜4“.Ñ  3§8¡8¨S¯U©U¨F°1©H°a¸¿
¹
À4Ó8HÑ6HÓ#IÑIÓIùs   ƒF'F*)r   r   Úsum_accuratelyr`   ra   )r   rE   r!   r>   rF   r.   rc   r    s   ` `    @r2   Úpcfwrr   ð   sk   ú€ ð, ×Ñ˜aÓ �D€A€qØ�‰�A‹€AöJð 	×Ñ˜5Ó!€AØ
×Ñ˜Ô × 1Ñ 1°!Ô 4Ø�G‰G�A‹JˆØ€Hr4   c                 ó  ‡‡‡— | j                  ‰«      rd‰‰z   z  S | j                  ‰dz   «      r:| j                  ‰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  s�   ú€ ð ‡{�{�1„~Ø�!�A‘#‰wˆØ
‡{�{�1�S‘5Ôð �;‰;�q˜‘sÔÜ%Ð&KÓLÐLõ	ð ˆs�}‰}˜Q  Ñ. vÑ.Ð.õð ˆ3�=‰=˜˜Q˜CÑ* 6Ñ*Ð*r4   c                 ó:  ‡‡‡— | j                  ‰«      sˆˆˆfd„} | j                  ||gfi |¤ŽS | j                  ‰«      sˆˆfd„} | j                  ||‰gfi |¤ŽS | j                  |‰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¥   ú€ à�;‰;�qŒ>ö	Uàˆs�}‰}˜Q  Ñ. vÑ.Ð.Ø�9‰9�QŒ<õ	Sàˆs�}‰}˜Q  A Ñ1¨&Ñ1Ð1à�<‰<˜˜!™˜AÓ  §¡¨Q¨B¨q°©s°1©u°Q©w°q¸±s¸A¸a¹CÀ¹7Ñ!MÀfÑ!MÑMÐMr4   c                 ó<   ‡‡— 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  s#   ù€ õ
@àˆ3�=‰=˜˜Q˜CÑ* 6Ñ*Ð*r4   c                 ó  — | j                  |«      r]t        |«      }||dk  z   dz  rG|s|S | j                  |«      }|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�   ^  s•   € à
‡y�y�„|Ü�‹Fˆà��Q‘‰K˜1ÒÙØ�Ø—'‘'˜!“*ˆCØ�R˜Ÿ™‘[ ‘^Ò#Ø�Ø�RŠxØ—’˜S˜DÑ •Øˆ3�:‰:�q�b˜˜1™˜Q  !¡ Q™wÑ1¨&Ñ1Ð1r4   c                 ó
  ‡— | j                  |«      }| j                  |«      }|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  sš   ø€ ð 	�‰�A‹€AØ�‰�A‹€AáØˆs�|‰|˜A˜qÑ+ FÑ+Ð+àˆq‚yô	ð ˆs�}‰}˜Q  1 Ñ0¨Ñ0Ð0Øˆq‚yô	ð ˆs�}‰}˜Q  1 Ñ0¨Ñ0Ð0Ü
Ð0Ó
1Ð1r4   c                 óz  ‡ ‡— ‰ j                  |«      }‰ j                  |«      }‰ j                  ‰«      Š‰dv r‰ j                  S |dk(  rˆ ˆfd„} ‰ j                  |||gfi |¤ŽS |dk(  rFt        ‰«      dkD  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$  •— ‰j                  |«      \  }}d|z  ‰j                  z  }|}d‰z   }d‰z
  }|dz  }d‰z
  dz  }	||||gdd|| gg d|z
  g|  | dz   gd|z
  g|	f}
| ||gd| |g| |z   dz   g| |z
  dz   |dz   g|  | dz   g|dz   g|	f}|
|fS ©Nr   r   rž   )rV   r   )r    r”   ÚcosÚsinr\   r[   rE   r€   r(   r)   r%   r/   r   r!   s               €€r2   rS   zlegenq.<locals>.h�  sí   ø€ Ø—‘ qÓ)‰HˆC�Ø�C‘˜#Ÿ&™&Ñ ˆAØˆAØ�!‘ˆAØ�!‘ˆAØ�!‘ˆAØ�1‘�a‘ˆAØ�Q˜˜1�  A q¨1¨"˜~¨r°A°a±C°5Ø��Q�q‘S�	˜A˜a™C˜5 !ð$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$ˆBà�r�6ˆMr4   rL   r   c                 óà   •— ‰j                  |«      d‰j                  ‰‰dz
  ‰dz   gd|  dz
  d|  |z
  dz
  d|z  d|z  g| |z   dz   g| dz   gdd| z   |z   z  dd| z   |z   z  g| dz   g‰dz  f}|gS )Nr   r   r   g      ø?rŠ   )rM   r   )r    r”   r%   r   r!   s      €€r2   rS   zlegenq.<locals>.h¢  s£   ø€ Ø—j‘j “m Q¨¯©°°1°Q±3¸¸!¹Ð<Ø˜!˜˜A™˜s Q B q¡D¨¡F¨C°©E°3°q±5Ð9Ø˜‘c˜!‘e�W˜q ™u˜gØ˜1˜Q™3˜q™5‘k 3¨¨!©¨A©¡;Ð/°!°C±%°¸!¸b¹'ðB�ð �t�r4   c                 ó<  •— d‰
j                  |«      z  ‰
j                  z  }‰
