§
    JŠtjrÉ  ã                   óŽ  — d dl mZ ddlmZmZ d„ ZdZeg df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„ ¦   «         Zed„ ¦   «         Zed„ ¦   «         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$d„¦   «         Zed„ ¦   «         Zed„ ¦   «         Zed „ ¦   «         Zed!„ ¦   «         Z ed"„ ¦   «         Z!	 ed#„ ¦   «         Z"dS )%é   )Úxrangeé   )ÚdefunÚdefun_wrappedc                 óê  — dx}}d}g }t          |¦  «        D �]U\  }}	|	\  }
}}}}}}d}t          |
¦  «        D ]4\  }}|s-|                      ||         ¦  «        dk    r||         rdx}}d}Œ5g d¢}t          |||g¦  «        D ]Ÿ\  }}t          |¦  «        D ]Š\  }}|                      |¦  «        \  }}|dk    rŒ$|| j        k    rMd}|dk    r1|D ].}|                      |¦  «        r|t          |¦  «        k    rd} nŒ/|rŒk||xx         dz  cc<   Œ||dk     r|| z  }d}Œ‹Œ |r4|d         |d         |d         z   k    r|s|                     |¦  «         �ŒBt          |¦  «        rdx}}�ŒW||||fS )NFé    T)r   r   r   r   r   éüÿÿÿ)Ú	enumerateÚreÚnint_distanceÚninfÚisnpintÚintÚappendÚsum)ÚctxÚtermsÚprecÚdiscard_known_zerosÚperturbÚ	recomputeÚ	extraprecÚdiscardÚ
term_indexÚtermÚw_sÚc_sÚalpha_sÚbeta_sÚa_sÚb_sÚzÚhave_singular_nongamma_weightÚkÚwÚ
pole_countÚ
data_indexÚdataÚiÚxÚnÚdÚokÚus                                ú]/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/mpmath/functions/hypergeometric.pyÚ_check_need_perturbr0      s  € ØÐ€GˆiØ€IØ€GÝ% eÑ,Ô,ð ('ñ ('Ñˆ
�DØ15Ñ.ˆˆS�'˜6 3¨¨QØ(-Ð%õ ˜c‘N”Nð 	9ð 	9‰DˆAˆqØð 9Ø—6’6˜#˜aœ&‘>”> QÒ&Ð&¨3¨q¬6Ð&Ø*.Ð.�G˜iØ48Ð1øØ�Y�Yˆ
å )¨7°F¸CÐ*@Ñ AÔ Að 	%ð 	%ÑˆJ˜Ý! $™œð %ð %‘��1Ø×(Ò(¨Ñ+Ô+‘��1à�q’5�5ØØ˜œ’=�=ð �BØ! Q’�Ø!$ð &ð &˜AØ"Ÿ{š{¨1™~œ~ð &°!µs¸1±v´v²+°+Ø%) Ø % øØð !Ø Ø˜zÐ*Ð*Ô*¨aÑ/Ð*Ð*Ñ*Ð*ð ˜’V�VØ ! ‘O�IØ $�Iøð-%ð. ð 	' :¨a¤=°:¸a´=À:ÈaÄ=Ñ3PÒ#PÐ#PØ1ð $Qà�NŠN˜:Ñ&Ô&Ð&Ñ&Ý�‰_Œ_ð 	'Ø"&Ð&ˆG�iùØ�I˜y¨'Ð1Ð1ó    a  
hypercomb() failed to converge to the requested %i bits of accuracy
using a working precision of %i bits. The function value may be zero or
infinite; try passing zeroprec=N or infprec=M to bound finite values between
2^(-N) and 2^M. Otherwise try a higher maxprec or maxterms.
Tc           
      ó~  ‡ ‡+— ‰ j         }‰ j        }‰ j        }‰ j        }|d d …         }	|                     dd¦  «        }
|                     d‰                      |¦  «        ¦  «        }||d<   |                     d¦  «        }|                     d¦  «        }d }d}	 	 ‰ xj         dz  c_         ‰ j         |k    rt          t          |‰ j         fz  ¦  «        ‚‰ j         }|	d d …         } ||Ž }|
rBt          ¦   «          t          d	¦  «         t          d
‰ j         ¦  «         t          d|¦  «         t          ‰ |||¦  «        \  }}}Š+‰ xj         |z  c_         |ršd|v r	|d         }n'‰ j
        rt          ‰ j         dz  ¦  «        }n|dz   |z   }‰                      ‰ j        | ¦  «        }|dz   |z   dz   ‰ _         t          t          |¦  «        ¦  «        D ]}||xx         |z  cc<   |||dz   z  z  }Œ|r ||Ž }|rˆ+fd„t!          |¦  «        D ¦   «         }|s‰ j        |‰ _         S g }t!          |¦  «        D �]¯\  }}|\  }}}}}} }!|
�rt          ¦   «          t          d|dz   t          |¦  «        t          |¦  «        t          | ¦  «        fz  ¦  «         t          d‰                      |¦  «        ‰                      |¦  «        ¦  «         t          d‰                      |¦  «        ‰                      |¦  «        ¦  «         t          d‰                      |¦  «        ‰                      | ¦  «        ¦  «         t          d‰                      |!¦  «        ¦  «         ‰                       ‰ j        || |!fi |¤Žgˆ fd„|D ¦   «         z   ˆ fd„|D ¦   «         z   ˆ fd„t)          ||¦  «        D ¦   «         z   ¦  «        }"|
rt          d|"¦  «         |                     |"¦  «         �Œ±t          |¦  «        dk    r|s
|d         }�n ‰ j
        r‰                      |¦  «        }�n‚‰                      |¦  «        }ˆ fd„|D ¦   «         }#t/          |#¦  «        }$‰                      |¦  «        }%|$|%z
  }&|
r8t          ¦   «          t          d|&d¦  «         t          d‰ j         |z
  d¦  «         |&‰ j         |z
  k     }'|€d}(n|$‰ j         z
  | k     }(|€d})n|$|k    })|'r|r‰                      |&¦  «        rnµ|'r_|€	|dz  }|}�Œd‰                      ||z
  ¦  «        ‰                      |¦  «        |z
  k    rnu|(r‰ j        }nk|)r‰ j        }nad|v rn\|dz  }|}nRt7          t/          |&|dz  ¦  «        t/          ||¦  «        ¦  «        }*‰ xj         |*z  c_         |
rt          d¦  «         �Œ
�Œ|‰ _         n# |‰ _         w xY w|
 S )NÚverboseFÚmaxprecÚzeroprecÚinfprecr   r   é
   zENTERING hypercomb main loopzprec =ÚhextraÚhmagg333333Ó?c                 ó"   •— g | ]\  }}|‰v¯	|‘ŒS © r;   )Ú.0r)   r   r   s      €r/   ú
<listcomp>zhypercomb.<locals>.<listcomp>k   s'   ø€ ÐTÐTÐT¡) 1 dÀ1ÈGÐCSÐCS˜ÐCSÐCSÐCSr1   z  Evaluating term %i/%i : %iF%iz
    powersz	    gammaz	    hyperz    zc                 ó:   •— g | ]}‰                      |¦  «        ‘ŒS r;   )Úgamma©r<   Úar   s     €r/   r=   zhypercomb.<locals>.<listcomp>~   s#   ø€ Ð3Ð3Ð3 a�S—Y’Y˜q‘\”\Ð3Ð3Ð3r1   c                 ó:   •— g | ]}‰                      |¦  «        ‘ŒS r;   )Úrgamma©r<   Úbr   s     €r/   r=   zhypercomb.<locals>.<listcomp>   s#   ø€ Ð3Ð3Ð3 q�S—Z’Z ‘]”]Ð3Ð3Ð3r1   c                 óB   •— g | ]\  }}‰                      ||¦  «        ‘ŒS r;   )Úpower)r<   r%   Úcr   s      €r/   r=   zhypercomb.<locals>.<listcomp>€   s)   ø€ Ð>Ð>Ð>©¨¨1�S—Y’Y˜q ‘^”^Ð>Ð>Ð>r1   z
    Value:c                 ó:   •— g | ]}‰                      |¦  «        ‘ŒS r;   )Úmag)r<   r*   r   s     €r/   r=   zhypercomb.<locals>.<listcomp>Ž   s#   ø€ ÐCÐCÐC¨a˜sŸwšw q™zœzÐCÐCÐCr1   z  Cancellation:Úbitsz  Increased precision:é   r   z*  Must start over with increased precision)r   Úzeror   r   ÚgetÚ_default_hyper_maxprecÚ
ValueErrorÚ_hypercomb_msgÚprintr0   Ú_fixed_precisionr   ÚldexpÚoneÚrangeÚlenr
   ÚnstrÚfprodÚhyperÚzipr   ÚfsumÚmaxrJ   ÚisnanÚinfÚmin),r   ÚfunctionÚparamsr   ÚkwargsÚorigÚsumvalueÚdistr   Úorig_paramsr3   r4   r5   r6   Úperturbed_reference_valuer8   Úorig2r   r   r   r   r9   Úhr$   Úevaluated_termsr   Ú	term_datar   r   r   r   r    r!   r"   ÚvÚterm_magnitudesÚmax_magnitudeÚsum_magnitudeÚcancellationÚprecision_okÚzero_okÚinf_okÚ	incrementr   s,   `                                          @r/   Ú	hypercombrv   :   s°  øø€ àŒ8€DØŒx€HØÔ€DØŒ8€DØ˜˜˜”)€KØ�jŠj˜ EÑ*Ô*€GØ�jŠj˜ C×$>Ò$>¸tÑ$DÔ$DÑEÔE€GØ€Fˆ9ÑØ�zŠz˜*Ñ%Ô%€HØ�jŠj˜Ñ#Ô#€GØ $ÐØ€Fðwðt	ØˆHŒH˜‰NˆHŒHØŒx˜'Ò!Ð!Ý ¥°4¸¼Ð2BÑ!BÑCÔCÐCØ”HˆEØ    ”^ˆFØ�H˜fÐ%ˆEØð (Ý‘”�ÝÐ4Ñ5Ô5Ð5Ý�h ¤Ñ)Ô)Ð)Ý�h Ñ'Ô'Ð'å# C¨°Ð6IÑJÔJñ 3ˆG�Y 	¨7àˆHŒH˜	Ñ!ˆHŒHØð !Ø˜VÐ#Ð#Ø! &œ>�D�DØÔ)ð .Ý˜sœx¨™|Ñ,Ô,�D�Dà "™9 vÑ-�DØ—I’I˜cœg¨ uÑ-Ô-�Ø  2™:¨Ñ,¨rÑ1�”Ý�s 6™{œ{Ñ+Ô+ð !ð !�AØ˜1�I�I”I ‘N�I�I‘Ið
 ˜˜A˜a™C™‘L�A�AØð *Ø ˜ &Ð)�Ø"ð UØTÐTÐTÐT­y¸Ñ/?Ô/?ÐTÑTÔT�Øð  Ø”xðd ˆŒˆðc !ˆOÝ)2°5Ñ)9Ô)9ð *ñ *Ñ%�
˜IØ9BÑ6��S˜' 6¨3°°QØñ 0Ý‘G”G�GÝÐ;Ø# A™¥s¨5¡z¤zµ3°s±8´8½SÀ¹X¼XÐFñGñ Hô Hð Hå˜,¨¯ª°©¬°s·x²xÀ±}´}ÑEÔEÐEÝ˜+ s§x¢x°Ñ'8Ô'8¸#¿(º(À6Ñ:JÔ:JÑKÔKÐKÝ˜+ s§x¢x°¡}¤}°c·h²h¸s±m´mÑDÔDÐDÝ˜' 3§8¢8¨A¡;¤;Ñ/Ô/Ð/ð
 —I’I˜y˜sœy¨¨c°1Ð?Ð?¸Ð?Ð?Ð@Ø3Ð3Ð3Ð3¨7Ð3Ñ3Ô3ñ4à3Ð3Ð3Ð3¨FÐ3Ñ3Ô3ñ4ð ?Ð>Ð>Ð>µ°S¸±´Ð>Ñ>Ô>ñ?ñ @ô @�ð ð +Ý˜,¨Ñ*Ô*Ð*Ø×&Ò& qÑ)Ô)Ð)Ñ)å�5‰zŒz˜QŠˆ¨ˆØ*¨1Ô-�ÙàÔ#ð ØŸ8š8 OÑ4Ô4�Ùà—x’x Ñ0Ô0ˆHØCÐCÐCÐC°?ÐCÑCÔCˆOÝ Ñ0Ô0ˆMØŸGšG HÑ-Ô-ˆMØ(¨=Ñ8ˆLØð IÝ‘”�ÝÐ'¨°vÑ>Ô>Ð>ÝÐ.°´¸4±ÀÑHÔHÐHà'¨#¬(°T©/Ò9ˆLàÐØ��à'¨#¬(Ñ2°h°YÒ>�ØˆØ��à&¨Ò0�àð  Wð °·²¸<Ñ1HÔ1Hð ØØð Ø,Ð4Ø˜b‘L�FØ08Ð-ÙØ—W’W˜XÐ(AÑAÑBÔBØŸš Ñ)Ô)¨DÑ0ò1ð 1àØð 
9Ø"œx�HØØð 9Ø"œw�HØØ˜vÐ%Ð%Øà˜a‘K�FØ08Ð-Ð-õ  ¥ L°$¸±'Ñ :Ô :½CÀ	È$Ñ<OÔ<OÑPÔP�	Ø�”˜IÑ%�”Øð HÝÐFÑGÔGÐGÙñit	ðl ˆŒˆø�4ˆŒˆˆˆˆØˆ9Ðs   ÂFV0 È'NV0 Ö0	V9c                 ó   ‡ — ‰                       |¦  «        }t          |¦  «        }t          |¦  «        }ˆ fd„|D ¦   «         }ˆ fd„|D ¦   «         }|                     dd¦  «        r‹|                     dd¦  «        }d}||k     rm|rk||         }	|	|v rR|s‰                      |	d         ¦  «        s5|                     |	¦  «         |                     |	¦  «         |dz  }|dz  }n|dz  }||k     r|°k|dk    r2|dk    r ‰ j        ||fi |¤ŽS |dk    r‰                      |¦  «        S �n|dk    rU|dk    r ‰ j        |||fi |¤ŽS |d	k    r ‰ j        |||fi |¤ŽS |dk    r"‰  	                    |d         d         |¦  «        S nª|d	k    rY|dk    r ‰ j
        |||fi |¤ŽS |d	k    r ‰ j        |||fi |¤ŽS |d
k    r ‰ j        |||fi |¤ŽS |dk    r ‰ j        |||fi |¤ŽS nK||dz   k    r ‰ j        |||||fi |¤ŽS ||dz   k    r'|                     d¦  «        s ‰ j        |||||fi |¤ŽS t!          ||z   Ž \  }
} ‰ j        ||||
|fi |¤ŽS )z0
    Hypergeometric function, general case.
    c                 ó:   •— g | ]}‰                      |¦  «        ‘ŒS r;   ©Ú_convert_paramr@   s     €r/   r=   zhyper.<locals>.<listcomp>Ê   ó'   ø€ Ð
.Ð
.Ð
. Qˆ3×Ò˜aÑ Ô Ð
.Ð
.Ð
.r1   c                 ó:   •— g | ]}‰                      |¦  «        ‘ŒS r;   ry   rD   s     €r/   r=   zhyper.<locals>.<listcomp>Ë   r{   r1   Ú	eliminateTÚeliminate_allFr   r   r   é   Úforce_series)ÚconvertrW   rN   r   ÚremoveÚ_hyp0f1ÚexpÚ_hyp1f1Ú_hyp1f2Ú_hyp1f0Ú_hyp2f1Ú_hyp2f2Ú_hyp2f3Ú_hyp2f0Ú_hypq1fqÚ
_hyp_borelr[   Úhypsum)r   r    r!   r"   rc   ÚpÚqÚelim_nonpositiver)   rE   ÚcoeffsÚtypess   `           r/   rZ   rZ   Â   s  ø€ ð
 	�Š�A‰Œ€AÝˆC‰Œ€AÝˆC‰Œ€AØ
.Ð
.Ð
.Ð
.¨#Ð
.Ñ
.Ô
.€CØ
.Ð
.Ð
.Ð
.¨#Ð
.Ñ
.Ô
.€Cà‡z‚z�+˜tÑ$Ô$ð Ø!Ÿ:š: o°uÑ=Ô=ÐØˆØ�!Šeˆe˜ˆeØ�A”ˆAØ�CˆxˆxÐ-ˆx°S·[²[ÀÀ1ÄÑ5FÔ5FˆxØ—
’
˜1‘”�Ø—
’
˜1‘”�Ø�Q‘�Ø�Q‘��à�Q‘�ð �!Šeˆe˜ˆeð 	ˆA‚v€vØ�!ŠVˆV˜K˜CœK¨¨QÐ9Ð9°&Ð9Ð9Ð9Ø�!ŠVˆV˜CŸGšG A™JœJÐ&‰VØ	
ˆaŠˆØ�!ŠVˆV˜K˜CœK¨¨S°!Ð>Ð>°vÐ>Ð>Ð>Ø�!ŠVˆV˜K˜CœK¨¨S°!Ð>Ð>°vÐ>Ð>Ð>Ø�!ŠVˆV˜CŸKšK¨¨A¬¨q¬	°1Ñ5Ô5Ð5ˆVØ	
ˆaŠˆØ�!ŠVˆV˜K˜CœK¨¨S°!Ð>Ð>°vÐ>Ð>Ð>Ø�!ŠVˆV˜K˜CœK¨¨S°!Ð>Ð>°vÐ>Ð>Ð>Ø�!ŠVˆV˜K˜CœK¨¨S°!Ð>Ð>°vÐ>Ð>Ð>Ø�!ŠVˆV˜K˜CœK¨¨S°!Ð>Ð>°vÐ>Ð>Ð>ˆVØ	
ˆa�‰cŠˆØˆsŒ|˜A˜q # s¨AÐ8Ð8°Ð8Ð8Ð8Ø	
ˆQˆq‰SŠˆ˜Ÿš NÑ3Ô3ˆØˆsŒ~˜a  C¨¨aÐ:Ð:°6Ð:Ð:Ð:Ý˜#˜c™'�O�M€FˆEØˆ3Œ:�a˜˜E 6¨1Ð7Ð7°Ð7Ð7Ð7r1   c                 ó$   —  | j         g |g|fi |¤ŽS ©N©rZ   )r   rE   r"   rc   s       r/   Úhyp0f1r—   í   s"   € àˆ3Œ9�R˜˜˜AÐ'Ð' Ð'Ð'Ð'r1   c                 ó&   —  | j         |g|g|fi |¤ŽS r•   r–   ©r   rA   rE   r"   rc   s        r/   Úhyp1f1rš   ñ   s$   € àˆ3Œ9�a�S˜!˜˜QÐ(Ð( Ð(Ð(Ð(r1   c                 ó(   —  | j         |g||g|fi |¤ŽS r•   r–   )r   Úa1Úb1Úb2r"   rc   s         r/   Úhyp1f2rŸ   õ   s&   € àˆ3Œ9�b�T˜2˜b˜' !Ð-Ð- fÐ-Ð-Ð-r1   c                 ó(   —  | j         ||g|g|fi |¤ŽS r•   r–   )r   rA   rE   rH   r"   rc   s         r/   Úhyp2f1r¡   ù   s&   € àˆ3Œ9�a˜�U˜A˜3˜qÐ*Ð* 6Ð*Ð*Ð*r1   c                 ó*   —  | j         ||g||g|fi |¤ŽS r•   r–   )r   rœ   Úa2r�   rž   r"   rc   s          r/   Úhyp2f2r¤   ý   s(   € àˆ3Œ9�b˜�W˜b ˜W QÐ0Ð0¨Ð0Ð0Ð0r1   c                 ó,   —  | j         ||g|||g|fi |¤ŽS r•   r–   )r   rœ   r£   r�   rž   Úb3r"   rc   s           r/   Úhyp2f3r§     s*   € àˆ3Œ9�b˜�W˜b  B˜Z¨Ð3Ð3¨FÐ3Ð3Ð3r1   c                 ó&   —  | j         ||gg |fi |¤ŽS r•   r–   r™   s        r/   Úhyp2f0r©     s$   € àˆ3Œ9�a˜�U˜2˜aÐ)Ð) &Ð)Ð)Ð)r1   c                 ó,   —  | j         |||g||g|fi |¤ŽS r•   r–   )r   rœ   r£   Úa3r�   rž   r"   rc   s           r/   Úhyp3f2r¬   	  s*   € àˆ3Œ9�b˜˜B�Z  B ¨Ð3Ð3¨FÐ3Ð3Ð3r1   c                 ó   — d|z
  | z  S ©Nr   r;   )r   rA   r"   s      r/   r‡   r‡     s   € àˆa‰C�a�R‰=Ðr1   c                 ó\  ‡ ‡‡	— |\  \  Š	}‰r‰                       ‰¦  «        }nd}|dk    rï|                     d¦  «        sÚ	 ‰ j        }	 ‰ xj        d|dz  z   z  c_        ˆ	ˆ ˆfd„}‰                      |g d¬¦  «        }‰                      ‰	¦  «        d‰                      ‰ j        ¦  «        z  z  |z  }|‰ _        n# |‰ _        w xY w‰                      ‰	¦  «        r*‰                      ‰¦  «        r‰                      |¦  «        }|
 S # ‰ j	        $ r Y nw xY w ‰ j
        dd	|f‰	g‰fi |¤ŽS )
Nr   é   r€   é   r   c                  ó   •— ‰                      ‰	 ¦  «        } ‰j        | z  }dd|z  z  }‰j        ‰z
  }‰                     d|z  ¦  «        }| |g|dgg g ‰‰j        z
  ‰j        ‰z
  gg | f}||g|dgg g ‰‰j        z
  ‰j        ‰z
  gg |f}||fS )Nr   é   r   éÿÿÿÿ)ÚsqrtÚjÚmpq_1_2r„   Úmpq_3_2)
r%   Újwr.   rH   ÚEÚT1ÚT2rE   r   r"   s
          €€€r/   rj   z_hyp0f1.<locals>.h!  s¶   ø€ ØŸš ! ™œ�AØœ˜q™�BØ˜1˜R™4™�AØœ a™�AØŸš  "¡™œ�AØ˜3˜q˜' A b 6¨2¨r°A°c´k±MÀ3Ä;ÈqÁ=Ð3QÐSUÐXYÐWYÐZ�BØ˜a˜& 1 Q %¨¨R°!°C´K±-ÀÄÈQÁÐ1OÐQSÐUVÐW�BØ˜r˜6�Mr1   T©r€   r   )rJ   rN   r   rv   r?   rµ   ÚpiÚ_is_real_typeÚ_reÚNoConvergencerŽ   )
r   r!   r"   rc   ÚbtypeÚmagzrd   rj   rm   rE   s
   ` `      @r/   rƒ   rƒ     s|  øøø€ à�K�J€QˆØð Ø�wŠw�q‰zŒzˆˆàˆØˆq‚y€y˜Ÿš NÑ3Ô3€yð	ð
 ”8ˆDð Ø�”˜B  q¡™LÑ(�”ð"ð "ð "ð "ð "ð "ð "ð —M’M ! R°d�MÑ;Ô;�Ø—I’I˜a‘L”L ! C§H¢H¨S¬VÑ$4Ô$4Ñ"4Ñ5°aÑ7�à�”�ø˜4�”����Ø× Ò  Ñ#Ô#ð ¨×(9Ò(9¸!Ñ(<Ô(<ð Ø—G’G˜A‘J”J�Ø�2ˆIøØÔ ð 	ð 	ð 	ØˆDð	øøøàˆ3Œ:�a˜˜U˜H q c¨1Ð7Ð7°Ð7Ð7Ð7s,   ÁD
 Á	A+B< Â4D
 Â<	CÃAD
 Ä

