§
    JŠtj2”  ã                   ó¦  — d dl mZmZ ed„ ¦   «         Zed„ ¦   «         Zed6d„¦   «         Zed6d„¦   «         Zed6d„¦   «         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„ ¦   «         Zd„ Zed„ ¦   «         Zed„ ¦   «         Zed„ ¦   «         Zed„ ¦   «         Zd„ Zd„ Zed6d „¦   «         Z ed6d!„¦   «         Z!d7d#„Z"ed6d$„¦   «         Z#ed8d%„¦   «         Z$d&„ Z%ed'„ ¦   «         Z&ed(„ ¦   «         Z'ei fd)„¦   «         Z(ed9d+„¦   «         Z)ei fd,„¦   «         Z*ed9d-„¦   «         Z+d.„ Z,d/„ Z-d0„ Z.d1i fd2„Z/ed6d3„¦   «         Z0ed6d4„¦   «         Z1d5S ):é   )ÚdefunÚdefun_wrappedc                 ó.   — |                       d|¦  «        S )zCComputes the Bessel function `J_0(x)`. See :func:`~mpmath.besselj`.é    ©Úbesselj©ÚctxÚxs     úU/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/mpmath/functions/bessel.pyÚj0r      ó   € ð �;Š;�q˜!ÑÔÐó    c                 ó.   — |                       d|¦  «        S )zDComputes the Bessel function `J_1(x)`.  See :func:`~mpmath.besselj`.r   r   r	   s     r   Új1r      r   r   r   c                 óF  ‡ ‡‡‡	‡
‡— t          ‰¦  «        t          u rd}nN‰                      ‰¦  «        Š‰                      ‰¦  «        }|r"t          ‰                      ‰¦  «        ¦  «        Š|r‰dk     rd‰z   ‰ j        ‰ ‰|fi |¤Žz  S ‰                      ‰¦  «        Š‰                      ‰¦  «        Š	|r×‰                      |¦  «        Š
‰                      ‰
¦  «        r“‰
dk    r�t          ‰
¦  «        Š
‰ j        }	 ‰ xj        dz  c_        ‰                      ˆ ˆ
ˆˆfd„t          ‰
dz   ¦  «        D ¦   «         ¦  «        }|‰ _        n# |‰ _        w xY w|‰  
                    d¦  «        ‰
 z  z  }�n+ˆ	ˆ ˆfd„} ‰ j        |‰‰
gfi |¤Ž}�n|sO|rMt          ‰	¦  «        d	k     r:t          ‰¦  «        d
k     r'	 ‰                      ‰‰¦  «        S # t          $ r Y nw xY w‰s=‰s‰ j        ‰z   ‰z   }n«‰                      ‰¦  «        dk    r‰‰z  }nŒ‰ j        ‰z   ‰z   }n~‰ j        }	 ‰ xj        t%          dt          ‰	¦  «        z  ‰ j        ¦  «        z  c_        ‰                      ‰dd¬¦  «        Šˆ	ˆ ˆfd„} ‰ j        |‰gfi |¤Ž}|‰ _        n# |‰ _        w xY w|
 }|S )NTr   éÿÿÿÿé   c              3   ó�   •K  — | ]@}d |z  ‰                      ‰|¦  «        z  ‰                     d|z  ‰z   ‰z
  ‰¦  «        z  V — ŒAdS )r   é   N)Úbinomialr   )Ú.0Úkr
   ÚdÚnÚzs     €€€€r   ú	<genexpr>zbesselj.<locals>.<genexpr>%   sm   øè è € ð )ð )Øð ! 1™W s§|¢|°A°aÑ'8Ô'8Ñ8¸3¿;º;ÀqÈÁsÈ1ÁuÈQÁwÈqÑ;QÔ;QÑQð )ð )ð )ð )ð )ð )r   r   r   c                 ó  •— ‰                      ‰                      ‰‰‰j        ‰z   ¬¦  «        dd¬¦  «        }d| |z
  dz   z  d| |z
  dz   z  g}d‰j        ‰g|d| z  z
  d| |z
  gg || dz   dz  | dz   dz  g|| dz   gz   |fg}|S )N©Úprecç      Ð¿T©Úexactç      à?r   r   ©Úfmulr    Úpi©r   r   ÚrÚBÚTÚMr
   r   s        €€€r   Úhzbesselj.<locals>.h+   s®   ø€ Ø—H’H˜SŸXšX a¨°´¸!±˜XÑ<Ô<¸eÈ4�HÑPÔP�Ø˜!˜A™#˜a™%‘[ # q¨¡s¨1¡u¡+Ð.�Ø˜œ �l A a¨¡c¡E¨#¨a°©c ?°2°a¸!¸A¹#¸s¹ÀAÀaÁCÈÁ9Ð8MÈaÐQRÐSTÑQTÐPUÉgÐVWÐXÐY�Ø�r   é
   é   é   r$   r"   c                 ó²   •— ‰                      ‰                     ‰‰t          d‰j        ‰z   ¦  «        ¬¦  «        d¬¦  «        }‰g| gg | dz   gg | dz   g|fgS )Nr   r   Tr"   r   )Úfnegr&   Úmaxr    )r   r)   r,   r
   Úws     €€€r   r-   zbesselj.<locals>.hG   sb   ø€ ØŸš §¢¨!¨QµS¸¸3¼8ÀA¹:Ñ5FÔ5F Ñ!GÔ!GÈt˜ÑTÔT�AØ˜S 1 # r¨A¨a©C¨5°"°q¸±s°e¸QÐ?Ð@Ð@r   )ÚtypeÚintÚconvertÚisintÚ_rer   Úmagr    ÚfsumÚrangeÚmpfÚ	hypercombÚabsÚ_besseljÚNotImplementedErrorÚoneÚreÚinfÚminr&   )r
   r   r   Ú