j                  |«      }d‰z   }‰dz
  }|dz  }d‰z
  dz  }||||gdd|| gg d|z
  g|  | dz   gd|z
  g|f}| |||gdd| |g| |z   dz   g| |z
  dz   |dz   g|  | dz   g|dz   g|f}	||	fS r    )Úsinpir   rM   )r    r”   r\   r[   rE   r€   r(   r)   r%   r/   r   r!   s             €€r2   rS   zlegenq.<locals>.h«  sð   ø€ Ø˜Ÿ	™	 !›Ñ$ s§v¡vÑ-�Ø—J‘J˜q“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¨   „  sÎ   ù€ ð 	�‰�A‹€AØ�‰�A‹€AØ�‰�A‹€AØˆG�|ð �w‰wˆØˆq‚yõ	ð ˆs�}‰}˜Q  A Ñ1¨&Ñ1Ð1Øˆq‚yô ˆq‹6�AŠ:õð !�3—=‘=  Q¨ FÑ5¨fÑ5Ð5õð !�3—=‘=  Q¨ FÑ5¨fÑ5Ð5Ü
Ð0Ó
1Ð1r4   c                 ó¬   — |s6| j                  |«      r%t        | j                  |«      «      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  sV   € á�3—9‘9˜Q”<¤C¨¯©°«
£O°aÑ$7¸1Ò$<Ø�1‰uˆØˆ3�:‰:�q�b˜˜5 ! A¡# q¡Ñ3¨FÑ3Ð3r4   c                 ó¾   — |s6| j                  |«      r%t        | j                  |«      «      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  sc   € á�3—9‘9˜Q”<¤C¨¯©°«
£O°aÑ$7¸1Ò$<Ø�1‰uˆØˆa‰C�:�3—:‘:˜q˜b ! A¡# u¨q°©s°A©gÑ@¸Ñ@Ñ@Ð@r4   c                 óî  ‡ ‡‡— ‰ j                  |«      }‰ j                  |«      }‰ j                  ‰«      Š‰ j                  ‰«      Š‰ j                  |«      }|xr |dk\  }‰ j                  |«      }|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	        |«      |kD  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           
      ó´  •— t        |«      }d‰j                  |‰z  «      d| z  dz   ‰j                  | |z   «      z  ‰j                  z  ‰j                  | |z
  «      z  ‰j	                  ‰«      dz  ‰j                  |«      dg}d|z  ‰j                  |«      dz   z  ddd|z  d| dz
  g}||g g || z
  | |z   dz   g|dz   g‰j	                  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  sò   ø€ Ü�q“6ˆDØ�S—X‘X˜a ™e“_Ø�A‘#�a‘%˜Ÿ™  4¡›Ñ(¨¯©Ñ/°·±¸¸$¹³Ñ?Ø—‘˜“ Ñ"Ø—‘˜“ ð#ˆAð �Q‘˜Ÿ™ › A™Ñ&¨¨3°°D±¸"¸t¸eÀA¹gÐFˆAØ˜˜2˜r D¨¡F¨A¨d©F°1©HÐ#5¸¸Q¹°xØ—‘˜˜E™	Ó" AÑ%ð'ð )ð )r4   c                 óà  •— ‰j                  | |z
  dz   «      s+‰j                  | |z   dz   «      s‰j                  d|z
  «      rdgdgg g g g dffS ‰j                  d‰z  «      \  }}d‰j                  |‰z  «      z  d| z  dz   ‰j                  z  ‰j	                  | |z
  dz   «      ‰j	                  | |z   dz   «      |dz  |dz  g}ddddd|z  d|z  g}||g d|z
  g|  | dz   gd|z
  g|dz  ffS )Nr   r   rž   r   r   g      à¿)r   Úcos_sinrl   r   ri   )	rX   r”   r¡   r¢   rp   r´   r   rµ   r¶   s	         €€€r2   rS   zspherharm.<locals>.hä  s  ø€ Ø�{‰{˜1˜Q™3˜q™5Ô! S§[¡[°°1±°Q±Ô%7¸3¿;¹;ÀqÈÁsÔ;KØ˜˜r˜d B¨¨B°°AÐ6Ð8Ð8Ø—{‘{ 3 u¡9Ó-‰HˆC�Ø�S—X‘X˜a ™e“_Ñ$ q¨¡s¨1¡u¨c¯f©f¡nØ—‘˜1˜Q™3˜q™5Ó! 3§9¡9¨Q¨q©S°©UÓ#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  sò   ú€ à�‰�A‹€AØ�‰�A‹€AØ�K‰K˜Ó€EØ
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Nó ð
Nð ñ+ó ð+ð ñ2ó ð2ð ò2ó ð2ð, ò32ó ð32ðj ñ4ó ð4ð
 ñAó ðAð
 ñ&-ó ñ&-r4   