DÄDc                 óÂ  ‡ ‡‡— |\  \  }}|\  \  }}‰s
‰ j         ‰z   S ‰                      ‰¦  «        }	|	dk    �r�‰                      |¦  «        r‰                      |¦  «        dk    �s`‰                      ‰¦  «        ra‰                      |¦  «        ‰                      |¦  «        cxk    r ‰                      ‰¦  «        cxk    rdk    r
n n‰ j        S ‰ j        ‰z  S 	 	 ‰ xj        |	z  c_        ‰  	                    ‰¦  «        dk     Šˆ ˆˆfd„}
‰  
                    |
||gd¬¦  «        }‰                      |¦  «        r?‰                      |¦  «        r*‰                      ‰¦  «        r‰                      |¦  «        }|
 ‰ xj        |	z  c_        S # ‰ j        $ r Y nw xY w	 ‰ xj        |	z  c_        n# ‰ xj        |	z  c_        w xY w ‰ j        dd||f||g‰fi |¤Ž}|S )Né   r   r   c                 ó,  •— ‰r+‰                      ‰                     | d¬¦  «        ¦  «        }n‰                      | ¦  «        }d‰z  }|‰gd|  g|g|| z
  g| d| z   |z
  gg | f}‰                     ‰¦  «        ‰gd| |z
  g|g| g|| z
  d| z
  gg |f}||fS )NT©Úexactr   )ÚexpjpiÚfnegr„   )	rA   rE   rº   Úrzr»   r¼   r   Úsectorr"   s	         €€€r/   rj   z_hyp1f1.<locals>.hE  s½   ø€ Øð *ØŸJšJ s§x¢x°¸ xÑ'>Ô'>Ñ?Ô?˜˜àŸJšJ q™MœM˜Ø˜1™�BØ˜Q˜% ! Q B ¨!¨¨q°©s¨e°a¸¸1¹¸Q¹°ZÀÀbÀSÐI�BØŸ7š7 1™:œ: a˜.¨1¨Q¨q©S¨'°A°3¸¸¸aÀ¹cÀ1ÀQÁ3¸ZÈÈRÐP�BØ˜r˜6�Mr1   Tr½   )rU   rJ   Úisintr   ÚisinfÚsignr_   Únanr   Ú_imrv   r¿   rÀ   rÁ   rŽ   )r   r    r!   r"   rc   rA   ÚatyperE   rÂ   rÃ   rj   rm   rÌ   s   `  `        @r/   r…   r…   5  s0  øøø€ à�K�J€QˆØ�K�J€QˆØð ØŒw�q‰yÐØ�7Š7�1‰:Œ:€DØˆq‚y�y˜#Ÿ)š) A™,œ,€y¨3¯6ª6°!©9¬9¸ª>©>Ø�9Š9�Q‰<Œ<ð 	Ø�xŠx˜‰{Œ{˜cŸhšh q™kœkÐ=Ð=Ò=Ð=¨S¯XªX°a©[¬[Ð=Ð=Ò=Ð=¸AÒ=Ð=Ð=Ð=Ð=Ø”w�Ø”7˜Q‘;Ðð	ðØ�”˜DÑ �”ØŸš ™œ aš�ð"ð "ð "ð "ð "ð "ð "ð —M’M ! a¨ U¸�MÑ>Ô>�Ø×$Ò$ QÑ'Ô'ð #¨C×,=Ò,=¸aÑ,@Ô,@ð #ÀS×EVÒEVÐWXÑEYÔEYð #ØŸš ™
œ
�AØ�rð ˆHŒH˜ÑˆHŒHˆHøð Ô$ð ð ð Ø�ðøøøØàˆHŒH˜ÑˆHŒHˆHøˆCˆHŒH˜ÑˆHŒHˆHˆHˆHˆHØˆŒ
�1�a˜% ˜¨!¨Q¨°Ð=Ð=°fÐ=Ð=€AØ€Hs%   Ã!B F Æ
FÆF4 ÆFÆF4 Æ4Gc                 óŽ  — ||||f\  }}}}	| j         }
|                     dd|
z  ¦  «        }d}	 |
|z   | _         |                      |	¦  «        }|                      d¦  «        }|                      d¦  «        }|                      d¦  «        }d}||z  |z  }|| j        z  }|dz   | j        z  }d|z
  }||z  }||z  |z  }||z
  |z
  }|dz
  }| j          dz
  }| j        }| j        }| j        }| j        }	 ||z   }||z   ||z   z  |z  d|dz   z  |z  ||z   z  z  }||||z   |z  |z  z
  z  } |||z  ||z   |z  z   z  }!||||z  ||z  z   |z
  z  |z
  z  d|z  |z  z  }"||z   |"z
  }#t          | ||#¦  «        ¦  «        } ||#|z
  ¦  «        |k     rn| |!|#}}}|dz  }Œ§| ||#¦  «        z
  }$|$|k     rn||$z  }||k    r| j
        ‚�Œ–|#S )Nr4   éd   r7   r   r   r   r³   )r   rN   r�   Úmpfr·   rX   ÚnprintrJ   r   r]   rÁ   )%r   rA   rE   rH   r"   rc   Ú_aÚ_bÚ_cÚ_zrd   r4   Úextrar,   ÚeÚfr$   ÚabzÚchÚc1hÚnzÚgÚabgÚcbaÚz2ÚtolrX   rÖ   rJ   ÚmaxmagÚkchÚkakbzÚd1Úe1ÚftÚf1rq   s%                                        r/   Ú_hyp2f1_gosperrî   Y  s\  € ð �Q˜˜1�*�K€B€rˆ"ˆRØŒ8€DØ�jŠj˜ C¨¡HÑ-Ô-€GØ€Eð,(Ø˜%‘<ˆŒð �KŠK˜‰OŒOˆØ�GŠG�A‰JŒJˆØ�GŠG�A‰JŒJˆØ�GŠG�A‰JŒJˆØˆð �‰c�!‰eˆØ�”‰_ˆØ�‰s�c”kÑ!ˆØˆq‰SˆØˆb‰DˆØ�‰c�!‰eˆØ�‰c�!‰eˆØˆq‰SˆØŒxˆi˜"‰nˆØŒxˆØ”ˆØŒgˆØ”ˆð	Ø�B‘$ˆCØ�q‘S˜1˜Q™3‘K ‘M Q¨¨!©¡W¨S¡[°!°C±%Ñ%8Ñ9ˆEØ˜˜1˜S™5 !™) A™+™Ñ&ˆBØ˜˜#™˜q ™s A™g™Ñ&ˆBØ�A�s˜1‘u˜Q˜r™T‘z !‘|Ñ$ SÑ(Ñ)¨1¨S©5°©8Ñ4ˆBØ�Q‘˜‘ˆBÝ˜   R¡¤Ñ)Ô)ˆFØˆs�2�a‘4‰yŒy˜3ŠˆØØ˜"˜b�!ˆqˆAØ�‰FˆAð	ð    B¡¤Ñ'ˆØ˜%ÒÐØà�\Ñ!ˆEØ�wŠˆØÔ'Ð'ñY,(ðZ €Ir1   c                 óò  ‡ ‡‡— |\  \  }}\  }}|\  \  Š}	‰dk    �r‰                       ‰|z
  |z
  ¦  «        dk    }
‰                      |¦  «        r|dk    p‰                      |¦  «        o|dk    }‰                      ‰¦  «        oP‰dk    oJ‰                      |¦  «        r‰|cxk    odk    nc p$‰                      |¦  «        o‰|cxk    odk    nc  }|
s|r*|s(‰                      ‰‰|z
  |z
  g‰|z
  ‰|z
  gd¬¦  «        S ‰                      ||‰d‰ j        dz  z
  ¦  «        ‰ j        z  S ‰s‰s|dk    s|dk    rd‰z   S ‰ j        S ‰                      ‰¦  «        rW‰dk    rQ‰                      |¦  «        r‰|cxk    rdk    s'n ‰                      |¦  «        r‰|cxk    rdk    rn nn‰ j        S t          ‰¦  «        }|dk    sB‰                      |¦  «        r|dk    r|dk    s!‰                      |¦  «        r$|dk    r|dk    r ‰ j        dd|||	f||‰g‰fi |¤ŽS ‰ j	        }	 ‰ xj	        dz  c_	        |d	k    rˆˆ ˆfd
„} ‰ j
        |||gfi |¤Ž}nƒt          d‰z
  ¦  «        dk    rˆˆfd„} ‰ j
        |||gfi |¤Ž}nUt          ‰‰dz
  z  ¦  «        dk    r+‰                      |‰|z
  ‰‰‰dz
  z  ¦  «        d‰z
  |z  z  }nt          ‰ ||‰‰fi |¤Ž}|‰ _	        n# |‰ _	        w xY w|
 S )Nr   r   T)Ú_infsignr   gš™™™™™é?iüÿÿr7   gÍÌÌÌÌÌô?c                 ó¾   •— ‰j         ‰z
  }| |z
  }d‰	z  }‰	 g|  g‰| g|‰| z
  g| || z   g‰j         |z   g|f}‰	 g| g‰|g| ‰|z
  g|||z   g‰j         |z
  g|f}||fS r®   )Úmpq_1)
rA   rE   ÚtÚabrË   r»   r¼   rH   r   r"   s
          €€€r/   rj   z_hyp2f1.<locals>.hÃ  s™   ø€ Ø”I˜a‘K� a¨¡c °°!±¨2Ø�r�d˜Q˜B˜4 ! R C ¨!¨A¨a©C¨°1°Q°q±S°'¸3¼9ÀR¹<¸.È2ÐN�Ø�r�d˜Q˜B˜4 ! B ¨¨1¨Q©3¨°!°A°a±C°¸#¼)ÀB¹,¸È"ÐM�Ø˜2�v�r1   g      è?c                 ó–   •— ‰| z
  |z
  }‰| z
  }‰|z
  }d‰	z
  }g g ‰|g||g| |gd|z
  g|f}|g|g‰| |z   ‰z
  g| |g||gd|z   g|f}||fS r®   r;   )
rA   rE   ró   ÚcaÚcbrË   r»   r¼   rH   r"   s
           €€r/   rj   z_hyp2f1.<locals>.hÌ  s‰   ø€ Ø�a‘C˜‘E�  !¡˜2¨!¨A©# R°A°a±C¨rØ˜˜a ˜U R¨ G¨a°¨U°Q°q±S°E¸2Ð=�Ø�T˜A˜3  1 Q¡3 q¡5 	¨A¨a¨5°2°b°'¸A¸a¹C¸5À"ÐD�Ø˜2�v�r1   )r   rÍ   Ú	gammaprodr¡   Úepsr_   rÐ   ÚabsrŽ   r   rv   rî   )r   r    r!   r"   rc   rA   rÒ   rE   rÂ   ÚctypeÚ
convergentÚfiniteÚzerodivÚabszrd   rj   rm   rH   s   `  `             @r/   rˆ   rˆ   �  sè  øøø€ à Ñ�J€Qˆ‘
��EØ�K�J€QˆØˆA‚v�và—V’V˜A˜a™C ™E‘]”] QÒ&ˆ
Ø—)’)˜A‘,”,Ð) 1¨¢6ÐG¨s¯yªy¸©|¬|Ð/FÀÀQÂˆØ—)’)˜A‘,”,ð O 1¨¢6ð OØ�iŠi˜‰lŒlÐ*˜q A˜{˜{š{˜{¨š{˜{˜{˜{ÐM°·	²	¸!±´Ð0LÀÀaÀÀÂÀÈ1ÂÀÀÀð/Oˆð ð 	H˜&ð 	H¨'ð 	HØ—=’= ! Q q¡S¨¡U ¨a°©c°1°Q±3¨ZÀ$�=ÑGÔGÐGð �zŠz˜!˜A˜a  #¤'¨!¡)¡Ñ,Ô,¨s¬wÑ6Ð6ð ð ð ð 	��Q’�˜!˜qš&˜&Ø�Q‘3ˆJàŒwˆð ‡y‚y��|„|ð ˜˜Qš˜Ø�IŠI�a‰LŒLð 	˜Q !˜[˜[š[˜[ qš[˜[˜[˜[Ø�IŠI�a‰LŒLð )Ø !˜[˜[š[˜[ qš[˜[˜[˜[˜[Øð ”7ˆNåˆq‰6Œ6€Dð ˆs‚{€{�s—y’y ‘|”|€{¨¨Qª¨°1¸²:°:Ø—y’y ‘|”|ð 4>Ø()¨Qª¨°1¸²:°:ØˆsŒz˜!˜Q ¨¨uÐ 5¸¸1¸a°yÀ!ÐNÐNÀvÐNÐNÐNàŒ8€DðØˆŒ�B‰ˆŒð �3Š;ˆ;ðð ð ð ð ð ð ð
 �”˜a ! A Ð1Ð1¨&Ð1Ð1ˆAˆAõ ��1‘‰XŒX˜ÒÐðð ð ð ð ð ð
 �”˜a ! A Ð1Ð1¨&Ð1Ð1ˆAˆAõ ��A�a‘C‘‰\Œ\˜TÒ!Ð!Ø—
’
˜1˜a ™c 1 a¨¨1©¡gÑ.Ô.°!°A±#¸±Ñ9ˆAˆAõ ˜s 1 Q q¨Ð4Ð4¨VÐ4Ð4ˆAð ˆŒˆø�4ˆŒˆˆˆˆØˆ2€Is   È0B2K* Ë*	K3c           
      óþ  ‡ ‡‡‡‡‡‡‡‡‡‡‡‡ — t          ‰Ž \  Š}t          ‰Ž \  Š}t          ‰¦  «        Št          ‰¦  «        Št          ‰¦  «        }	d}
‰D ]!}‰                      |¦  «        r
|dk    rd}
 nŒ"|	dk     s|
r2	  ‰ j        ‰‰||z   ‰‰z   ‰fi |¤ŽS # ‰ j        $ r |	dk    s|
r‚ Y nw xY w‰dk    rP‰                      t          ‰¦  «        t          ‰¦  «        z
  ¦  «        }|dk    r ‰ j        ‰‰dfi |¤Ž‰ j	        z  S ‰‰fdk    rØt          ‰dz
  ¦  «        dk     rÂ‰\  ŠŠŠ‰\  ŠŠ‰‰z   ‰z
  }‰  
                    ‰‰z
  ‰‰z
  ‰‰g‰‰z
  ‰‰z
  d|g¦  «        }d|ifˆˆˆˆˆˆ ˆfd	„	Š 	 ‰                      ‰ d‰ j	        g|                     d
¦  «        |                     dd¦  «        ¬¦  «        }|‰  
                    ‰‰g‰‰‰g¦  «        z  S # ‰ j        $ r Y nw xY w|	dk     �rõ‰                      ‰¦  «        dk    �rÛd‰ j        ifˆˆˆ ˆˆˆ ˆfd„	Š |                     dd¦  «        }	 ‰                      ‰ d‰ j	        g|                     d
¦  «        |                     dd¦  «        |                     dd¦  «        ¬¦  «        S # ‰ j        $ r d|vr‚ Y nw xY w|                     d