derivativeÚkwargsÚn_isintÚorigÚvr-   r,   r   r4   s   ```      @@@r   r   r      sK  øøøøøø€ åˆA�w„w•#€~€~Øˆˆà�KŠK˜‰NŒNˆØ—)’)˜A‘,”,ˆØð 	 Ý�C—G’G˜A‘J”J‘”ˆAØð B�1�q’5�5Ø�Q‰w˜˜œ a R¨¨JÐAÐA¸&ÐAÐAÑAÐAØ�Š�A‰Œ€AØ�Š�‰
Œ
€AØð 3Ø�KŠK˜
Ñ#Ô#ˆð
 �9Š9�Q‰<Œ<ð 	2˜A šF˜FÝ�A‘”ˆAØ”8ˆDð Ø�”˜B‘�”Ø—H’Hð )ð )ð )ð )ð )ð )ð )Ý" 1 Q¡3™ZœZð)ñ )ô )ñ )ô )�ð  �”�ø˜4�”����Ø�—’˜‘”˜q˜bÑ!Ñ!ˆA‰Aðð ð ð ð ð ð ð
 �”˜a ! A Ð1Ð1¨&Ð1Ð1ˆA‰Að ð 	 ð 	­C°©F¬F°RªK¨K½CÀ¹F¼FÀRºK¸KðØ—|’| A qÑ)Ô)Ð)øÝ&ð ð ð Ø�ðøøøàð 	 Øð $Ø”G˜a‘K ‘M��Ø—’˜‘”˜Q’�Ø�a‘C��à”G˜a‘K !‘O��ð ”8ˆDð
 ð �”�C ¥# a¡&¤&¡¨#¬(Ñ3Ô3Ñ3�”Ø—H’H˜Q ¨4�HÑ0Ô0�ðAð Að Að Að Að Að Að "�C”M ! a SÐ3Ð3¨FÐ3Ð3�à�”�ø˜4�”����ØˆBˆØ€Hs,   Ä AE Å	EÆ;G Ç
GÇGÈ(A"J Ê	Jc                 ó  ‡ ‡‡	— ‰                       |¦  «        }‰                       ‰¦  «        Š‰ss|rt          ‚|sd|z   ‰z   S ‰                      |¦  «        rd|‰z   z  S ‰                      |¦  «        }|dk    r‰ j        |‰z   z  S |dk    rd|‰z   z  S ‰ j        |‰z   z   S ‰                      ‰¦  «        Š	|r.‰                       |¦  «        }ˆ	ˆ ˆfd„} ‰ j        |||gfi |¤Ž}nˆ	ˆ ˆfd„} ‰ j        ||gfi |¤Ž}|S )Nr   r   c                 ó  •— ‰                      ‰                      ‰‰‰j        ‰z   ¬¦  «        dd¬¦  «        }d| |z
  dz   z  d| |z
  dz   z  | dz   g}d‰j        ‰g|d| z  z
  d| |z
  g| dz   g|| dz   dz  | dz   dz  g||fg}|S )Nr   ç      Ð?Tr"   r$   r   r   r%   r(   s        €€€r   r-   zbesseli.<locals>.hf   s«   ø€ Ø—’˜Ÿš ! Q¨S¬X°a©Z˜Ñ8Ô8¸$Àd�ÑKÔKˆAØ�a˜‘c˜!‘e‘˜c 1 Q¡3 q¡5™k¨1¨Q©3Ð/ˆAØ�S”V˜A�,  ! A¡#¡ c¨!¨A©#˜°°!±¨u°Q¸¸1¹¸c¹	À1ÀQÁ3ÈÁ)Ð7LÈQÈqÐQÐRˆAØˆHr   c           	      ó¸   •— ‰                      ‰dd¬¦  «        }‰                      ||t          d‰j        ‰z   ¦  «        ¬¦  «        }|g| gg | dz   gg | dz   g|fgS )Nr$   Tr"   r   r   r   )r&   r3   r    )r   r4   r)   r,   r
   r   s      €€€r   r-   zbesseli.<locals>.hm   si   ø€ Ø—’˜˜C t�Ñ,Ô,ˆAØ—’˜˜A¥C¨¨#¬(°1©*Ñ$5Ô$5�Ñ6Ô6ˆAØ�S˜1˜#˜r A a¡C 5¨"¨q°©s¨e°QÐ7Ð8Ð8r   )r7   Ú
ValueErrorr8   rC   ÚnanrD   r:   r>   )
r
   r   r   rF   rG   r)   r   r-   rJ   r,   s
   ` `      @r   ÚbesselirQ   P   sp  øøø€ à�Š�A‰Œ€AØ�Š�A‰Œ€AØð !Øð 	ÝÐØð 	à�Q‘3�q‘5ˆLØ�9Š9�Q‰<Œ<ð 	Ø�a˜‘c‘7ˆNØ�FŠF�1‰IŒIˆØ�Š6ˆ6Ø”7˜A˜a™C‘=Ð Ø�ŠUˆUØ�a˜‘c‘7ˆNà”7˜A˜a™C‘=Ð Ø�Š�‰
Œ
€AØð ,Ø�KŠK˜
Ñ#Ô#ˆð	ð 	ð 	ð 	ð 	ð 	ð 	ð
 ˆCŒM˜!˜a ˜UÐ-Ð- fÐ-Ð-ˆˆð	9ð 	9ð 	9ð 	9ð 	9ð 	9ð 	9ð ˆCŒM˜!˜a˜SÐ+Ð+ FÐ+Ð+ˆØ€Hr   c                 óì  — |sË|rt           ‚|s| j         ||z   z   S |                      |¦  «        r| j        ||z   z  S |                      |¦  «        }|dz   }|                      |¦  «        r|dk    r| j         ||z   z   S d||z   z  S |dk     r2t          |                      |¦  «        ¦  «        dz  r| j        ||z   z   S | j        ||z   z   S | xj	        dz  c_	        |  
                    |¦  «        \  }}|| j	         k     r| j        
 }	| xj	        dz  c_	        ||	z  }n|dk     r| xj	        |z  c_	        |                      |¦  «        \  }
} | j        |||fi |¤Ž|
z   | j        | ||fi |¤Žz
  |z  S )Nr$   r   r   r.   )rO   rD   ÚimrP   rC   r8   r6   ÚfloorÚninfr    Únint_distanceÚepsÚcospi_sinpir   )r
   r   r   rF   rG   r)   ÚqÚmr   r-   ÚcosÚsins               r   Úbesselyr]   t   sÀ  € àð $Øð 	åÐØð 	$à”G�8˜q ™sÑ#Ð#Ø�6Š6�!‰9Œ9ð 	#Ø”7˜a ™c‘?Ð"Ø�FŠF�1‰IŒIˆØˆc‰EˆØ�9Š9�Q‰<Œ<ð 	!Ø�1ŠuˆuØœ�x 1 Q¡3Ñ'Ð'à˜A˜a™C‘yÐ ØˆqŠ5ˆ5•S˜Ÿš 1™œÑ&Ô&¨Ñ*ˆ5Ø”7˜a ™c‘?Ð"à”8˜q ™sÑ#Ð#à€H„H��N€H„HØ×Ò˜QÑÔ�D€A€qØˆCŒHˆ9‚}€}ØŒWˆHˆØˆŒ�A‰ˆŒØ	ˆQ‰ˆˆØ	
ˆQŠˆØˆŒ�A‰ˆŒà�Š˜qÑ!Ô!�H€CˆØˆCŒK˜˜!˜JÐ0Ð0¨Ð0Ð0°Ñ4ØˆŒ�Q�B�q˜Ð-Ð- fÐ-Ð-ñ.Ø/2ñ3ð 3r   c                 ó¦   ‡ ‡— ‰s‰ j         S ‰                      ‰¦  «        }|dk     rˆfd„}n‰ xj        |z  c_        ˆ ˆfd„} ‰ j        ||gfi |¤ŽS )Nr   c                 óv   •— ‰dz  dz  }‰dg|  | dz