¦  «        rt!          d¦  «         	 ˆˆˆ ˆfd„Šˆˆˆ ˆfd„}‰ j        dz  }‰ j        }	 d‰ j        z  }‰ xj        dz  c_        t)          d¦  «        D ]·}‰                      ˆ fd„t)          |¦  «        D ¦   «         ¦  «        }‰                      ‰ |‰ j	        g| ||¦  «        |                     d
¦  «        dd¬¦  «        \  }}||k     r||z   } n9|dz  }‰ xj        ‰ j        dz  z  c_        |dk    r‰                      d¦  «        ‚Œ¸|‰ _        n# |‰ _        w xY w|
 S ˆ ˆˆˆfd„} ‰ j        |‰‰z   fi |¤ŽS ) z&
    Evaluates 3F2, 4F3, 5F4, ...
    Fr   Tr   gš™™™™™ñ?gÍÌÌÌÌÌì?)r   r   gš™™™™™©?c                 óÔ   •— ‰‰z   ‰z
  | z   }| |v r	||          }n5|| dz
           }|‰| z   ‰z
  dz
  ‰| z   ‰z
  dz
  z  z  }|| |dz
  z  z  }||| <   |‰	                      ‰‰|‰
¦  «        z  S r®   )r¡   )r$   Ú_cacher.   ró   rœ   r£   r«   r�   rž   r   r"   s       €€€€€€€r/   r   z_hypq1fq.<locals>.term  s“   ø€ Ø�2‘�b‘˜‘
ˆAØ�Fˆ{ˆ{Ø˜1”I��à˜1˜Q™3”K�Ø�b˜‘d˜2‘g˜a‘i " Q¡$ r¡'¨!¡)Ñ,Ñ,�Ø�Q˜˜!™‘W‘�Ø��q‘	Ø�s—z’z " R¨¨!Ñ,Ô,Ñ,Ð,r1   r3   Ústrict)r3   r  c                 óæ  •— t          | ¦  «        }|| k    rl‰‰
                     | ¦  «        z  ‰
                     | ¦  «        z  }‰D ]}|‰
                     || ¦  «        z  }Œ‰	D ]}|‰
                     || ¦  «        z  }Œ|S ||v r||         S  ‰|dz
  ¦  «        }|dz
  }t	          ‰¦  «        D ]}|‰|         |z   z  }Œt	          ‰¦  «        D ]}|‰	|         |z   z  }Œ|‰z  }||z  }|||<   |S r®   )r   rÕ   ÚfacÚrfr   )Úkkr  r$   ró   rA   rE   Úmr¶   r    r!   r   r�   r�   r   r"   s           €€€€€€€r/   r   z_hypq1fq.<locals>.term'  s  ø€ Ý�B‘”ˆAØ�BŠwˆwØ˜Ÿš ™œÑ$ s§w¢w¨r¡{¤{Ñ2�ØÐ/Ð/�A˜a 3§6¢6¨!¨B¡<¤<Ñ/˜a˜aØÐ/Ð/�A˜a 3§6¢6¨!¨B¡<¤<Ñ/˜a˜aØ�Ø�Fˆ{ˆ{Ø˜a”yÐ Ø��Q�q‘S‘	”	ˆAØ�!‘ˆAÝ˜A‘Y”YÐ/Ð/�  c¨!¤f¨Q¡h¡  Ý˜A‘Y”YÐ/Ð/�  c¨!¤f¨Q¡h¡  Ø�‰FˆAØ�‰FˆAØˆF�1‰IØˆHr1   Ú
sum_methodzr+s+erÜ   Ú )r3   r  Úmethodz$Attempting Euler-Maclaurin summationc              3   óŒ  •‡ ‡K  — ‰dgz   }t          ˆˆ fd„‰D ¦   «         ¦  «        t          ˆˆ fd„|D ¦   «         ¦  «        z
  ‰ ‰                     ‰¦  «        z  z   V — dŠ	 t          ˆˆˆ fd„‰D ¦   «         ¦  «        t          ˆˆˆ fd„|D ¦   «         ¦  «        z
  }‰dk    r|‰                     ‰¦  «        z  }|V — ‰dz  ŠŒc)Nr   c              3   óH   •K  — | ]}‰                      |‰z   ¦  «        V — Œd S r•   ©Úloggamma)r<   rA   r   Úk0s     €€r/   ú	<genexpr>z._hypq1fq.<locals>.log_diffs.<locals>.<genexpr>i  s3   øè è € Ð6Ð6¨Q�c—l’l 1 R¡4Ñ(Ô(Ð6Ð6Ð6Ð6Ð6Ð6r1   c              3   óH   •K  — | ]}‰                      |‰z   ¦  «        V — Œd S r•   r  )r<   rE   r   r  s     €€r/   r  z._hypq1fq.<locals>.log_diffs.<locals>.<genexpr>j  s3   øè è € Ð3Ð3¨1�C—L’L  2¡Ñ&Ô&Ð3Ð3Ð3Ð3Ð3Ð3r1   r   c              3   óJ   •K  — | ]}‰                      ‰|‰z   ¦  «        V — Œd S r•   ©Úpsi)r<   rA   r   r)   r  s     €€€r/   r  z._hypq1fq.<locals>.log_diffs.<locals>.<genexpr>m  s3   øè è € Ð5Ð5¨A˜Ÿš  ! B¡$™œÐ5Ð5Ð5Ð5Ð5Ð5r1   c              3   óJ   •K  — | ]}‰                      ‰|‰z   ¦  «        V — Œd S r•   r  )r<   rE   r   r)   r  s     €€€r/   r  z._hypq1fq.<locals>.log_diffs.<locals>.<genexpr>n  s3   øè è € Ð4Ð4¨A˜Ÿš  ! B¡$™œÐ4Ð4Ð4Ð4Ð4Ð4r1   )r   Úlog)r  rž   rm   r)   r    r!   r   r"   s   `  @€€€€r/   Ú	log_diffsz_hypq1fq.<locals>.log_diffsg  s  øøøè è € Ø˜�s‘ˆBÝÐ6Ð6Ð6Ð6Ð6°#Ð6Ñ6Ô6Ñ6Ô6ÝÐ3Ð3Ð3Ð3Ð3°Ð3Ñ3Ô3Ñ3Ô3ñ4Ø68¸¿ºÀ¹¼±mñDð Dð Dð DàˆAðÝÐ5Ð5Ð5Ð5Ð5Ð5°Ð5Ñ5Ô5Ñ5Ô5ÝÐ4Ð4Ð4Ð4Ð4Ð4°Ð4Ñ4Ô4Ñ4Ô4ñ5�à˜’6�6Ø˜Ÿš ™œ‘O�AØ���Ø�Q‘�ðr1   c              3   ó´   •K  — ‰                      d„ ‰D ¦   «         d„ ‰D ¦   «         ¦  «        }‰                      ‰| ¦  «        ¦  «        D ]}||z  }|V — Œd S )Nc                 ó   — g | ]}|‘ŒS r;   r;   ©r<   rE   s     r/   r=   z1_hypq1fq.<locals>.hyper_diffs.<locals>.<listcomp>u  s   € Ð.Ð.Ð. Q˜qÐ.Ð.Ð.r1   c                 ó   — g | ]}|‘ŒS r;   r;   ©r<   rA   s     r/   r=   z1_hypq1fq.<locals>.hyper_diffs.<locals>.<listcomp>u  s   € Ð0@Ð0@Ð0@°q°Ð0@Ð0@Ð0@r1   )rø   Ú	diffs_exp)r  ÚCr,   rm   r    r!   r   r  s       €€€€r/   Úhyper_diffsz_hypq1fq.<locals>.hyper_diffst  sy   øè è € Ø—’Ð.Ð.¨#Ð.Ñ.Ô.Ð0@Ð0@¸CÐ0@Ñ0@Ô0@ÑAÔAˆAØ—]’] 9 9¨R¡=¤=Ñ1Ô1ð ð �Ø˜‘E�Ø����ðð r1   i   é2   rL   é   c              3   ó.   •K  — | ]} ‰|¦  «        V — Œd S r•   r;   )r<   r$   r   s     €r/   r  z_hypq1fq.<locals>.<genexpr>€  s+   øè è € Ð?Ð?¨A   Q¡¤Ð?Ð?Ð?Ð?Ð?Ð?r1   )ræ   Úadiffsr3   ÚerrorÚ_fast_abortr   r³   z*Euler-Maclaurin summation did not convergec            
      ó>  •‡
‡‡‡— t          | d ‰…         ¦  «        Š
t          | ‰d …         ¦  «        Šg }‰j        ‰z  }‰                     ‰d¬¦  «        }t          ‰dz   ¦  «        D ]³Š‰
‰         Š|g}‰ g}‰‰gz   ˆ
ˆˆfd„t          ‰dz   ¦  «        D ¦   «         z   }‰
ˆˆfd„t          ‰¦  «        D ¦   «         z   }‰gˆˆfd„t          ‰¦  «        D ¦   «         z   }ˆ
ˆˆfd„t          ‰dz   ¦  «        D ¦   «         }	|                     ||||||	|f¦  «         Œ´|S )NTrÇ   r   c                 ó2   •— g | ]}|‰k    ¯‰|         ‰z
  ‘ŒS r;   r;   ©r<   r¶   r    Úakr$   s     €€€r/   r=   z'_hypq1fq.<locals>.h.<locals>.<listcomp>¡  s&   ø€ ÐGÐGÐG¨QÀÀQÂÀ˜s 1œv b™yÀÀÀr1   c                 ó&   •— g | ]}‰|         ‰z
  ‘ŒS r;   r;   ©r<   r¶   r*  r!   s     €€r/   r=   z'_hypq1fq.<locals>.h.<locals>.<listcomp>¢  s!   ø€ Ð4Ð4Ð4 a˜˜Aœ˜r™	Ð4Ð4Ð4r1   c                 ó,   •— g | ]}‰‰|         z
  d z   ‘ŒS ©r   r;   r,  s     €€r/   r=   z'_hypq1fq.<locals>.h.<locals>.<listcomp>£  s%   ø€ Ð7Ð7Ð7¨˜˜C œF™ 1™Ð7Ð7Ð7r1   c                 ó8   •— g | ]}|‰k    ¯d ‰|         z
  ‰z   ‘ŒS r.  r;   r)  s     €€€r/   r=   z'_hypq1fq.<locals>.h.<locals>.<listcomp>¤  s*   ø€ Ð<Ð<Ð< !°Q¸!²V°V�!�C˜”F‘(˜2‘+°V°V°Vr1   )ÚlistrU   rÊ   rV   r   )ÚargsÚTsÚreczÚnegzr  ÚCpÚGnÚGdÚFnÚFdr    r*  r!   r$   r   r�   r�   r"   s             @@@@€€€€r/   rj   z_hypq1fq.<locals>.h—  s]  øøøøø€ Ý�4˜˜˜”8‰nŒnˆÝ�4˜˜˜”8‰nŒnˆØˆØŒw�q‰yˆØ�xŠx˜ ˆxÑ&Ô&ˆÝ�q˜‘s‘”ð 	5ð 	5ˆAØ�Q”ˆBØ�ˆAØ�#�ˆBØ˜�t‘ÐGÐGÐGÐGÐGÐGµ%¸¸!¹±*´*ÐGÑGÔGÑGˆBØÐ4Ð4Ð4Ð4Ð4­5°©8¬8Ð4Ñ4Ô4Ñ4ˆBØ�Ð7Ð7Ð7Ð7Ð7­e°A©h¬hÐ7Ñ7Ô7Ñ7ˆBØ<Ð<Ð<Ð<Ð<Ð<¥u¨Q¨q©S¡z¤zÐ<Ñ<Ô<ˆBØ�IŠI�q˜"˜b " b¨"¨dÐ3Ñ4Ô4Ð4Ð4Øˆ	r1   )r[   r0  rú   rÍ   rŽ   rÁ   r   r   rZ   r_   rø   ÚnsumrN   rÀ   rU   ÚreplacerR   rù   r   Údpsr   r\   Úsumemrv   )!r   r�   r�   r    r!   r"   rc   Úa_typesÚb_typesrÿ   ÚispolyrA   ÚSr.   Úinitialr	  r   ræ   r   Útruncr)   ÚheadÚtailÚerrrm   rj   rœ   r£   r«   r�   rž   r  r   s!   ``````                    @@@@@@@r/   rŒ   rŒ   ß  s©  øøøøøøøøøøøøø€ õ
 ˜�9�L€CˆÝ˜�9�L€CˆÝ
ˆs‰)Œ)€CÝ
ˆs‰)Œ)€CÝˆq‰6Œ6€DØ€FØð ð ˆØ�9Š9�Q‰<Œ<ð 	˜A šF˜FØˆFØˆEøàˆa‚x€x�6€xð	Ø�3”:˜a  G¨G¡O°S¸±W¸aÐJÐJÀ6ÐJÐJÐJøØÔ ð 	ð 	ð 	Ø�cŠzˆz˜VˆzØð ˆzð	øøøð$ 	ˆA‚v€vð �FŠF•3�s‘8”8�C ™HœHÑ$Ñ%Ô%ˆØ�Š6ˆ6à�3”9˜S # sÐ5Ð5¨fÐ5Ð5¸¼Ñ?Ð?Ø	ˆ!€u�‚~€~�#˜a ™c™(œ( Tš/˜/à‰ˆˆ2ˆbØ‰ˆˆ2Øˆr‰E�"‰HˆØ—-’-  B¡ r¨"¡u¨R°Ð 3°R¸±U¸2¸b¹5ÀÀ1Ð4EÑFÔFˆØ˜g˜;ð 		-ð 		-ð 		-ð 		-ð 		-ð 		-ð 		-ð 		-ð 		-ð 		-ð 		-ð 		-ð	Ø—’˜  #¤'˜{°F·J²J¸yÑ4IÔ4IØ—z’z (¨DÑ1Ô1ð ñ 3ô 3ˆAà�s—}’} b¨ W¨b°°B¨ZÑ8Ô8Ñ8Ð8øØÔ ð 	ð 	ð 	ØˆDð	øøøð ˆc‚z�z�c—g’g˜a‘j”j A’o‘oà˜sœw˜Kð 	ð 	ð 	ð 	ð 	ð 	ð 	ð 	ð 	ð 	ð 	ð 	ð$ —Z’Z ¨gÑ6Ô6ˆ
ð	Ø—8’8˜D 1 S¤W +°v·z²zÀ)Ñ7LÔ7LØ—z’z (¨DÑ1Ô1Ø!×)Ò)¨#¨bÑ1Ô1ð ñ 3ô 3ð 3øð Ô ð 	ð 	ð 	Ø˜*Ð$Ð$ØØˆDð	øøøð
 �:Š:�iÑ Ô ð 	:ÝÐ8Ñ9Ô9Ð9ð	ð>	ð 	ð 	ð 	ð 	ð 	ð 	ð 	ð	ð 	ð 	ð 	ð 	ð 	ð 	ð 	ð Œg˜‰nˆØŒxˆð	Ø˜œ‘LˆEØˆHŒH˜‰NˆHŒHÝ˜A‘Y”Yð Fð F�Ø—x’xÐ?Ð?Ð?Ð?µ¸±´Ð?Ñ?Ô?Ñ?Ô?�ØŸIšI d¨U°C´GÐ,<À#Ø&˜; uÑ-Ô-Ø"ŸJšJ yÑ1Ô1ØØ $ð	 &ñ &ô &‘	��cð
 ˜’9�9Ø˜t™�AØ�EØ˜‘
�ð �”˜CœH a™KÑ'�”Ø˜’6�6Ø×+Ò+ØDñFô Fð Fð ð ˆCŒHˆHø�tˆCŒHˆOˆOˆOˆOØˆrˆ	ð
ð ð ð ð ð ð ð ð  ˆ3Œ=˜˜C ™GÐ.Ð. vÐ.Ð.Ð.sE   ÂB ÂB0Â/B0Å4A$G Ç
G&Ç%G&È5AJ ÊJ#Ê"J#Ë-C!O Ï	Oc                 ól  ‡ ‡‡‡‡— ‰rt          ‰Ž \  Š}t          ‰¦  «        Šng dcŠ}‰rt          ‰Ž \  Š}t          ‰¦  «        Šng dcŠ}|                     d‰ j        ¦  «        |d<   	  ‰ j        ||||z   ‰‰z   ‰fi |¤ŽS # ‰ j        $ r Y nw xY w‰ j        }		 |                     d‰ j        dz  ¦  «        }
‰ xj        dz  c_        d‰ j        ifˆˆˆˆfd„	Š‰ j        }t          d‰ j        ¦  «        D ]0} ‰|¦  «        }||z  }t          |¦  «        |
k    r|c |	‰ _        S Œ1	 |	‰ _        n# |	‰ _        w xY w||d	z   k    rê|                     d
¦  «        }|s�‰  
                    ‰¦  «        dk     r_‰t          dt          ‰¦  «        ¦  «        z  }‰  
                    ‰¦  «        dk    rddd|z  d|z  ‰ j        g}nddd|z  d|z  ‰ j        g}n	d‰ j        g}|                     di ¦  «        }ˆˆˆ ˆfd„} ‰ j        ||fddi|¤Ž\  }}|t          |¦  «        ‰ j        z  dz  k    r|S ‰ j        ‚)Nr;   ÚmaxtermsÚ	asymp_tolr³   r7   r   c                 óš   •— | |v r||          S  ‰| dz
  ¦  «        }‰D ]}||| dz
  z   z  }Œ‰D ]}||| dz
  z   z  }Œ|‰z  }|| z  }||| <   |S r®   r;   )	r$   Úcacheró   rA   rE   r    r!   r   r"   s	        €€€€r/   r   z_hyp_borel.<locals>.term¿  s€   ø€ Ø�EˆzˆzØ˜Q”x�Ø��Q�q‘S‘	”	ˆAØÐ(Ð(�˜!  1 Q¡3¡™.˜!˜!ØÐ(Ð(�˜!  1 Q¡3¡™.˜!˜!Ø�‰FˆAØ�‰FˆAØˆE�!‰HØˆHr1   r   r   Úcontourg      Ð?y               @y       @       @r   y       €       Ày       @       ÀÚquad_kwargsc                 ón   •— ‰                      |  ¦  «        ‰                     ‰‰dgz   | ‰z  ¦  «        z  S r®   )r„   rZ   )ró   r    r!   r   r"   s    €€€€r/   râ   z_hyp_borel.<locals>.gà  s4   ø€ Ø—7’7˜A˜2‘;”;˜sŸyšy¨¨c°1°#©g°q¸±sÑ;Ô;Ñ;Ð;r1   r%  Tr°   )r[   r0  rN   r   rŽ   rÁ   rù   rU   r   rú   Úargr]   r_   Úquad)r   r�   r�   r    r!   r"   rc   r>  r?  r   ræ   Úsr$   ró   rL  r.   rM  râ   ÚIrF  r   s   `  ```              @r/   r�   r�   ©  sè  øøøøø€ à
ð Ý˜C�y‰ˆˆWÝ�3‰iŒiˆˆà˜2ˆˆˆWØ
ð Ý˜C�y‰ˆˆWÝ�3‰iŒiˆˆà˜2ˆˆˆWØŸš J°´Ñ9Ô9€Fˆ:ÑðØˆsŒz˜!˜Q ¨¡°°S±¸!ÐFÐF¸vÐFÐFÐFøØÔð ð ð ØˆðøøøàŒ8€DðØ�jŠj˜ c¤g¨a¡iÑ0Ô0ˆØˆŒ�B‰ˆŒà˜SœW˜+ð 		ð 		ð 		ð 		ð 		ð 		ð 		ð 		ð 		ð ŒGˆÝ˜˜3œ8Ñ$Ô$ð 	ð 	ˆAØ��Q‘”ˆAØ�‰FˆAÝ�1‰vŒv˜Š}ˆ}Ø��àˆŒˆð ð	ð ˆŒˆø�4ˆŒˆˆˆˆØˆAˆa‰C‚x€xØ—*’*˜YÑ'Ô'ˆØð 	'Ø�wŠw�q‰zŒz˜DÒ Ð Ø�˜A�s 1™vœv™œÑ&�Ø—7’7˜1‘:”: ’?�?Ø  " t¨Q¡h°°!±°S´WÐ=�G�Gà  #¨¨q¡y°!°A±#°s´wÐ?�G�Gð
 ˜cœg˜,�Ø—j’j °Ñ3Ô3ˆð	<ð 	<ð 	<ð 	<ð 	<ð 	<ð 	<ð 	<à�”˜!˜WÐ@Ð@¨DÐ@°KÐ@Ð@‰ˆˆ3Ø•#�a‘&”&˜œ‘. Ñ"Ò"Ð"ØˆHØ
Ô
Ðs%   Á+B Â
BÂBÂBD0 Ä&D0 Ä0	D9c           	      ój  ‡ ‡— |\  \  }}\  }}|\  \  }	}
\  }}t          ‰¦  «        }‰                      ‰¦  «        }‰ j        }|}|                     d¦  «         o‰                      |¦  «        dk    }|r¤	 	 ‰ xj        |z  c_        ˆ ˆfd„}‰                      ||||	|gdd‰ j        z  ¬¦  «        }t          ˆ fd„|||	|‰fD ¦   «         ¦  «        dk    r‰                      |¦  «        }||‰ _        S # ‰ j        $ r Y nw xY w	 |‰ _        n# |‰ _        w xY w ‰ j        d	d	|||
|f|||	|g‰fi |¤ŽS )
Nr€   r   c                 ó¸  •— | |z   |z
  |z
  }| |z   }||z   }i }‰j         |d<   |dz
  |z  ||z  z   | |z  z
  |d<   d}d}	d}
	 |	|vr•d|z
  d| z  z   | dz  z   d|z  z   |dz  z   ||z  z
  | |z  z   ||z  z   d|z  d|dz   z  z
  |	z  z   d|	dz  z  z   }|	|z
  |z   dz
  |	|z
  |z   dz
  z  |	|z
  dz
  z  }‰j         |	z  |||	dz
           z  |||	dz
           z  z
  z  ||	<   ||	         ‰|	 z  z  }t          |¦  «        d‰j        z  k     rn>|	dk    r*t          |
¦  «        t          |¦  «        z  dk     r‰j        ‚||z  }|}
|	dz  }	�Œ‰                     ‰¦  «        |z  }‰|g|dg||g| |gg g df}‰ g|  g|||| z
  g||| z
  || z
  g| | |z
  dz   | |z
  dz   g| |z
  dz   gd‰z  f}‰ g| g||| |z
  g| ||z
  ||z
  g|||z
  dz   ||z
  dz   g|  |z   dz   gd‰z  f}|||fS )	Nr   r   r   r   çš™™™™™¹?r"  ç      ø?r´   )rU   rú   rù   rÁ   r„   )rœ   r£   r�   rž   ÚXÚA2ÚB2rH   Ús1r$   ÚtprevÚuu1Úuu2Út1rA  r»   r¼   ÚT3r   r"   s                     €€r/   rj   z_hyp2f2.<locals>.h  sÙ  ø€ Ø˜2™˜b™ ™�AØ˜B™�BØ˜B™�BØ�AØœ7�A�a‘DØ˜q™D !™8 B r¡E™>¨"¨R©%Ñ/�A�a‘DØ�BØ�AØ�EðØ A˜:˜:Ø"# B¡$ q¨¡t¡)¨B°©E¡/°!°B±$Ñ"6°r¸1±uÑ"<¸RÀ¹UÑ"BÀ2ÀbÁ5Ñ"HÈÈBÉÑ"NÐPQÐRTÑPTÐUVÐXZÐ[\ÑX\ÑU]ÑP]Ð_`ÑO`Ñ"`ÐabÐcdÐfgÑcgÑagÑ"g˜CØ#$ R¡4¨¡7¨1¡9¨q°©t°B©w°q©yÑ"9¸1¸Q¹3¸q¹5Ñ"A˜CØ#&¤7¨1¡9°°A°a¸±c´F±
¸3¸qÀÀ1Á¼v¹:Ñ0EÑ#F˜A˜a™DØ˜qœT A¨¨¡G™^˜Ý˜r™7œ7 S¨¬¡[Ò0Ð0à!à˜qš5˜5¥S¨¡Z¤Zµ#°b±'´'Ñ%9¸CÒ%?Ð%?à"%Ô"3Ð3Ø˜b™˜Ø "˜Ø˜Q™˜ñð  Ÿš ™
œ
 2™�AØ˜A˜  1 ¨¨2 w°°2¨w°r¸"¸QÐ>�BØ˜"˜ ˜s˜e R¨¨2¨b©5 M°2°b¸±e¸B¸r¹EÐ2BÀBÀrÈ"ÁuÈQÁwÈrÐRTÉuÐUVÉwÐCWÐY[Ð\^ÑY^Ð_`ÑY`ÐXaÐbdÐefÑbfÐf�BØ˜"˜ ˜s˜e R¨¨2¨b©5 M°2°b¸±e¸B¸r¹EÐ2BÀBÀrÈ"ÁuÈQÁwÈrÐRTÉuÐUVÉwÐCWÐZ\ÐY\Ð]_ÑY_Ð`aÑYaÐXbÐceÐfgÑcgÐg�BØ˜r 2˜:Ð%r1   Tr³   ©r€   rH  c              3   óB   •K  — | ]}‰                      |¦  «        V — Œd S r•   ©r¿   ©r<   r.   r   s     €r/   r  z_hyp2f2.<locals>.<genexpr>!  s1   øè è € ÐEÐE°�s×(Ò(¨Ñ+Ô+ÐEÐEÐEÐEÐEÐEr1   r"  r   )	rú   rJ   r   rN   rv   r   r   rÁ   rŽ   )r   r    r!   r"   rc   rœ   Úa1typer£   Úa2typer�   Úb1typerž   Úb2typerÿ   rÃ   rd   Úasymp_extraprecÚcan_use_asymptoticrj   rm   s   `  `                r/   r‰   r‰   è  s´  øø€ à!$Ñ�L€Rˆ‘,�2�vØ!$Ñ�L€Rˆ‘,�2�våˆq‰6Œ6€DØ�7Š7�1‰:Œ:€DØŒ8€Dð €Oð %Ÿjšj¨Ñ8Ô8Ð8ð Ø	�Š�‰Œ˜Ò	ð ð
 ð -ð+	ð(Ø�”˜OÑ+�”ð&ð &ð &ð &ð &ð &ð> —M’M ! b¨¨B¨r ]ÀÐPQÐRUÔRZÑPZ�MÑ[Ô[�ÝÐEÐEÐEÐE°b¸¸B¸rÀ!°_ÐEÑEÔEÑEÔEÈÒJÐJØŸš˜q™	œ	�AØð ˆCŒHˆHøð Ô$ð ð ð Ø�ðøøøØàˆCŒHˆHø�tˆCŒHˆOˆOˆOˆOàˆ3Œ:�a˜˜V V¨V°VÐ<¸rÀ2ÀrÈ2Ð>NÐPQÐ\Ð\ÐU[Ð\Ð\Ð\s%   Á9A5C6 Ã6
DÄ D ÄDÄD Ä	Dc                 óœ  ‡ ‡— |\  \  }}|\  \  }}\  }	}
t          ‰¦  «        }‰                      ‰¦  «        }‰ j        }‰o|dz  }|                     d¦  «         o4‰                      |¦  «        dk    o‰                      |¦  «        d|z  k    }|r¢	 	 ‰ xj        |z  c_        ˆ ˆfd„}‰                      ||||	gdd‰ j        z  ¬¦  «        }t          ˆ fd	„|||	‰fD ¦   «         ¦  «        dk    r‰                      |¦  «        }||‰ _        S # ‰ j        $ r Y nw xY w	 |‰ _        n# |‰ _        w xY w ‰ j	        d
d|||
f|||	g‰fi |¤ŽS )Nr   r€   é   rV  c                 ó  •— ‰j         | |z
  |z
  ‰j         z   z  }i }‰j        |d<   d‰j        d| z  |z   |z   dz
  z  | |z
  |z
  z  ||z  z   ‰j        z
  z  |d<   d||z  ‰j        | |z
  |z
  z  d| z  |z   |z   dz
  z  z   ‰j        z
  dz  z  ‰j        dd| z  dz
  z  |z  |z  d| |z
  |z
  z  d| dz  z  d| z  z   |z   |z   dz
  z  z   dz
  z  z   |d<   d}d}d}d}	 ||vr²d|dz  z  d	| z  d|z  z   d|z  z   dz
  |z  z   d| dz  z  z   ||z
  dz  z
  d| z  ||z   dz
  z  z
  ‰j        z   }	|| z
  |z   |z
  ‰j         z
  || z
  |z
  |z   ‰j         z
  z  || z
  |z   |z   ‰j        z
  z  }
‰j        d|z  z  |	||dz
           z  |
||dz
           z  z
  z  ||<   ||         ‰ d
|z  z  z  }‰j         |z  ‰                     d¦  «        | z  z  |z  }‰j        |z  ‰                     d¦  «        | z  z  |z  }t          |¦  «        d‰j	        z  k     rnC|dk    r*t          |¦  «        t          |¦  «        z  dk     r‰j
        ‚||z  }||z  }|}|dz  }�Œv‰                     ‰j        |z  d‰                     ‰ ¦  «        z  z   ¦  «        |z  ‰                     ‰j        |z  d‰                     ‰ ¦  «        z  z    ¦  «        |z  z   }d|z  ‰j        ‰ gdd
|g||g| gg g df}‰ g|  g||g|| z
  || z
  g| | |z
  dz   | |z
  dz   gg d‰z  f}||fS )Nr   r   r   r   éðÿÿÿr³   éøÿÿÿé   iúÿÿÿç      à¿rU  r"  rV  ç      à?)r·   rU   Úmpq_1_4Úmpq_3_16Úmpq_1_16Úmpq_5_2r¶   rÕ   rú   rù   rÁ   Úexpjr¾   rµ   )rœ   r�   rž   rW  rH   rZ  Ús2r$   r[  r\  r]  r%   r^  Út2rA  r»   r¼   r   r"   s                    €€r/   rj   z_hyp1f2.<locals>.hI  s÷  ø€ Øœ R¨¡U¨2¡X¨c¬kÑ%9Ñ:�AØ�AØœ7�A�a‘DØ˜cœk¨1¨R©4°©7°2©:°a©<Ñ8¸"¸R¹%À¹(ÑCÀBÀrÁEÑIÈ#Ì,ÑVÑW�A�a‘DØ˜b ™e C¤K°°B±°r±Ñ$:¸A¸b¹DÀ¹GÀB¹JÀq¹LÑ$IÑIÈ#Ì,ÑVÐYZÑZÑZØœ c¨1¨R©4°©6¡l°2¡o°bÑ&8Ø˜2˜b™5 ™8™ b¨¨Q©¡h¨r°"©u¡n°RÑ&7¸Ñ&:¸1Ñ&<Ñ=ñ'>Ø>?ñ'@ñ AñA�A�a‘Dð �BØ�BØ�AØ�EðØ A˜:˜:Ø#$ Q¨¡T¡6¨2¨b©5°°2±©:°a¸±d©?¸1Ñ+<¸aÑ*?Ñ#?À!ÀBÈÁEÁ'Ñ#IØ!# B¡¨¡
ñ$+Ø-.¨r©T°2°b±5¸±7©^ñ$<Ø>A¼kñ$J˜Cà#$ R¡4¨¡7¨2¡:¨c¬kÑ#9¸A¸b¹DÀ¹GÀB¹JÀsÄ{Ñ<RÑ"SØ!" 2¡ b¡¨¡¨C¬KÑ!7ñ#9˜Cà#&¤7¨A¨a©C¡=°#°a¸¸!¹´f±*¸SÀÀ1ÀQÁ3Ä¹ZÑ2GÑ#H˜A˜a™DØ˜aœD Q B¨$¨q©&¡>Ñ1˜Ø"œu˜f q™[¨3¯7ª7°1©:¬:¸¸Ñ+;Ñ;¸aÑ?˜Ø œU A™X¨¯ª°©
¬