  g| gg g d| z
  g|f}‰dg| |  dz
  g|  gg g d| z   g|f}||fS )Nr   r   © )r   r)   ÚT1ÚT2r   s       €r   r-   zbesselk.<locals>.hŸ   so   ø€ Ø�1‘�q‘ˆAØ�Q�˜1˜"˜a ™c˜ Q C¨¨R°!°A±#°¸Ð9ˆBØ�Q�˜!˜a˜R ™T˜ a R D¨"¨b°1°Q±3°%¸Ð:ˆBØ�r�6ˆMr   c           	      óv   •— ‰j         dz  ‰‰                     ‰ ¦  «        gg d¢g g | dz   d| z
  gg dd‰z  z  fgS )Nr   )r$   ç      à¿r   r$   r   )r'   Úexp©r   r
   r   s    €€r   r-   zbesselk.<locals>.hª   sS   ø€ Ø”f˜Q‘h  3§7¢7¨A¨2¡;¤;Ð/°°°¸rÀ2Ø�3‘˜˜A™�  B¨¨!©¡Hð.ð /ð /r   )rD   r:   r    r>   )r
   r   r   rG   r,   r-   s   ` `   r   Úbesselkrg   ˜   s˜   øø€ àð ØŒwˆØ�Š�‰
Œ
€AØˆ1‚u€uð	ð 	ð 	ð 	ð 	ð 	ð 	ˆŒ�A‰ˆŒð	/ð 	/ð 	/ð 	/ð 	/ð 	/ð ˆ3Œ=˜˜Q˜CÐ*Ð* 6Ð*Ð*Ð*r   c                 óP   —  | j         ||fi |¤Ž| j         | j        ||fi |¤Žz  z   S ©N©r   Újr]   ©r
   r   r   rG   s       r   Úhankel1rm   ¯   ó@   € àˆ3Œ;�q˜Ð$Ð$˜VÐ$Ð$ s¤u¨[¨S¬[¸¸1Ð-FÐ-F¸vÐ-FÐ-FÑ'FÑFÐFr   c                 óP   —  | j         ||fi |¤Ž| j         | j        ||fi |¤Žz  z
  S ri   rj   rl   s       r   Úhankel2rp   ³   rn   r   c                 óB  — |dk    rH|                       |¦  «        dk    r|S |                       |¦  «        dk     r
| j        |z   S | j        |z  S |                      d|d¬¦  «        }d|z   }|                      |¦  «        ||z  z   | j        ||z
  dd|z  z   |fi |¤Žz  S )Nr   rd   Tr"   r$   r   r   )rC   rD   rP   r&   re   Úhyp1f1)r
   r   rZ   r   rG   r   Úys          r   Úwhitmrt   ·   s´   € àˆA‚v€và�6Š6�!‰9Œ9�tÒÐØˆHØ�VŠV�A‰YŒY˜ÒÐØ”7˜Q‘;Ðà”7˜Q‘;ÐØ�Š��q ˆÑ%Ô%€AØˆA‰€AØ�7Š7�1‰:Œ:˜˜1™Ñ˜z˜sœz¨!¨A©#¨q°°1±©u°aÐBÐB¸6ÐBÐBÑBÐBr   c                 ó:  — |dk    rDt          |                      |¦  «        ¦  «        }|dk     r|S |dk    r
| j        |z   S | j        |z  S |                      d|d¬¦  «        }d|z   }|                      |¦  «        ||z  z   | j        ||z
  dd|z  z   |fi |¤Žz  S )Nr   r$   rd   Tr"   r   r   )r?   rC   rD   rP   r&   re   Úhyperu)r
   r   rZ   r   rG   Úgr   rs   s           r   Úwhitwrx   Å   s±   € àˆA‚v€vÝ�—’�q‘	”	‰NŒNˆØˆsŠ7ˆ7ØˆHØ�ŠWˆWØ”7˜Q‘;Ðà”7˜Q‘;ÐØ�Š��q ˆÑ%Ô%€AØˆA‰€AØ�7Š7�1‰:Œ:˜˜1™Ñ˜z˜sœz¨!¨A©#¨q°°1±©u°aÐBÐB¸6ÐBÐBÑBÐBr   c                 ó`  ‡ ‡— ‰                       |¦  «        \  }}‰                       |¦  «        \  }}‰                      ‰¦  «        Š‰sD‰                      |¦  «        dk    r!‰                      d|z
  g||z
  dz   g¦  «        S ‰ j        ‰z   S d|z   |z
  }‰                       |¦  «        \  }}	 ‰ j        }		 ‰ xj        dz  c_        ‰                      dd||f||gd‰z  ‰ j        ¬¦  «        }
|
‰|z  z  |	‰ _        S # |	‰ _        w xY w# ‰ j        $ r Y nw xY wˆ ˆfd„} ‰ j        |||gfi |¤ŽS )Nr   r.   r   r   r   )Úmaxtermsc                 ó¾   •— ‰                      |¦  «        }‰j        |gddgg | |z
  dz   |g| g|g‰f}‰j         |‰gddd|z
  gg | d|z
  g| |z
  dz   gd|z
  g‰f}||fS )Nr   r   r   )Úsinpir'   )ÚaÚbr4   ra   rb   r
   r   s        €€r   r-   zhyperu.<locals>.hé   s‰   ø€ Ø�IŠI�a‰LŒLˆØŒv�aˆj˜!˜B˜  A a¡C¨¡E¨! 9¨a¨S°!°°QÐ7ˆØ”ˆw�q˜ˆm˜Q˜r ! A¡#˜J r¨1¨Q¨q©S¨'°1°Q±3°q±5°'¸1¸Q¹3¸%ÀÐBˆØ�2ˆvˆr   )	Ú_convert_paramr7   rC   Ú	gammaprodrD   r    ÚhypsumÚNoConvergencer>   )r
   r}   r~   r   rG   ÚatypeÚbtypeÚbbÚbbtyperI   rJ   r-   s   `  `        r   rv   rv   Ó   s�  øø€ à×!Ò! !Ñ$Ô$�H€A€uØ×!Ò! !Ñ$Ô$�H€A€uØ�Š�A‰Œ€AØð Ø�6Š6�!‰9Œ9˜Š>ˆ>Ø—=’= ! A¡# ¨¨!©¨A© wÑ/Ô/Ð/à”7˜Q‘;ÐØ	
ˆ1‰ˆQ‰€BØ×#Ò# BÑ'Ô'�J€Bˆð	ØŒxˆð	ØˆHŒH˜‰NˆHŒHØ—
’
˜1˜a %¨ °1°b°'¸2¸a¹4È#Ì(�
ÑSÔSˆAØ�q˜!‘t‘8àˆCŒHˆHø�tˆCŒHˆOˆOˆOˆOøØÔð ð ð Øˆðøøøðð ð ð ð ð ð
 ˆ3Œ=˜˜Q˜q˜EÐ,Ð, VÐ,Ð,Ð,s*   Â/D	 Â7>C= Ã5D	 Ã=	DÄD	 Ä	
DÄDc                 ó†   ‡ ‡— ‰                       |¦  «        }‰                       ‰¦  «        Šˆ ˆfd„} ‰ j        ||gfi |¤ŽS )Nc                 ó„   •— ‰dz  d‰                      ‰j        ¦  «        z  g| dz   dgg | dz   gdgd| dz   g‰dz  dz   fgS ©Nr   r$   r   r   ç      ø?©Úsqrtr'   rf   s    €€r   r-   zstruveh.<locals>.hõ   sb   ø€ Ø�A‘#�s˜3Ÿ8š8 C¤FÑ+Ô+Ñ+Ð,¨q°©s°B¨i¸¸aÀ¹e¸WÀqÀcÈCÐQRÐSVÑQVÈ<Ð[\Ð]^Ñ[^ÐabÑZbÐYbÐcÐdÐdr   ©r7   r>   ©r
   r   r   rG   r-   s   ` `  r   Ústruvehr�   ð   sd   øø€ à�Š�A‰Œ€AØ�Š�A‰Œ€Aðeð eð eð eð eð eàˆ3Œ=˜˜Q˜CÐ*Ð* 6Ð*Ð*Ð*r   c                 ó†   ‡ ‡— ‰                       |¦  «        }‰                       ‰¦  «        Šˆ ˆfd„} ‰ j        ||gfi |¤ŽS )Nc                 ó‚   •— ‰dz  d‰                      ‰j        ¦  «        z  g| dz   dgg | dz   gdgd| dz   g‰dz  dz  fgS r‰   r‹   rf   s    €€r   r-   zstruvel.<locals>.hþ   s_   ø€ Ø�A‘#�s˜3Ÿ8š8 C¤FÑ+Ô+Ñ+Ð,¨q°©s°B¨i¸¸aÀ¹e¸WÀqÀcÈCÐQRÐSVÑQVÈ<ÐZ[Ð\]ÑZ]Ð`aÑYaÐbÐcÐcr   r�   rŽ   s   ` `  r   Ústruvelr’   ù   sd   øø€ à�Š�A‰Œ€AØ�Š�A‰Œ€Aðdð dð dð dð dð dàˆ3Œ=˜˜Q˜CÐ*Ð* 6Ð*Ð*Ð*r   c                 ó–   ‡ ‡‡— ‰                       |¦  «        d         }‰                      ‰¦  «        Šˆ ˆˆfd„} ‰ j        ||gfi |¤ŽS )Nr   c                 ó:  •— ‰j         }| |z  }|dz  }||z
  ||z   d|z
  d|z   f\  }}}}‰                     |¦  «        \  }}	‰dk    r
|‰z  |	g|g}}
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   r   s       €€r   r-   zlommels1.<locals>.h"  s�   ø€ ØŒKˆØ×Ò˜q uÐÑ-Ô-ˆØ�1‘�Q‘˜˜!™˜A™˜qÐ! B¨¨A¨a©C =°"°b¸1¸#Ø��!‘�A‘‰Y�q˜!˜A™#˜a™%‘yÐ! 1ð&ð 'ð 	'r   r¢   ©r
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ˆˆdØˆe�Q˜ˆ]˜Q  !¡ Q q¡S˜M¨A¨a©C¨5°"°b¸3ÀÀQÀqÁSÁ	È1ÈSÐQRÉUÉ7Ð:SÐUVÐVˆØˆe�Q˜ˆ]˜Q  !¡ a R˜L¨1¨#¨r°2¸¸SÀ!ÀAÁ#¹YÈÈ#ÈaÉ%ÉÐ7PÐRSÐSˆØˆe�Q˜ˆ]˜Q   1¡ a˜L¨A¨2¨$°°B¸¸cÀ1ÀQÁ3¹iÈÈaÐPQÉcÉÐ8SÐUVÐVˆØˆe�Q˜ˆ]˜Q   1¡ a¨¡c˜N¨a¨R°©T¨F°B¸¸SÀ#ÀqÈÁsÁ)ÈSÐRSÐTUÑRUÉYÐ<WÐYZÐZˆØ�2�r˜2ˆ~Ðr   r�   rŽ   s   ` `  r   ÚkeirÎ   p  rË   r   c                 ó$   ‡ ‡— ‰ j         Šˆ ˆfd„}|S )Nc                 ó    •— | j         }| j        }|                     ‰d¦  «        \  }}||k    r|
 S | ‰| ¦  «        f|‰<   |‰         d         S )N)r   r   r   )Ú_misc_const_cacher    Úget)r
   Úcacher    ÚprJ   ÚfÚnames        €€r   Ú	f_wrappedzc_memo.<locals>.f_wrappedƒ  s\   ø€ ØÔ%ˆØŒxˆØ�iŠi˜˜fÑ%Ô%‰ˆˆ!Ø�Š9ˆ9Ø�2ˆIà   3¡¤˜.ˆE�$‰KØ˜”;˜q”>Ð!r   )Ú__name__)rÕ   r×   rÖ   s   ` @r   Úc_memorÙ   �  s1   øø€ ØŒ:€Dð"ð "ð "ð "ð "ð "ð Ðr   c                 óŠ   — d|                       d¦  «        |                      |                      d¦  «        dz  ¦  «        z  z  S )Nr   é	   r   r0   ©ÚcbrtÚgammar=   ©r
   s    r   Ú
_airyai_C1rà   Ž  s6   € à�—’˜‘”˜cŸiši¨¯ª°©
¬
°1©Ñ5Ô5Ñ5Ñ6Ð6r   c                 óŠ   — d|                       d¦  «        |                      |                      d¦  «        dz  ¦  «        z  z  S )Nr   r0   r   rÜ   rß   s    r   Ú
_airyai_C2râ   ’  s6   € à�—’˜!‘”˜sŸyšy¨¯ª°©¬°A©Ñ6Ô6Ñ6Ñ7Ð7r   c                 óŒ   — d|                       dd¦  «        |                      |                      d¦  «        dz  ¦  «        z  z  S )Nr   r0   é   r   ©ÚnthrootrÞ   r=   rß   s    r   Ú
_airybi_C1rç   –  s:   € à�—’˜A˜aÑ Ô  3§9¢9¨S¯WªW°Q©Z¬Z¸©\Ñ#:Ô#:Ñ:Ñ;Ð;r   c                 ó†   — |                       dd¦  «        |                      |                      d¦  «        dz  ¦  «        z  S )Nr0   rä   r   rå   rß   s    r   Ú
_airybi_C2ré   š  s5   € à�;Š;�q˜ÑÔ˜cŸiši¨¯ª°©
¬
°1©Ñ5Ô5Ñ5Ð5r   c                 ó®   — | j         }	 |                      dd¦  «        |                      d¦  «        z  d| j        z  z  }|| _         n# || _         w xY w|