°a°RÑ(8Ñ8¸1Ñ<˜Ý˜r™7œ7 S¨¬¡[Ò0Ð0à!à˜qš5˜5¥S¨¡Z¤Zµ#°b±'´'Ñ%9¸CÒ%?Ð%?à"%Ô"3Ð3Ø˜b™˜Ø˜b™˜Ø "˜Ø˜Q™˜ñ)ð* Ÿš ¤¨¡¨!¨C¯HªH°a°R©L¬L©.Ñ!8Ñ9Ô9¸"Ñ<ØŸš 3¤6¨!¡8¨A¨c¯hªh¸°r©l¬l©NÑ#:Ð!;Ñ<Ô<¸RÑ?ñ@�Aà˜a™% ¤¨!¨Ð,¨q°$¸¨l¸RÀ¸HÀrÀdØ˜B ð"�Bà˜"˜  ˜u r¨" g¨r°"©u°R¸±U¨mØ˜B˜r™E !™G B r¡E¨!¡GÐ,¨b°!°A±#ð6�Bà˜r˜6�Mr1   Tr³   r`  c              3   óB   •K  — | ]}‰                      |¦  «        V — Œd S r•   rb  rc  s     €r/   r  z_hyp1f2.<locals>.<genexpr>r  s1   øè è € ÐBÐB°�s×(Ò(¨Ñ+Ô+ÐBÐBÐBÐBÐBÐBr1   r   ©
rú   rJ   r   rN   rµ   rv   r   r   rÁ   rŽ   )r   r    r!   r"   rc   rœ   rd  r�   rf  rž   rg  rÿ   rÃ   rd   rh  ri  rj   rm   s   `  `              r/   r†   r†   -  sÄ  øø€ à�M�L€RˆØ!$Ñ�L€Rˆ‘,�2�våˆq‰6Œ6€DØ�7Š7�1‰:Œ:€DØŒ8€Dð �m˜D !™G€Oð %Ÿjšj¨Ñ8Ô8Ð8ð $Ø	�Š�‰Œ˜Ò	ð$à	�Š�$‰Œ˜#˜d™(Ò	"ð ð ð 6ð4	ð1Ø�”˜OÑ+�”ð'"ð '"ð '"ð '"ð '"ð '"ðP —M’M ! b¨¨B Z¸dÈQÈsÌxÉZ�MÑXÔX�ÝÐBÐBÐBÐB°b¸¸B¸q°\ÐBÑBÔBÑBÔBÀaÒGÐGØŸš˜q™	œ	�AØð ˆCŒHˆHøð Ô$ð ð ð Ø�ðøøøØàˆCŒHˆHø�tˆCŒHˆOˆOˆOˆOð ˆ3Œ:�a˜˜V V¨VÐ4°r¸2¸r°lÀAÐPÐPÈÐPÐPÐPs%   ÂA3D Ä
DÄD* ÄDÄD* Ä*	D3c           
      ó¼  ‡ ‡— |\  \  }}\  }}|\  \  }	}
\  }}\  }}t          ‰¦  «        }‰                      ‰¦  «        }‰o|dz  }‰ j        }|                     d¦  «         o4‰                      |¦  «        dk    o‰                      |¦  «        d|z  k    }|r¦	 	 ‰ xj        |z  c_        ˆ ˆfd„}‰                      ||||	||gdd‰ j        z  ¬¦  «        }t          ˆ fd	„|||	||‰fD ¦   «         ¦  «        d
k    r‰                      |¦  «        }||‰ _        S # ‰ j        $ r Y nw xY w	 |‰ _        n# |‰ _        w xY w ‰ j	        dd|||
||f|||	||g‰fi |¤ŽS )Nr   r€   rk  rV  c           	      ó  •— ‰j         | |z   |z
  |z
  |z
  ‰j         z   z  }| |z   }||z   |z   }| |z  }||z  ||z  z   ||z  z   }	||z  |z  }
i }‰j        |d<   d|	|z
  ‰j        d|z  |z   dz
  z  ||z
  z  z   ‰j        z
  z  |d<   ‰j         |d         dz  z  ‰j        dd|z  dz
  z  |	|z
  z  d|
z  z   dd|dz  z  d	|z  z   d
|z  z   |z   dz
  z  ||z
  z  z   dz
  z  z   |d<   d}d}d}d}	 ||v�r3|d|z  z
  dz
  |d|z  z
  d|z  z
  dz
  z  |d|z  z
  d|z  z
  dz
  z  |d|z  z
  d|z  z
  dz
  z  }d|dz
  dz  z  dd|z  |z   z  |dz
  dz  z  z
  dd|dz  z  d|z  |z  z   d|	z  z   |z   dz
  z  |dz
  z  z   d|dz  z  z
  d|z  |dz  z  z
  d|	z  z
  d
|
z  z
  dd|	z  |z   dz
  z  |z  z
  d|z  z   dz
  }d|dz
  dz  z  dd|z  |z   d|z  z
  dz   z  |dz
  z  z   d|d         z  z   }‰j        d|z  z  |||dz
           z  |||dz
           z  z
  |||dz
           z  z   z  ||<   ||         ‰                     ‰ d|z  ¦  «        z  }‰j         |z  ‰                     d¦  «        | z  z  |z  }‰j        |z  ‰                     d¦  «        | z  z  |z  }t          |¦  «        d‰j	        z  k     rnC|dk    r*t          |¦  «        t          |¦  «        z  dk     r‰j
        ‚||z  }||z  }|}|dz  }�Œ	‰                     ‰j        |z  d‰                     ‰ ¦  «        z  z   ¦  «        |z  ‰                     ‰j        |z  d‰                     ‰ ¦  «        z  z    ¦  «        |z  z   }d|z  ‰j        ‰ gdd|g|||g| |gg g df}‰ g|  g||||| z
  g||| z
  || z
  || z
  g| | |z
  dz   | |z
  dz   | |z
  dz   g| |z
  dz   gd‰z  f}‰ g| g|||| |z
  g| ||z
  ||z
  ||z
  g|||z
  dz   ||z
  dz   ||z
  dz   g|  |z   dz   gd‰z  f}|||fS )Nr   r   r   r   rm  é    r³   rn  ro  r°   é   é   r±   r"  iöÿÿÿrp  rU  rV  rq  )r·   rU   rr  rs  rt  rG   r¶   rÕ   rú   rù   rÁ   rv  r¾   rµ   )rœ   r£   r�   rž   r¦   rW  rX  ÚB3ÚAÚBÚRrH   rZ  rw  r$   r[  r\  r]  Úuu3r%   r^  rx  rA  r»   r¼   r_  r   r"   s                             €€r/   rj   z_hyp2f3.<locals>.h˜  s   ø€ Øœ R¨¡U¨2¡X¨b¡[°¡^°C´KÑ%?Ñ@�AØ˜B™�BØ˜B™˜r™�BØ˜2™�AØ˜2™˜b ™e™ B r¡EÑ)�AØ˜2™˜b™�AØ�AØœ7�A�a‘DØ˜a !™e c¤k°1°R±4¸±7¸1±9Ñ&=¸rÀ"¹uÑ&EÑEÈÌÑTÑU�A�a‘DØœ; q¨¤t¨Q¡wÑ.°´¸sÀAÀbÁDÈÁF¹|ÈQÈqÉSÑ?QÐTVÐWXÑTXÑ?XØ˜2˜b !™e™8 b¨¡eÑ+¨a°©cÑ1°BÑ6¸Ñ:Ñ;¸RÀ¹UÑCñ@DØDEñ@Fñ 2Gñ G�A�a‘Dà�BØ�BØ�AØ�EðØ A˜:™:Ø#$ Q q¡S¡5¨¡7¨Q¨q°©s©U°1°R±4©Z¸©\Ñ":¸A¸aÀ¹c¹EÀ!ÀBÁ$¹JÀq¹LÑ"IØ!" 1 Q¡3¡ q¨¡t¡¨A¡ñ#/˜Cà#$ a¨¡c¨A¡X¡:°°1°Q±3°r±6±
¸A¸a¹CÀ!¹8Ñ0CÑ#CØ ! 2 a¨¡d¡7¨2¨b©5°©7¡?°1°Q±3Ñ#6°rÑ#9¸!Ñ#;Ñ <¸aÀ¹cÑ Bñ$CØEGÈÈ1ÉÁWñ$Mà " 2¡ a¨¡d¡
ñ$+à-.¨q©Sñ$1à34°Q±3ñ$7à9:¸A¸a¹CÀ¹FÀ1¹H¹Àa¹ñ$HàJKÈBÉ$ñ$OàOPñ$Q˜Cð $% a¨¡c¨A¡X¡:¨a°°Q±°r±¸!¸B¹$±¸q±Ñ.AÀ1ÀQÁ3Ñ.GÑ#GÈÈ!ÈAÌ$ÉÑ#N˜CØ#&¤7¨A¨a©C¡=°#°a¸¸!¹´f±*¸SÀÀ1ÀQÁ3Ä¹ZÑ2GÈÈAÈaÐPQÉcÌFÉ
Ñ2RÑ#S˜A˜a™DØ˜aœD 3§9¢9¨a¨R°°a±Ñ#8Ô#8Ñ8˜Ø"œu˜f q™[¨3¯7ª7°1©:¬:¸¸Ñ+;Ñ;¸aÑ?˜Ø œU A™X¨¯ª°©
¬
°a°RÑ(8Ñ8¸1Ñ<˜Ý˜r™7œ7 S¨¬¡[Ò0Ð0Ø!à˜qš5˜5¥S¨¡Z¤Zµ#°b±'´'Ñ%9¸CÒ%?Ð%?Ø"%Ô"3Ð3Ø˜b™˜Ø˜b™˜Ø "˜Ø˜Q™˜ñ)ð* Ÿš ¤¨¡¨!¨C¯HªH°a°R©L¬L©.Ñ!8Ñ9Ô9¸"Ñ<ØŸš 3¤6¨!¡8¨A¨c¯hªh¸°r©l¬l©NÑ#:Ð!;Ñ<Ô<¸RÑ?ñ@�Aà˜a™% ¤¨!¨Ð,¨q°$¸¨l¸RÀÀR¸LÈ2ÈrÈ(Ø˜B ð"�Bà˜"˜  ˜u r¨"¨R°°2±Ð&6¸¸2¸b¹5ÀÀBÁÀrÈ"ÁuÐ7MØ˜B˜r™E !™G B r¡E¨!¡G¨B¨r©E°!©GÐ4°r¸"±u¸Q±w°iÀÀ1ÁðE�Bà˜"˜  ˜u r¨"¨R°°2±Ð&6¸¸2¸b¹5ÀÀBÁÀrÈ"ÁuÐ7MØ˜B˜r™E !™G B r¡E¨!¡G¨B¨r©E°!©GÐ4°r°c¸"±f¸Q±h°ZÀÀ1ÁðE�Bà˜r 2˜:Ð%r1   Tr³   r`  c              3   óB   •K  — | ]}‰                      |¦  «        V — Œd S r•   rb  rc  s     €r/   r  z_hyp2f3.<locals>.<genexpr>Ç  s1   øè è € ÐHÐH°�s×(Ò(¨Ñ+Ô+ÐHÐHÐHÐHÐHÐHr1   r~  r   rz  )r   r    r!   r"   rc   rœ   rd  r£   re  r�   rf  rž   rg  r¦   Úb3typerÿ   rÃ   rh  rd   ri  rj   rm   s   `  `                  r/   rŠ   rŠ     sê  øø€ à!$Ñ�L€Rˆ‘,�2�vØ/2Ñ,�L€Rˆ‘,�2�v¡  Våˆq‰6Œ6€DØ�7Š7�1‰:Œ:€Dð �m˜D !™G€OØŒ8€Dð
 %Ÿjšj¨Ñ8Ô8Ð8ð =Ø	�Š�‰Œ˜Ò	ð=Ø"%§(¢(¨4¡.¤.°3°t±8Ò";ð ð ð <ð:	ð7Ø�”˜OÑ+�”ð-&ð -&ð -&ð -&ð -&ð -&ð\ —M’M ! b¨¨B¨r°"Ð%5ÀDÐSTÐUXÔU]ÑS]�MÑ^Ô^�ÝÐHÐHÐHÐH°b¸¸B¸rÀ"ÀQÐ5GÐHÑHÔHÑHÔHÈAÒMÐMØŸš˜q™	œ	�AØð ˆCŒHˆHøð Ô$ð ð ð Ø�ðøøøØàˆCŒHˆHø�tˆCŒHˆOˆOˆOˆOàˆ3Œ:�a˜˜V V¨V°V¸VÐDÀrÈ2ÈrÐSUÐWYÐFZÐ\]ÐhÐhÐagÐhÐhÐhs%   ÂA7D Ä
D*Ä'D6 Ä)D*Ä*D6 Ä6	D?c                 ó4  ‡ ‡— |\  \  }}\  }}	 |                      ¦   «         }	|	                     d‰ j        ¦  «        |	d<    ‰ j        dd||f||g‰fi |	¤ŽS # ‰ j        $ r |                     d¦  «        r‚ Y nw xY wˆ ˆfd„}
 ‰ j        |
|d|z   |z
  gfi |¤ŽS )NrH  r   r   r€   c                 óÒ   •— ‰                      |¦  «        }d‰z  }‰j        ||gdd| gg | |z
  dz   |g| g|g|f}‰j         ||gddd| z   |z
  gg | d|z
  g| |z
  dz   gd|z
  g|f}||fS )Nr´   r   r   )Úsinpir¾   )rA   rE   r%   rË   r»   r¼   r   r"   s         €€r/   rj   z_hyp2f0.<locals>.hÞ  sš   ø€ Ø�IŠI�a‰LŒLˆØ�‰TˆØŒv�a˜ˆm˜Q˜r !˜H R¨¨1©¨Q©¨q¨	°1°#°q°c¸"Ð=ˆØ”ˆw�q˜ˆn˜a  1 Q¡3 q¡5˜\¨"¨a°°!±¨W°a¸±c¸!±e°W¸aÀ¹c¸UÀ2ÐFˆØ�2ˆvˆr1   r   )ÚcopyrN   r   rŽ   rÁ   rv   )r   r    r!   r"   rc   rA   rÒ   rE   rÂ   Úkwargsbrj   s   `  `       r/   r‹   r‹   Ñ  sî   øø€ à Ñ�J€Qˆ‘
��EðØ—+’+‘-”-ˆØ%Ÿkšk¨*°c´hÑ?Ô?ˆ�
ÑØˆsŒz˜!˜Q  u °°!¨u°aÐCÐC¸7ÐCÐCÐCøØÔð ð ð Ø�:Š:�nÑ%Ô%ð 	ØØˆðøøøðð ð ð ð ð ð ˆ3Œ=˜˜Q  !¡ A¡˜JÐ1Ð1¨&Ð1Ð1Ð1s   �AA Á A:Á9A:Nc                 ó@  ‡ ‡‡‡‡‡‡— |\  }}|\  }	}
t          |¦  «        Š‰t          |¦  «        z   Št          |	¦  «        Š‰t          |
¦  «        z   Š||z   }|	|
z   }ˆ fd„|D ¦   «         }ˆ fd„|D ¦   «         }‰                      ‰¦  «        Š|€7‰‰k     rd}‰‰k    rd}‰‰k    r!‰‰z   ‰k    rt          ‰¦  «        dk    rd}nd}|                     d¦  «        rt	          d‰‰‰‰|¦  «         |dk    rˆ ˆˆˆˆˆˆfd„}nˆ ˆˆˆˆˆˆfd„} ‰ j        |||z   fi |¤ŽS )	Nc                 ó:   •— g | ]}‰                      |¦  «        ‘ŒS r;   ©r�   ©r<   Ú_r   s     €r/   r=   zmeijerg.<locals>.<listcomp>ð  ó#   ø€ Ð#Ð#Ð#˜Aˆ�Š�Q‰ŒÐ#Ð#Ð#r1   c                 ó:   •— g | ]}‰                      |¦  «        ‘ŒS r;   rŽ  r�  s     €r/   r=   zmeijerg.<locals>.<listcomp>ñ  r‘  r1   r   r   r3   zMeijer G m,n,p,q,series =c            
      ó\  •‡	‡
‡— | d ‰…         Š	| ‰d …         Š
g }t          ‰¦  «        D �] Š‰g}‰
‰         ‰z  g}ˆ
ˆfd„t          ‰¦  «        D ¦   «         }|ˆ	ˆ
ˆfd„t          ‰¦  «        D ¦   «         z  }ˆ	ˆ
ˆfd„t          ‰‰¦  «        D ¦   «         }|ˆ
ˆfd„t          ‰‰¦  «        D ¦   «         z  }ˆ	ˆ
ˆfd„t          ‰¦  «        D ¦   «         }ˆ
ˆfd„t          ‰¦  «        D ¦   «         }‰j         ‰‰z
  ‰z
  z  ‰‰j        ‰z  z  z  }|                     |||||||f¦  «         �Œ|S )Nc                 ó>   •— g | ]}|‰k    ¯‰|         ‰‰         z