 S )Nr0   ú2/3r   )r    ÚpowerrÞ   r'   )r
   r    rJ   s      r   Ú_airybi_n2_infrí   ž  s\   € ØŒ8€DðØ�IŠI�a˜ÑÔ˜sŸyšy¨Ñ/Ô/Ñ/°°3´6±Ñ:ˆàˆŒˆø�4ˆŒˆˆˆˆØˆ2€Is   ‰7A Á	Ac                 ó*  — |dk    �r|dk     r|S | j         }| j        }	 | xj        dz  c_        |                      |dz   |z  ¦  «        |                      d||z  ¦  «        z  | j        z  }|dk    r;||                      d|dz   z  |z  ¦  «        z  }||                      dd¦  «        z  }nG|t          |                      d|dz   z  |z  ¦  «        ¦  «        z  }||                      dd¦  «        z  }|| _        n# || _        w xY w|
 |z   S t          ‚)	NÚZr   r.   r   r0   r   rë   z1/6)Úmpq_1_3r    rÞ   rì   r'   r|   r?   rA   )r
   r   r   Úntyper¡   r)   r    rJ   s           r   Ú_airyderiv_0rò   ¨  s$  € Ø�‚|�|ØˆqŠ5ˆ5ØˆHØŒKˆØŒxˆð
	ØˆHŒH˜‰NˆHŒHØ—	’	˜1˜Q™3 ™'Ñ"Ô" S§Y¢Y¨q°°1±Ñ%5Ô%5Ñ5¸¼Ñ>ˆAØ˜ŠzˆzØ�S—Y’Y˜q ! A¡#™w q™yÑ)Ô)Ñ)�Ø�S—Y’Y˜q Ñ'Ô'Ñ'��à•S˜Ÿš 1 a¨¡c¡7¨1¡9Ñ-Ô-Ñ.Ô.Ñ.�Ø�S—Y’Y˜q Ñ'Ô'Ñ'�àˆCŒHˆHø�tˆCŒHˆOˆOˆOˆOØˆr�A‰vˆõ "Ð!s   ŸCC< Ã<	Dc           	      óº  ‡ ‡‡— ‰                       ‰¦  «        Š|r‰                      |¦  «        \  }}nd}‰                      ‰¦  «        sµ‰r³|r…|dk    r|dk    rR‰‰ j        k    r‰                      d¦  «        dz  d‰z  z   S ‰‰ j        k    r‰                      d¦  «        dz  d‰z  z   S |dk     r!‰‰ j        k    r‰S ‰‰ j        k    r	d|z  ‰ z  S |s‰‰ j        k    s‰‰ j        k    rd‰z  S t          d¦  «        ‚‰r4t          dt          d‰  	                    ‰¦  «        z  ¦  «        ¦  «        ŠndŠ|r�|dk    rˆ ˆˆfd	„} ‰ j
        |g fi |¤ŽS ‰dk    rt          ‰ ‰||d¦  «        S ˆ ˆˆfd
„} ‰ j
        ||gfi |¤Ž}‰                      ‰¦  «        r*‰                      |¦  «        r‰                      |¦  «        }|S ˆ ˆˆfd„} ‰ j
        |g fi |¤ŽS )Nr   rï   r   r   r0   éþÿÿÿzessential singularity of Ai(z)rŠ   c                  ó.  •— ‰                      ‰
¦  «        dk    r‹‰xj        ‰	z  c_        ‰
dz  } d| z  }d| z  dz  }‰xj        ‰	z  c_        ‰                     |¦  «         d‰                     ‰j        ¦  «        z  z  ‰                     ‰
d¦  «        z  }|gdgg g dd	gg |ffS ‰xj        ‰	z  c_        ‰
dz  d
z  } ‰xj        ‰	z  c_        t          ‰¦  «        dz  }t          ‰¦  «        }|‰
gddgg g g ‰j        g| f}|gdgg g g ‰j	        g| f}||fS )Nr¸   rŠ   r¹   rô   r0   r   r   )r   rä   )é   rä   rÛ   r$   )
r9   r    re   rŒ   r'   ræ   rà   râ   Úmpq_5_3rð   ©r4   r)   r™   ÚCÚC1ÚC2ra   rb   r
   Ú	extraprecr   s           €€€r   r-   zairyai.<locals>.hÝ  s>  ø€ à—7’7˜1‘:”: ’>�>Ø�H”H 	Ñ)�H”HØ˜3™�A E¨!¡G °°A±°a±¨QØ�H”H 	Ñ)�H”HØŸš ™œ˜ Q s§x¢x°´Ñ'7Ô'7Ñ%7Ñ8¸¿ºÀQÀqÑ9IÔ9IÑI�AØ˜C   B r¨6°%¨.¸¸AÐ>Ð?Ð?ð �H”H 	Ñ)�H”HØ˜1™˜q™�AØ�H”H 	Ñ)�H”HÝ# C™œ¨3Ñ.�BÝ# C™œ�BØ˜Q˜  1  b¨¨B°´¨}¸QÐ>�BØ˜˜q˜c " R¨¨C¬K¨=¸Ð:�BØ˜r˜6�Mr   c                 ó`  •— ‰xj         ‰z  c_         ‰dz  dz  }‰xj         ‰z  c_         ‰j        ‰j        ‰j        }}}|}d}d| z
  |z  }d| z
  |z  }d| |z  z
  }	d‰g| |z
  |  g|g|||	g||g|||	g|f}
|}d| z
  |z  }d| |z  z
  }d| z
  |z  }	d‰‰ g| |z
  |  dg|g|||	g||g|||	g|f}|
|fS ©Nr0   rÛ   r   r   r¸   )r    rð   Úmpq_2_3Úmpq_4_3)r   r4   Úq13Úq23Úq43rš   r›   rœ   r�   Úb3ra   rb   r
   rü   r   s               €€€r   r-   zairyai.<locals>.hô  s  ø€ Ø�”˜IÑ%�”Ø�q‘D˜‘F�Ø�”˜IÑ%�”Ø!œk¨3¬;¸¼˜�C�Ø�˜1˜ ! A¡# s¡˜b°°!±°S©y¨B¸Q¸qÀ¹u¹W¸"Ø˜�V˜a ™e a R˜[¨2¨$°°B°r°
Ø˜�G˜b  B˜Z¨ð+�à�˜A˜a™C ™9˜¨¨1¨S©5© b°a¸±c¸3±Y°"Ø˜˜Q˜B�Z ! C¡%¨!¨¨Q °"°¸¸2¸b°zØ˜�G˜b  B˜Z¨ð+�à˜2�v�r   c                  ó(  •— ‰                      ‰
¦  «        dk    rŠ‰xj        ‰	z  c_        ‰