  ‘ŒS r;   r;   ©r<   r¶   rE   r$   s     €€r/   r=   z&meijerg.<locals>.h.<locals>.<listcomp>  ó*   ø€ Ð<Ð<Ð< A°Q¸!²V°V�a˜”d˜1˜Qœ4‘i°V°V°Vr1   c                 ó8   •— g | ]}d ‰|         z
  ‰‰         z   ‘ŒS r.  r;   ©r<   r¶   rA   rE   r$   s     €€€r/   r=   z&meijerg.<locals>.h.<locals>.<listcomp>  ó)   ø€ Ð5Ð5Ð5 q�q˜˜1œ‘v˜a œd‘{Ð5Ð5Ð5r1   c                 ó2   •— g | ]}‰|         ‰‰         z
  ‘ŒS r;   r;   r˜  s     €€€r/   r=   z&meijerg.<locals>.h.<locals>.<listcomp>  ó%   ø€ Ð4Ð4Ð4 A�a˜”d˜1˜Qœ4‘iÐ4Ð4Ð4r1   c                 ó8   •— g | ]}d ‰|         z
  ‰‰         z   ‘ŒS r.  r;   r•  s     €€r/   r=   z&meijerg.<locals>.h.<locals>.<listcomp>  ó)   ø€ Ð7Ð7Ð7 q�q˜˜1œ‘v˜a œd‘{Ð7Ð7Ð7r1   c                 ó8   •— g | ]}d ‰|         z
  ‰‰         z   ‘ŒS r.  r;   r˜  s     €€€r/   r=   z&meijerg.<locals>.h.<locals>.<listcomp>	  ó)   ø€ Ð4Ð4Ð4 a�a˜˜!œ‘f˜Q˜qœT‘kÐ4Ð4Ð4r1   c                 óD   •— g | ]}|‰k    ¯d ‰|         z
  ‰‰         z   ‘ŒS r.  r;   r•  s     €€r/   r=   z&meijerg.<locals>.h.<locals>.<listcomp>
  ó.   ø€ Ð>Ð>Ð> a°q¸A²v°v�a˜˜!œ‘f˜Q˜qœT‘k°v°v°vr1   )rV   rU   r   ©r1  r   ÚbasesÚexptsÚgnÚgdÚhnÚhdÚhzrA   rE   r$   r   r  r+   r�   r�   Úrr"   s            @@@€€€€€€€r/   rj   zmeijerg.<locals>.hþ  s…  øøøø€ Ø�R�a�R”ˆAØ�Q�R�R”ˆAØˆEÝ˜1‘X”Xð 
Añ 
A�Ø˜�Ø˜1œ˜a™˜�Ø<Ð<Ð<Ð<Ð<­¨q©¬Ð<Ñ<Ô<�ØÐ5Ð5Ð5Ð5Ð5Ð5­E°!©H¬HÐ5Ñ5Ô5Ñ5�Ø4Ð4Ð4Ð4Ð4Ð4­¨q°©¬Ð4Ñ4Ô4�ØÐ7Ð7Ð7Ð7Ð7­E°!°A©J¬JÐ7Ñ7Ô7Ñ7�Ø4Ð4Ð4Ð4Ð4Ð4­5°©8¬8Ð4Ñ4Ô4�Ø>Ð>Ð>Ð>Ð>­5°©8¬8Ð>Ñ>Ô>�Ø”w�h ! A¡# a¡%Ñ(¨1¨s¬w°q©y©>Ñ9�Ø—’˜e U¨B°°B¸¸BÐ?Ñ@Ô@Ð@Ñ@ØˆLr1   c            
      ó®  •‡	‡
‡— | d ‰…         Š	| ‰d …         Š
g }t          ‰¦  «        D �])Š‰g}‰dk    r‰	‰         dz
  g}n"‰	‰         dz
  ‰                     ‰¦  «        z  g}ˆ	ˆfd„t          ‰¦  «        D ¦   «         }|ˆ	ˆ
ˆfd„t          ‰¦  «        D ¦   «         z  }ˆ	ˆ
ˆfd„t          ‰‰¦  «        D ¦   «         }|ˆ	ˆfd„t          ‰‰¦  «        D ¦   «         z  }ˆ	ˆ
ˆfd„t          ‰¦  «        D ¦   «         }ˆ	ˆfd„t          ‰¦  «        D ¦   «         }‰j         ‰‰z
  ‰z
  z  ‰‰j        ‰z  z  z  }|                     |||||||f¦  «         �Œ+|S )Nr   c                 ó>   •— g | ]}|‰k    ¯‰‰         ‰|         z
  ‘ŒS r;   r;   ©r<   r¶   rA   r$   s     €€r/   r=   z&meijerg.<locals>.h.<locals>.<listcomp>  r–  r1   c                 ó8   •— g | ]}d ‰‰         z
  ‰|         z   ‘ŒS r.  r;   r˜  s     €€€r/   r=   z&meijerg.<locals>.h.<locals>.<listcomp>  r™  r1   c                 ó2   •— g | ]}‰‰         ‰|         z
  ‘ŒS r;   r;   r˜  s     €€€r/   r=   z&meijerg.<locals>.h.<locals>.<listcomp>  r›  r1   c                 ó8   •— g | ]}d ‰‰         z
  ‰|         z   ‘ŒS r.  r;   r­  s     €€r/   r=   z&meijerg.<locals>.h.<locals>.<listcomp>  r�  r1   c                 ó8   •— g | ]}d ‰‰         z
  ‰|         z   ‘ŒS r.  r;   r˜  s     €€€r/   r=   z&meijerg.<locals>.h.<locals>.<listcomp>  rŸ  r1   c                 óD   •— g | ]}|‰k    ¯d ‰|         z   ‰‰         z
  ‘ŒS r.  r;   r­  s     €€r/   r=   z&meijerg.<locals>.h.<locals>.<listcomp>  r¡  r1   )rV   r�   rU   r   r¢  s            @@@€€€€€€€r/   rj   zmeijerg.<locals>.h  s¯  øøøø€ Ø�R�a�R”ˆAØ�Q�R�R”ˆAØˆEÝ˜1‘X”Xð Añ A�Ø˜�Ø˜’6�6Ø˜qœT !™V˜H�E�Eà œd 1™f c§k¢k°!¡n¤nÑ4Ð5�EØ<Ð<Ð<Ð<Ð<­¨q©¬Ð<Ñ<Ô<�ØÐ5Ð5Ð5Ð5Ð5Ð5­E°!©H¬HÐ5Ñ5Ô5Ñ5�Ø4Ð4Ð4Ð4Ð4Ð4­¨q°©¬Ð4Ñ4Ô4�ØÐ7Ð7Ð7Ð7Ð7­E°!°A©J¬JÐ7Ñ7Ô7Ñ7�Ø4Ð4Ð4Ð4Ð4Ð4­5°©8¬8Ð4Ñ4Ô4�Ø>Ð>Ð>Ð>Ð>­5°©8¬8Ð>Ñ>Ô>�Ø”w�h ! A¡# a¡%Ñ(¨1¨s¬w°q©y©>Ñ9�Ø—’˜e U¨B°°B¸¸BÐ?Ñ@Ô@Ð@Ñ@ØˆLr1   )rW   r�   rú   rN   rR   rv   )r   r    r!   r"   rª  Úseriesrc   ÚanÚapÚbmÚbqrA   rE   rj   r  r+   r�   r�   s   `  ``         @@@@r/   Úmeijergr¸  æ  s¼  øøøøøøø€ à�F€BˆØ�F€BˆÝˆB‰Œ€AØ	�C�‰GŒG‰€AÝˆB‰Œ€AØ	�C�‰GŒG‰€AØ
ˆ2‰€AØ
ˆ2‰€AØ#Ð#Ð#Ð# Ð#Ñ#Ô#€AØ#Ð#Ð#Ð# Ð#Ñ#Ô#€AØ�Š�A‰Œ€AØ€~ØˆqŠ5ˆ5˜1�&ØˆqŠ5ˆ5˜1�&Ø�Š6ˆ6Ø�‰s�aŠxˆx�C ™FœF QšJ˜JØ��à�Ø‡z‚z�)ÑÔð ;ÝÐ)¨1¨Q¨q°°6Ñ:Ô:Ð:Ø�‚{€{ð	ð 	ð 	ð 	ð 	ð 	ð 	ð 	ð 	ð 	ð 	ð 	ð"	ð 	ð 	ð 	ð 	ð 	ð 	ð 	ð 	ð 	ð 	ð& ˆ3Œ=˜˜A˜a™CÐ*Ð* 6Ð*Ð*Ð*r1   c           	      óò  — t          |¦  «        t          |¦  «        k    r||}}||}}d„ }|                      |¦  «        rnœ|                      |¦  «        rn†|                      |¦  «        r||||f\  }}}}ne ||¦  «        sZ||z
  |dz
  z  }	 ||	¦  «        st          d¦  «        ‚d|z
  | z  d|z
  ||z
  |z
  z  z   | j        ||z
  |||z
  |z
  ||	|fi |¤Žz  S  | j        |g|g|gdœd|gi||fi |¤ŽS )Nc                 ó(   — t          | ¦  «        dk     S )Ng®Gáz®ï?)rú   )r*   s    r/   r-   zappellf1.<locals>.ok+  s   € Ý�1‰vŒv˜Š}Ðr1   r   z%Analytic continuation not implemented©úm+nr  r+   r¼  )rú   r   rP   Úappellf1Úhyper2d)
r   rA   r�   rž   rH   r*   Úyrc   r-   Úu1s
             r/   r½  r½  $  sa  € õ ˆ1�v„v•�A‘”‚€Ø�!ˆ1ˆØ�RˆBˆðð ð ð ‡{‚{�1�~„~ð =ØØ	�Š�R‰Œð =ØØ	�Š�R‰Œð =Ø˜!˜R �|‰ˆˆ1ˆb�"�"ð
 ˆr�!‰uŒuð 	=Ø�A‘#˜˜!™‘ˆBØ�2�b‘6”6ð JÝ Ð!HÑIÔIÐIà�a‘C˜B˜3‘<  1¡¨¨!©¨B©¡Ñ/Ø�”˜Q˜q™S  A b¡D¨¡G¨A¨b°Ð<Ð<°VÐ<Ð<ñ=ð =àˆ3Œ;˜q˜c r d°¨tÐ4Ð4°u¸a¸S°kÀ1ÀQÐQÐQÈ&ÐQÐQÐQr1   c                 ó<   —  | j         |g|g|gdœ|g|gdœ||fi |¤ŽS )Nr»  ©r  r+   ©r¾  )	r   rA   r�   rž   Úc1Úc2r*   r¿  rc   s	            r/   Úappellf2rÆ  A  sL   € ð ˆ3Œ;˜q˜c r d°¨tÐ4Ð4ØˆT�r�dÐÐ˜Q˜qð,ð ,Ø$*ð,ð ,ð ,r1   c                 óF  — |                       |¦  «        p|                       |¦  «        }	|                       |¦  «        p|                       |¦  «        }
|	s1|
s t          |¦  «        t          |¦  «        k    r||}}||||f\  }}}} | j        ||g||gdœd|gi||fi |¤ŽS )NrÂ  r¼  )r   rú   r¾  )r   rœ   r£   r�   rž   rH   r*   r¿  rc   Úouter_polynomialÚinner_polynomials              r/   Úappellf3rÊ  G  s¸   € à—{’{ 2‘”Ð9¨#¯+ª+°b©/¬/ÐØ—{’{ 2‘”Ð9¨#¯+ª+°b©/¬/ÐØð &Øð 	&�s 1™vœv­¨A©¬š˜Ø�aˆqˆAØ˜R  2˜+‰KˆBˆr�"�RØˆ3Œ;˜R ˜G¨¨B¨Ð0Ð0°5¸!¸°+¸aÀÐKÐKÀFÐKÐKÐKr1   c                 ó6   —  | j         d||gi|g|gdœ||fi |¤ŽS )Nr¼  rÂ  rÃ  )r   rA   rE   rÄ  rÅ  r*   r¿  rc   s           r/   Úappellf4rÌ  Q  s8   € ð ˆ3Œ;˜˜q ˜e�}¨B¨4°R°DÐ&9Ð&9¸!¸AÐGÐGÀÐGÐGÐGr1   c                 ó~  ‡ ‡"— ‰                       |¦  «        }‰                       |¦  «        }ˆ fd„}t          |¦  «        }t          |¦  «        } ||d¦  «        }	 ||d¦  «        }
 ||d¦  «        } ||d¦  «        } ||d¦  «        } ||d¦  «        } ||d¦  «        } ||d	¦  «        } ||d¦  «        } ||d¦  «        } ||d¦  «        }|r*t          d
|                     ¦   «         d         z  ¦  «        ‚|r*t          d
|                     ¦   «         d         z  ¦  «        ‚d}‰ j        }‰                      d¦  «        Š"d}‰ j        }|                     dd|z  ¦  «        }	 ‰ xj        dz  c_        ‰ j        
 }	 d}d}t          |
¦  «        }t          |¦  «        }ˆ"fd„|	D ¦   «         }ˆ"fd„|D ¦   «         }|D ]1}|‰"z   }| 
                    |¦  «         | 
                    |¦  «         Œ2|D ]1}|‰"z   }| 
                    |¦  «         | 
                    |¦  «         Œ2|D ]5}| 
                    |‰"z
  ¦  «         | 
                    |‰"z
  dz
  ¦  «         Œ6|D ]C}|dz  }|d‰"z  z  }| 
                    d|z
  ‰"z
  ¦  «         | 
                    | ‰"z
  ¦  «         ŒD|D ]D}| 
                    |d‰"z  z   ¦  «         | 
                    |d‰"z  z   d|z   d‰"z  z   z  ¦  «         ŒE|D ]L}|dz  }| 
                    d|z
  d‰"z  z
  ¦  «         | 
                    |d‰"z  z   d|z   d‰"z  z   z  ¦  «         ŒM|D ][}|dz  }| 
                    d|‰"z
  z  ¦  «         | 
                    d|‰"z
  dz   z  ¦  «         | 
                    |‰"z
  dz
  ¦  «         Œ\ ‰ j        ||||z  fd‰ j        i|¤Ž} || z  |z  }!t          |!¦  «        |k     r|dz  }nd}|dk    s|snC||!z  }|D ]}||z  }Œ|D ]}||z  }Œ‰"dz  Š"||z  ‰"z  }‰"|k    r‰                      d¦  «        ‚�Œ¡	 |‰ _        n# |‰ _        w xY w|
 S )az  
    Sums the generalized 2D hypergeometric series

    .. math ::

        \sum_{m=0}^{\infty} \sum_{n=0}^{\infty}
            \frac{P((a),m,n)}{Q((b),m,n)}
            \frac{x^m y^n} {m! n!}

    where `(a) = (a_1,\ldots,a_r)`, `(b) = (b_1,\ldots,b_s)` and where
    `P` and `Q` are products of rising factorials such as `(a_j)_n` or
    `(a_j)_{m+n}`. `P` and `Q` are specified in the form of dicts, with
    the `m` and `n` dependence as keys and parameter lists as values.
    The supported rising factorials are given in the following table
    (note that only a few are supported in `Q`):

    +------------+-------------------+--------+
    | Key        |  Rising factorial | `Q`    |
    +============+===================+========+
    | ``'m'``    |   `(a_j)_m`       | Yes    |
    +------------+-------------------+--------+
    | ``'n'``    |   `(a_j)_n`       | Yes    |
    +------------+-------------------+--------+
    | ``'m+n'``  |   `(a_j)_{m+n}`   | Yes    |
    +------------+-------------------+--------+
    | ``'m-n'``  |   `(a_j)_{m-n}`   | No     |
    +------------+-------------------+--------+
    | ``'n-m'``  |   `(a_j)_{n-m}`   | No     |
    +------------+-------------------+--------+
    | ``'2m+n'`` |   `(a_j)_{2m+n}`  | No     |
    +------------+-------------------+--------+
    | ``'2m-n'`` |   `(a_j)_{2m-n}`  | No     |
    +------------+-------------------+--------+
    | ``'2n-m'`` |   `(a_j)_{2n-m}`  | No     |
    +------------+-------------------+--------+