dz  } d| z  }d| z  dz  }‰xj        ‰	z  c_        ‰                     |¦  «        d‰                     ‰j        ¦  «        z  ‰                     ‰
d¦  «        z  z  }|gdgg g dd	gg |ffS ‰xj        ‰	z  c_        ‰
dz  d
z  } ‰xj        ‰	z  c_        t          ‰¦  «        }t          ‰¦  «        }|gdgg g g ‰j        g| f}‰
|z  gdgg g g ‰j	        g| f}||fS )Nr¸   rŠ   r¹   rô   r0   r   r   )r   rä   )é   rä   rÛ   )
r9   r    re   rŒ   r'   ræ   rà   râ   rÿ   r   rø   s           €€€r   r-   zairyai.<locals>.h  s7  ø€ Ø�wŠw�q‰zŒz˜AŠ~ˆ~ð �”˜IÑ%�”Ø�s‘F�  a¡˜A¨R°©T°!©V¨Ø�”˜IÑ%�”Ø—G’G˜A‘J”J  #§(¢(¨3¬6Ñ"2Ô"2Ñ 2°3·;²;¸qÀÑ3CÔ3CÑ CÑD�Ø˜˜Q˜C  2 u¨U m°B°qÐ9Ð:Ð:à�”˜IÑ%�”Ø�q‘D˜1‘H�Ø�”˜IÑ%�”Ý ‘_”_�Ý ‘_”_�Ø�T˜1˜#˜b  B¨¬ }°QÐ6�Ø˜‘d�V˜Q˜C  2 b¨#¬+¨°qÐ8�Ø˜2�v�r   )r7   r   ÚisnormalrD   r=   rU   rO   r3   r6   r:   r>   rò   Ú_is_real_typer8   r9   ©	r
   r   rF   rG   r   rñ   r-   rJ   rü   s	   ``      @r   Úairyair
  ¾  s‡  øøø€ à�Š�A‰Œ€AØð Ø×%Ò% jÑ1Ô1‰ˆˆ5ˆ5àˆà�<Š<˜‰?Œ?ð ;˜qð ;Øð 
	*�˜#’�Ø�BŠwˆwØ˜œ’<�<ØŸ7š7 1™:œ: a™<¨!¨A©#Ñ-Ð-Ø˜œ’=�=ØŸ7š7 2™;œ; q™=¨1¨Q©3Ñ.Ð.Ø�2ŠvˆvØ˜œ’<�<Ø�HØ˜œ’=�=Ø ™7 q b™>Ð)Øð 	�q˜CœG’|�| q¨C¬H¢} }Ø�Q‘3ˆJåÐ9Ñ:Ô:Ð:àð Ý˜�3˜s 3§7¢7¨1¡:¤:™~Ñ.Ô.Ñ/Ô/ˆ	ˆ	àˆ	Øð =.Ø�Š6ˆ6ð"ð "ð "ð "ð "ð "ð "ð$ !�3”=  BÐ1Ð1¨&Ð1Ð1Ð1à�AŠvˆvÝ# C¨¨A¨u°aÑ8Ô8Ð8ðð ð ð ð ð ð ð �”˜a ! Ð/Ð/¨Ð/Ð/ˆAØ× Ò  Ñ#Ô#ð ¨¯	ª	°!©¬ð Ø—G’G˜A‘J”J�ØˆHð	ð 	ð 	ð 	ð 	ð 	ð 	ð& ˆsŒ}˜Q Ð-Ð- fÐ-Ð-Ð-r   c           	      óN  ‡ ‡‡— ‰                       ‰¦  «        Š|r‰                      |¦  «        \  }}nd}‰                      ‰¦  «        s‰r}|rM|dk    rG‰‰ j        k    r‰S ‰‰ j        k    r/|dk    rd‰z  S |dk    rt          ‰ ¦  «        S |dk     r	d|z  ‰ z  S |s‰‰ j        k    r‰S ‰‰ j        k    rd‰z  S t          d¦  «        ‚‰r4t          dt          d‰  	                    ‰¦  «        z  ¦  «        ¦  «        ŠndŠ|r�|dk    rˆ ˆˆfd„} ‰ j
        |g fi |¤ŽS ‰dk    rt          ‰ ‰||d¦  «        S ˆ ˆˆfd	„} ‰ j
        ||gfi |¤Ž}‰                      ‰¦  «        r*‰                      |¦  «        r‰                      |¦  «        }|S ˆ ˆˆfd
„} ‰ j
        |g fi |¤ŽS )Nr   rï   r   r   rô   zessential singularity of Bi(z)rŠ   c                  óæ   •— ‰xj         ‰z  c_         ‰dz  dz  } ‰xj         ‰z  c_         t          ‰¦  «        dz  }t          ‰¦  «        }|‰gddgg g g ‰j        g| f}|gdgg g g ‰j        g| f}||fS )Nr0   rÛ   r$   r   r   )r    rç   ré   r÷   rð   ©r4   rú   rû   ra   rb   r
   rü   r   s        €€€r   r-   zairybi.<locals>.h;  s�   ø€ Ø�”˜IÑ%�”Ø�q‘D˜1‘H�Ø�”˜IÑ%�”Ý ‘_”_ SÑ(�Ý ‘_”_�Ø˜�V˜Q˜q˜E " R¨¨C¬K¨=¸Ð:�Ø�T˜1˜#˜b  B¨¬ }°QÐ6�Ø˜2�v�r   c                 óz  •— ‰xj         ‰z  c_         ‰dz  dz  }‰xj         ‰z  c_         ‰j        ‰j        ‰j        }}}‰j        }‰j        }|}d}d| z
  |z  }	d| z
  |z  }
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  }
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|g|f}||fS rþ   )r    rð   rÿ   r   Úmpq_1_6Úmpq_5_6)r   r4   r  r  r  Úq16Úq56rš   r›   rœ   r�   r  ra   rb   r
   rü   r   s                 €€€r   r-   zairybi.<locals>.hH  s   ø€ Ø�”˜IÑ%�”Ø�q‘D˜‘F�Ø�”˜IÑ%�”Ø!œk¨3¬;¸¼˜�C�Ø”k�Ø”k�Ø�˜1˜ ! A¡# s¡˜b°°!±°S©y¨B¸Q¸qÀ¹u¹W¸"Ø˜�V˜a ™e a R˜[¨2¨$°°B°r°
Ø˜�G˜b  B˜Z¨ð+�à�˜A˜a™C ™9˜¨¨1¨S©5© b°a¸±c¸3±Y°"Ø˜�V˜a ™e Q q¡S˜\¨B¨4°"°R¸°Ø˜�G˜b  B˜Z¨ð+�à˜2�v�r   c                  óâ   •— ‰xj         ‰z  c_         ‰dz  dz  } ‰xj         ‰z  c_         t          ‰¦  «        }t          ‰¦  «        }|gdgg g g ‰j        g| f}‰|z  gdgg g g ‰j        g| f}||fS )Nr0   rÛ   r   )r    rç   ré   rÿ   r   r  s        €€€r   r-   zairybi.<locals>.h[  s‹   ø€ ØˆHŒH˜	Ñ!ˆHŒHØ�1‘�q‘ˆAØˆHŒH˜	Ñ!ˆHŒHÝ˜C‘”ˆBÝ˜C‘”ˆBØ��q�c˜"˜R  C¤K =°Ð2ˆBØ�B‘$�˜˜˜B˜r " c¤k ]°1Ð4ˆBØ�r�6ˆMr   )r7   r   r  rD   rU   rí   rO   r3   r6   r:   r>   rò   r  r8   r9   r	  s	   ``      @r   Úairybir    s^  øøø€ à�Š�A‰Œ€AØð Ø×%Ò% jÑ1Ô1‰ˆˆ5ˆ5àˆà�<Š<˜‰?Œ?ð ;˜qð ;Øð 		*�˜#’�Ø�C”GŠ|ˆ|Ø�Ø�C”HŠ}ˆ}Ø˜’7�7Ø˜Q™3�JØ˜’7�7Ý)¨#Ñ.Ô.Ð.Ø�r’6�6Ø ™7 q b™>Ð)Øð 	Ø�C”GŠ|ˆ|Ø�Ø�C”HŠ}ˆ}Ø˜‘s�
åÐ9Ñ:Ô:Ð:Øð Ý˜�3˜s 3§7¢7¨1¡:¤:™~Ñ.Ô.Ñ/Ô/ˆ	ˆ	àˆ	Øð ,.Ø�Š6ˆ6ðð ð ð ð ð ð ð !�3”=  BÐ1Ð1¨&Ð1Ð1Ð1à�AŠvˆvÝ# C¨¨A¨u°aÑ8Ô8Ð8ðð ð ð ð ð ð ð �”˜a ! Ð/Ð/¨Ð/Ð/ˆAØ× Ò  Ñ#Ô#ð ¨¯	ª	°!©¬ð Ø—G’G˜A‘J”J�ØˆHð	ð 	ð 	ð 	ð 	ð 	ð 	ð ˆsŒ}˜Q Ð-Ð- fÐ-Ð-Ð-r   Fc                 óV  ‡ — d„ }d„ }t          |¦  «        }|dk     rt          d¦  «        ‚|dvrt          d¦  «        ‚|dk    rr|r7‰                      ˆ fd„ |d	‰ j        z  d
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|z  dz
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|z  dz
  z  dz  ¦  «         ¦  «        S ‰                      ‰ j         |d	‰ j        z  d
|z  d	z
  z  dz  ¦  «         ¦  «        S |dk    rì|dk    rè|rqd	‰ j        z  d
|z  d	z
  z  dz  d‰ j        z  z   }‰                      ‰                      d¦  «        d	z  ¦  «         ||¦  «        z  }‰                      ˆ fd„|¦  «        S d	‰ j        z  d
|z  dz
  z  dz  d‰ j        z  z   }‰                      ‰                      d¦  «        d	z  ¦  «         ||¦  «        z  }‰                      ‰ j        |¦  «        S d S d S )Nc                 ó*   — | dz  dd| dz  dz  z  z
  z  S )NçUUUUUUå?r   rö   r   é0   r`   ©Úts    r   ÚUz_airy_zero.<locals>.Uh  ó    € �Q˜‘Y  ! Q¨¡T¨"¡W¡+¡Ñ.Ð.r   c                 ó*   — | dz  dd| dz  dz  z  z   z  S )Nr  r   r  r   r  r`   r  s    r   r+   z_airy_zero.<locals>.Ti  r  r   r   zk cannot be less than 1©r   r   z%Derivative should lie between 0 and 1r   c                 ó0   •— ‰                      | d¦  «        S r«   )r
  ©r   r
   s    €r   ú<lambda>z_airy_zero.<locals>.<lambda>q  ó   ø€ ¨#¯*ª*°Q°q©/¬/€ r   r0   r¸   é   Fc                 ó0   •— ‰                      | d¦  «        S r«   ©r  r   s    €r   r!  z_airy_zero.<locals>.<lambda>v  r"  r   Ty              è?c                 ó0   •— ‰                      | d¦  «        S r«   r%  r   s    €r   r!  z_airy_zero.<locals>.<lambda>}  r"  r   )	r6   rO   Úfindrootr'   r
  r  Úln2Úexpjpir=   )	r
   r¡   r   rF   Úcomplexr  r+   r  rŸ   s	   `        r   Ú
_airy_zeror+  f  sˆ  ø€ à.Ð.Ð.Ø.Ð.Ð.ÝˆA‰Œ€AØˆ1‚u€uÝÐ2Ñ3Ô3Ð3Ø˜ÐÐÝÐ@ÑAÔAÐAØ�‚z€zØð 	(Ø—<’<Ð 9Ð 9Ð 9Ð 9Ø��1�S”V‘8˜Q˜q™S ™UÑ# AÑ%Ñ&Ô&Ð&ñ(ô (ð (à�|Š|˜CœJ¨¨¨1¨S¬V©8°Q°q±S¸±UÑ+;¸AÑ+=Ñ)>Ô)>Ð(>Ñ?Ô?Ð?Ø�‚z€z�g Ò&Ð&Øð 	(Ø—<’<Ð 9Ð 9Ð 9Ð 9Ø��1�S”V‘8˜Q˜q™S ™UÑ# AÑ%Ñ&Ô&Ð&ñ(ô (ð (à�|Š|˜CœJ¨¨¨1¨S¬V©8°Q°q±S¸±UÑ+;¸AÑ+=Ñ)>Ô)>Ð(>Ñ?Ô?Ð?Ø�‚z€z�g ’o�oØð 	>Ø�#”&‘˜!˜A™#˜a™%Ñ  Ñ" U¨3¬7¡]Ñ2ˆAØ—
’
˜3Ÿ7š7 1™:œ: a™<Ñ(Ô(¨1¨1¨Q©4¬4Ñ/ˆAØ—<’<Ð 9Ð 9Ð 9Ð 9¸1Ñ=Ô=Ð=ØˆcŒf‰H�a˜‘c˜!‘eÑ˜QÑ  s¤w¡Ñ.ˆØ�JŠJ�s—w’w˜q‘z”z !‘|Ñ$Ô$ q q¨¡t¤tÑ+ˆØ�|Š|˜CœJ¨Ñ*Ô*Ð*ð €z�o�or   c                 ó(   — t          | d||d¦  «        S )Nr   F©r+  )r
   r   rF   s      r   Ú
airyaizeror.  ‚  s   € å�c˜1˜a ¨UÑ3Ô3Ð3r   c                 ó(   — t          | d|||¦  «        S r«   r-  )r
   r   rF   r*  s       r   Ú
airybizeror0  †  s   € å�c˜1˜a ¨WÑ5Ô5Ð5r   c           	      ó|  ‡ ‡‡‡‡— ‰                       ‰¦  «        Š‰                      ‰¦  «        r=‰‰ j        k    r‰dk    rd‰z  S ‰dk    r‰S ‰‰ j        k    rd‰z  S t	          d¦  «        ‚‰r4t          dt          d‰                      ‰¦  «        z  ¦  «        ¦  «        ŠndŠ‰                     d¦  «        rt          ‚	 ‰                      ‰¦  «        dk    rº‰dk    rUt          ‰                      ‰¦  «        ¦  «        ‰ j        dz  dz  k     r$ˆ ˆfd„}‰                      |g ‰ j        d	¬