    For example, the Appell F1 and F4 functions

    .. math ::

        F_1 = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty}
              \frac{(a)_{m+n} (b)_m (c)_n}{(d)_{m+n}}
              \frac{x^m y^n}{m! n!}

        F_4 = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty}
              \frac{(a)_{m+n} (b)_{m+n}}{(c)_m (d)_{n}}
              \frac{x^m y^n}{m! n!}

    can be represented respectively as

        ``hyper2d({'m+n':[a], 'm':[b], 'n':[c]}, {'m+n':[d]}, x, y)``

        ``hyper2d({'m+n':[a,b]}, {'m':[c], 'n':[d]}, x, y)``

    More generally, :func:`~mpmath.hyper2d` can evaluate any of the 34 distinct
    convergent second-order (generalized Gaussian) hypergeometric
    series enumerated by Horn, as well as the Kampe de Feriet
    function.

    The series is computed by rewriting it so that the inner
    series (i.e. the series containing `n` and `y`) has the form of an
    ordinary generalized hypergeometric series and thereby can be
    evaluated efficiently using :func:`~mpmath.hyper`. If possible,
    manually swapping `x` and `y` and the corresponding parameters
    can sometimes give better results.

    **Examples**

    Two separable cases: a product of two geometric series, and a
    product of two Gaussian hypergeometric functions::

        >>> from mpmath import *
        >>> mp.dps = 25; mp.pretty = True
        >>> x, y = mpf(0.25), mpf(0.5)
        >>> hyper2d({'m':1,'n':1}, {}, x,y)
        2.666666666666666666666667
        >>> 1/(1-x)/(1-y)
        2.666666666666666666666667
        >>> hyper2d({'m':[1,2],'n':[3,4]}, {'m':[5],'n':[6]}, x,y)
        4.164358531238938319669856
        >>> hyp2f1(1,2,5,x)*hyp2f1(3,4,6,y)
        4.164358531238938319669856

    Some more series that can be done in closed form::

        >>> hyper2d({'m':1,'n':1},{'m+n':1},x,y)
        2.013417124712514809623881
        >>> (exp(x)*x-exp(y)*y)/(x-y)
        2.013417124712514809623881

    Six of the 34 Horn functions, G1-G3 and H1-H3::

        >>> from mpmath import *
        >>> mp.dps = 10; mp.pretty = True
        >>> x, y = 0.0625, 0.125
        >>> a1,a2,b1,b2,c1,c2,d = 1.1,-1.2,-1.3,-1.4,1.5,-1.6,1.7
        >>> hyper2d({'m+n':a1,'n-m':b1,'m-n':b2},{},x,y)  # G1
        1.139090746
        >>> nsum(lambda m,n: rf(a1,m+n)*rf(b1,n-m)*rf(b2,m-n)*\
        ...     x**m*y**n/fac(m)/fac(n), [0,inf], [0,inf])
        1.139090746
        >>> hyper2d({'m':a1,'n':a2,'n-m':b1,'m-n':b2},{},x,y)  # G2
        0.9503682696
        >>> nsum(lambda m,n: rf(a1,m)*rf(a2,n)*rf(b1,n-m)*rf(b2,m-n)*\
        ...     x**m*y**n/fac(m)/fac(n), [0,inf], [0,inf])
        0.9503682696
        >>> hyper2d({'2n-m':a1,'2m-n':a2},{},x,y)  # G3
        1.029372029
        >>> nsum(lambda m,n: rf(a1,2*n-m)*rf(a2,2*m-n)*\
        ...     x**m*y**n/fac(m)/fac(n), [0,inf], [0,inf])
        1.029372029
        >>> hyper2d({'m-n':a1,'m+n':b1,'n':c1},{'m':d},x,y)  # H1
        -1.605331256
        >>> nsum(lambda m,n: rf(a1,m-n)*rf(b1,m+n)*rf(c1,n)/rf(d,m)*\
        ...     x**m*y**n/fac(m)/fac(n), [0,inf], [0,inf])
        -1.605331256
        >>> hyper2d({'m-n':a1,'m':b1,'n':[c1,c2]},{'m':d},x,y)  # H2
        -2.35405404
        >>> nsum(lambda m,n: rf(a1,m-n)*rf(b1,m)*rf(c1,n)*rf(c2,n)/rf(d,m)*\
        ...     x**m*y**n/fac(m)/fac(n), [0,inf], [0,inf])
        -2.35405404
        >>> hyper2d({'2m+n':a1,'n':b1},{'m+n':c1},x,y)  # H3
        0.974479074
        >>> nsum(lambda m,n: rf(a1,2*m+n)*rf(b1,n)/rf(c1,m+n)*\
        ...     x**m*y**n/fac(m)/fac(n), [0,inf], [0,inf])
        0.974479074

    **References**

    1. [SrivastavaKarlsson]_
    2. [Weisstein]_ http://mathworld.wolfram.com/HornFunction.html
    3. [Weisstein]_ http://mathworld.wolfram.com/AppellHypergeometricFunction.html

    c                 ó”   •— |                       |g ¦  «        }	 t          |¦  «        }n# t          $ r |g}Y nw xY wˆfd„|D ¦   «         S )Nc                 ó:   •— g | ]}‰                      |¦  «        ‘ŒS r;   rŽ  )r<   rO  r   s     €r/   r=   z*hyper2d.<locals>.parse.<locals>.<listcomp>æ  s%   ø€ Ð1Ð1Ð1 S�—’˜CÑ Ô Ð1Ð1Ð1r1   )Úpopr0  Ú	TypeError)ÚdctÚkeyr1  r   s      €r/   Úparsezhyper2d.<locals>.parseà  sh   ø€ Ø�wŠw�s˜BÑÔˆð	Ý˜‘:”:ˆDˆDøÝð 	ð 	ð 	Ø�6ˆDˆDˆDð	øøøà1Ð1Ð1Ð1¨DÐ1Ñ1Ô1Ð1s   ™) ©9¸9r  r+   r¼  zm-nzn-mz2m+nz2m-nz2n-mzunsupported key: %rr   rH  rL   r7   r   c                 ó   •— g | ]}|‰z   ‘ŒS r;   r;   )r<   rA   r  s     €r/   r=   zhyper2d.<locals>.<listcomp>  ó   ø€ Ð(Ð(Ð(˜q�q˜‘sÐ(Ð(Ð(r1   c                 ó   •— g | ]}|‰z   ‘ŒS r;   r;   )r<   rE   r  s     €r/   r=   zhyper2d.<locals>.<listcomp>  rÖ  r1   r´   r   r³   rq  r5   r   zmaxterms exceeded in hyper2d)r�   ÚdictrP   ÚkeysrU   rÕ   r   rN   rù   r0  r   rZ   rú   rÁ   )#r   rA   rE   r*   r¿  rc   rÔ  r    r!   Úa_mÚa_nÚ	a_m_add_nÚ	a_m_sub_nÚ	a_n_sub_mÚ
a_2m_add_nÚ
a_2m_sub_nÚ
a_2n_sub_mÚb_mÚb_nÚ	b_m_add_nrQ  ÚouterÚok_countr   rH  ræ   Ú
inner_signÚ
outer_signÚinner_aÚinner_bÚouter_aÚouter_bÚinnerr   r  s#   `                                 @r/   r¾  r¾  V  sI  øø€ ðP 	�Š�A‰Œ€AØ�Š�A‰Œ€Að2ð 2ð 2ð 2ð 2õ ˆq‰'Œ'€CÝ
ˆq‰'Œ'€CØ
ˆ%��3‰-Œ-€CØ
ˆ%��3‰-Œ-€CØ��a˜‘”€IØ��a˜‘”€IØ��a˜‘”€IØ��q˜&Ñ!Ô!€JØ��q˜&Ñ!Ô!€JØ��q˜&Ñ!Ô!€JØ
ˆ%��3‰-Œ-€CØ
ˆ%��3‰-Œ-€CØ��a˜‘”€IØÐ?•
Ð0°1·6²6±8´8¸A´;Ñ>Ñ?Ô?Ð
?ØÐ?•
Ð0°1·6²6±8´8¸A´;Ñ>Ñ?Ô?Ð
?Ø	€AØŒG€EØ�Š�‰
Œ
€AØ€HØŒ8€DØ�zŠz˜* b¨¡gÑ.Ô.€Hð>ØˆŒ�B‰ˆŒØŒwˆhˆð9	HØˆJØˆJÝ˜3‘i”iˆGÝ˜3‘i”iˆGØ(Ð(Ð(Ð( CÐ(Ñ(Ô(ˆGØ(Ð(Ð(Ð( CÐ(Ñ(Ô(ˆGàð "ð "�Ø�a‘C�Ø—’˜qÑ!Ô!Ð!Ø—’˜qÑ!Ô!Ð!Ð!àð "ð "�Ø�a‘C�Ø—’˜qÑ!Ô!Ð!Ø—’˜qÑ!Ô!Ð!Ð!àð &ð &�Ø—’˜q ™sÑ#Ô#Ð#Ø—’˜q ™s 1™uÑ%Ô%Ð%Ð%àð %ð %�Ø˜rÑ"�
Ø˜r Q™iÑ'�
Ø—’˜q ™s 1™uÑ%Ô%Ð%Ø—’ ˜r !™tÑ$Ô$Ð$Ð$àð 2ð 2�Ø—’˜q  1¡™uÑ%Ô%Ð%Ø—’  ! A¡#¡¨¨!©¨A¨a©C©Ñ0Ñ1Ô1Ð1Ð1àð 2ð 2�Ø˜rÑ"�
Ø—’˜q ™s 1 Q¡3™wÑ'Ô'Ð'Ø—’  ! A¡#¡¨¨!©¨A¨a©C©Ñ0Ñ1Ô1Ð1Ð1àð &ð &�Ø˜a‘�
Ø—’˜s A a¡C™yÑ)Ô)Ð)Ø—’˜s A a¡C¨¡E™{Ñ+Ô+Ð+Ø—’˜q ™s 1™uÑ%Ô%Ð%Ð%Ø�C”I˜g w°
¸1±ð -ð -Øœð-Ø%+ð-ð -ˆEà˜5‘= :Ñ-ˆDÝ�4‰yŒy˜3ŠˆØ˜A‘��à�Ø˜1Š}ˆ} Eˆ}ØØ�‰IˆAØÐ(Ð(�˜e q™j˜e˜eØÐ(Ð(�˜e q™j˜e˜eØ�‰FˆAØ˜A‘I ‘MˆEØ�8Š|ˆ|Ø×'Ò'Ð(FÑGÔGÐGñs9	Hðd ð ˆŒˆø�4ˆŒˆˆˆˆØˆ2€Is   Å-J:P0 Ð0	P9c                 óè   ‡‡‡	— |                       ‰¦  «        Š||z   }t          |¦  «        Š	t          |¦  «        }‰	|fdk    s‰	|fdk    r
| j        ‰z  S ‰	|z
  dz  Šˆˆ	ˆfd„} | j        ||fi |¤ŽS )a`  
    Evaluates the bilateral hypergeometric series

    .. math ::

        \,_AH_B(a_1, \ldots, a_k; b_1, \ldots, b_B; z) =
            \sum_{n=-\infty}^{\infty}
            \frac{(a_1)_n \ldots (a_A)_n}
                 {(b_1)_n \ldots (b_B)_n} \, z^n

    where, for direct convergence, `A = B` and `|z| = 1`, although a
    regularized sum exists more generally by considering the
    bilateral series as a sum of two ordinary hypergeometric
    functions. In order for the series to make sense, none of the
    parameters may be integers.

    **Examples**

    The value of `\,_2H_2` at `z = 1` is given by Dougall's formula::

        >>> from mpmath import *
        >>> mp.dps = 25; mp.pretty = True
        >>> a,b,c,d = 0.5, 1.5, 2.25, 3.25
        >>> bihyper([a,b],[c,d],1)
        -14.49118026212345786148847
        >>> gammaprod([c,d,1-a,1-b,c+d-a-b-1],[c-a,d-a,c-b,d-b])
        -14.49118026212345786148847

    The regularized function `\,_1H_0` can be expressed as the
    sum of one `\,_2F_0` function and one `\,_1F_1` function::

        >>> a = mpf(0.25)
        >>> z = mpf(0.75)
        >>> bihyper([a], [], z)
        (0.2454393389657273841385582 + 0.2454393389657273841385582j)
        >>> hyper([a,1],[],z) + (hyper([1],[1-a],-1/z)-1)
        (0.2454393389657273841385582 + 0.2454393389657273841385582j)
        >>> hyper([a,1],[],z) + hyper([1],[2-a],-1/z)/z/(a-1)
        (0.2454393389657273841385582 + 0.2454393389657273841385582j)

    **References**

    1. [Slater]_ (chapter 6: "Bilateral Series", pp. 180-189)
    2. [Wikipedia]_ http://en.wikipedia.org/wiki/Bilateral_hypergeometric_series

    )r   r   )r   r   r   c                  ót  •— t          | d ‰
…         ¦  «        }t          | ‰
d …         ¦  «        }d„ |D ¦   «         }d„ |D ¦   «         }d‰	z  ‰z  gd„ |D ¦   «         z   d„ |D ¦   «         z   }dgdgt          |¦  «        z  z   dgt          |¦  «        z  z   }g g g g |dgz   |‰f}||g g |dgz   |d‰	z  ‰z  f}||fS )Nc                 ó   — g | ]}d |z
  ‘ŒS ©r   r;   r  s     r/   r=   z&bihyper.<locals>.h.<locals>.<listcomp>~  ó   € Ð!Ð!Ð!˜��!‘Ð!Ð!Ð!r1   c                 ó   — g | ]}d |z
  ‘ŒS rñ  r;   r  s     r/   r=   z&bihyper.<locals>.h.<locals>.<listcomp>  rò  r1   r´   c                 ó   — g | ]}d |z
  ‘ŒS r.  r;   r  s     r/   r=   z&bihyper.<locals>.h.<locals>.<listcomp>€  s   € Ð1Ð1Ð1¨  !¡Ð1Ð1Ð1r1   c                 ó   — g | ]}d |z
  ‘ŒS r.  r;   r  s     r/   r=   z&bihyper.<locals>.h.<locals>.<listcomp>€  s   € Ð4FÐ4FÐ4F¸Q°Q°q±SÐ4FÐ4FÐ4Fr1   r   )r0  rW   )r   r    r!   Úaa_sÚbb_sÚrpÚrcr»   r¼   Únegr�   r"   s            €€€r/   rj   zbihyper.<locals>.h{  sô   ø€ Ý�3�r˜�r”7‰mŒmˆÝ�3�q�r�r”7‰mŒmˆØ!Ð!˜SÐ!Ñ!Ô!ˆØ!Ð!˜SÐ!Ñ!Ô!ˆØ�C‰i˜!‰mˆ_Ð1Ð1¨SÐ1Ñ1Ô1Ñ1Ð4FÐ4FÀ#Ð4FÑ4FÔ4FÑFˆØˆT�Q�C�˜C™œ‘LÑ  B 4­¨C©¬¡=Ñ0ˆØ��R˜˜S A 3™Y¨¨QÐ.ˆØ��R˜˜T Q C™Z¨°°S©y¸1©}Ð<ˆØ�2ˆvˆr1   )r�   rW   rM   rv   )
r   r    r!   r"   rc   r   r�   rj   rú  r�   s
      `    @@r/   Úbihyperrû  D  s©   øøø€ ð` 	�Š�A‰Œ€AØ
�‰)€CÝˆC‰Œ€AÝˆC‰Œ€AØ	ˆ1€v�‚€˜1˜a˜& Eš/˜/ØŒx˜!‰|ÐØˆQ‰3�!‰)€Cð	ð 	ð 	ð 	ð 	ð 	ð 	ð ˆ3Œ=˜˜CÐ*Ð* 6Ð*Ð*Ð*r1   )r   N)#Úlibmp.backendr   Ú	functionsr   r   r0   rQ   rv   rZ   r—   rš   rŸ   r¡   r¤   r§   r©   r¬   r‡   rƒ   r…   rî   rˆ   rŒ   r�   r‰   r†   rŠ   r‹   r¸  r½  rÆ  rÊ  rÌ  r¾  rû  r;   r1   r/   ú<module>rþ     s†  ðØ "Ð "Ð "Ð "Ð "Ð "Ø +Ð +Ð +Ð +Ð +Ð +Ð +Ð +ð-2ð -2ð -2ð^€ð Ø$&¸Dð Eð Eð Eñ „ðEðN ð(8ð (8ñ „ð(8ðT ð(ð (ñ „ð(ð ð)ð )ñ „ð)ð ð.ð .ñ „ð.ð ð+ð +ñ „ð+ð ð1ð 1ñ „ð1ð ð4ð 4ñ „ð4ð ð*ð *ñ „ð*ð ð4ð 4ñ „ð4ð ðð ñ „ðð ð!8ð !8ñ „ð!8ðF ð!ð !ñ „ð!ðF4ð 4ð 4ðl ðMð Mñ „ðMð^ ðG/ð G/ñ „ðG/ðR ð;ð ;ñ „ð;ð| ð@]ð @]ñ „ð@]ðH ðMQð MQñ „ðMQðb ðOið Oiñ „ðOiðb ð2ð 2ñ „ð2ð( ð;+ð ;+ð ;+ñ „ð;+ðz ðRð Rñ „ðRð8 ð,ð ,ñ „ð,ð
 ðLð Lñ „ðLð ðHð Hñ „ðHð ðdð dñ „ðdðLð ð@+ð @+ñ „ð@+ð @+ð @+r1   