¦  «        S ‰dk    rYt          ‰                      ‰ ¦  «        ¦  «        d‰ j        z  dz  dz  k     r$ˆ ˆfd„}‰                      |g ‰ j        d	¬
¦  «        S n# ‰ j        $ r Y nw xY wˆ ˆˆˆˆfd„} ‰ j        |g fi ‰¤ŽS )Nr   r   zessential singularityrŠ   rF   r0   g+‡ÙÎ÷ï?c            	      ó:   •— ‰ j         ‰gddgg g g d¢g d‰dz  z  ffS ©Nr   ))r   r0   )r   r0   r   rÛ   r0   ©r'   ©r
   r   s   €€r   r-   z_scorer.<locals>.hž  s1   ø€ Ø!œf Q˜Z¨¨B¨°°2°o°o°oÀbÈÈ1ÈaÉ4ÉÐPÐRÐRr   T)rz   Úforce_seriesr   c            	      ó<   •— ‰ j          ‰gddgg g g d¢g d‰dz  z  ffS r3  r4  r5  s   €€r   r-   z_scorer.<locals>.h¢  s3   ø€ Ø"œv˜g a˜[¨"¨R¨°°B°°°ÀrÈ!ÈAÈqÉDÉ&ÐQÐSÐSr   c                  óü   •—  ‰j         ‰	fi ‰¤Ždz  } d‰j        z  }‰dk    r
| dz  } |dz  }‰xj        ‰z  c_        ‰	dz  dz  }‰xj        ‰z  c_        | gdgg g g g df}|‰	gddgg g dg‰j        ‰j        g|f}||fS )Nr0   rô   r   r   r   rÛ   r   )r  r'   r    r   r÷   )
r    r*   r4   ra   rb   r
   rü   rG   r¡   r   s
        €€€€€r   r-   z_scorer.<locals>.h§  s¼   ø€ ØˆCŒJ�qÐ#Ð#˜FÐ#Ð# AÑ%ˆØˆsŒv‰IˆØ�AŠ:ˆ:Ø�‰FˆAØ�‰GˆAØˆŒ�IÑˆŒØˆq‰D�‰FˆØˆŒ�IÑˆŒØˆS�1�#�r˜2˜r 2 qÐ(ˆØ�ˆU�R˜�F˜B  Q C¨#¬+°c´kÐ)BÀAÐEˆØ�2ˆvˆr   )r7   ÚisinfrD   rU   rO   r3   r6   r:   rÒ   rA   r?   Úargr'   r>   r    r‚   )r
   r   r¡   rG   r-   rü   s   ```` @r   Ú_scorerr;  Š  s-  øøøøø€ Ø�Š�A‰Œ€AØ
‡y‚y��|„|ð 2Ø�”Š<ˆ<Ø˜Šzˆz ! A¡#˜:Ø˜Šzˆz !˜8Ø�”Š=ˆ=Ø�Q‘3ˆJÝÐ0Ñ1Ô1Ð1Øð Ý˜�3˜s 3§7¢7¨1¡:¤:™~Ñ.Ô.Ñ/Ô/ˆ	ˆ	àˆ	Ø‡z‚z�,ÑÔð "Ý!Ð!ðØ�7Š7�1‰:Œ:˜Š>ˆ>Ø˜Šzˆz�c #§'¢'¨!¡*¤*™oœo°´°q±¸5Ñ0@Ò@Ð@ðSð Sð Sð Sð Sð Sà—}’} Q¨°S´XÈD�}ÑQÔQÐQØ˜Šzˆz�c #§'¢'¨1¨"¡+¤+Ñ.Ô.°°3´6±¸!±¸eÑ1CÒCÐCðTð Tð Tð Tð Tð Tà—}’} Q¨°S´XÈD�}ÑQÔQÐQøøØÔð ð ð Øˆðøøøðð ð ð ð ð ð ð ð ð ˆ3Œ=˜˜BÐ)Ð) &Ð)Ð)Ð)s   ÃA3F Ä6AF Æ
F#Æ"F#c                 ó&   — t          | |d|¦  «        S r¥   ©r;  ©r
   r   rG   s      r   Úscorergir?  µ  ó   € å�3˜˜1˜fÑ%Ô%Ð%r   c                 ó&   — t          | |d|¦  «        S r«   r=  r>  s      r   ÚscorerhirB  ¹  r@  r   c                 ó"  — ||f|v r*|||f         d         | j         k    r|||f         d         
 S |                      d|z  dz   ¦  «        }|                      d|z   | j        |z  z   ¦  «        }|                      d|z   | j        |z  z
  ¦  «        }d|z  |                      | j         |z  |z   |z   dz  |z
  ¦  «        z  }|                      |¦  «        s*|                      |¦  «        s|                      |¦  «        }| j         |f|||f<   |S )Nr   r   r   )r    Úloggammark   re   r'   rS   rC   )r
   ÚlÚetaÚ_cacheÚG3ÚG1ÚG2rJ   s           r   ÚcoulombcrK  ½  s  € à	ˆ3€x�6ÐÐ˜f Q s Uœm¨AÔ.°#´(Ò:Ð:Ø�q˜�u”˜aÔ Ð Ð Ø	�Š�a˜‘c˜!‘eÑ	Ô	€BØ	�Š�a˜‘c˜#œ% ™)‘mÑ	$Ô	$€BØ	�Š�a˜‘c˜#œ% ™)‘mÑ	$Ô	$€BØ	ˆ1‰ˆs�wŠw˜œ˜ ™ B™ rÑ)¨1Ñ,¨rÑ1Ñ2Ô2Ñ2€AØ�FŠF�1‰IŒIð ˜Ÿš ™œð Ø�FŠF�1‰IŒIˆØ”X˜q�M€Fˆ1ˆSˆ5�MØ€Hr   Tc                 ó  ‡ ‡‡— ˆ ˆˆfd„} ‰ j         |||gfi |¤Ž}|rm‰                      |¦  «        sX‰                      |¦  «        sC‰                      ‰¦  «        s.‰                      ‰¦  «        dk    r‰                      |¦  «        }|S )Nc                 óP  •— 	 ‰j         ‰z  }‰                     |‰	d¬¦  «        }‰                     |dd¬¦  «        }‰                     | |¦  «        }|‰	‰                     |¦  «        gd| dz   dgg g d| z   ||z  z   gd| z  dz   g|f}n# t          $ r dgdgg g g g df}Y nw xY w|fS )NTr"   rô   r   r   r   r   )rk   r&   rK  re   rO   )
rE  rF  ÚjwÚjwzÚjwz2rù   ra   r
   r4   r   s
          €€€r   r-   zcoulombf.<locals>.hÐ  sÞ   ø€ ð	.Ø”�q‘ˆBØ—(’(˜2˜q¨�(Ñ-Ô-ˆCØ—8’8˜C ¨4�8Ñ0Ô0ˆDØ—’˜Q Ñ$Ô$ˆAØ�Q˜Ÿš ™œÐ%¨¨1¨Q©3° {°B¸¸Q¸q¹SÀÀCÁ¹Z¸LØ�1‘�Q‘�˜ðˆBˆBøåð 	.ð 	.ð 	.Ø��r�d˜B  B¨¨AÐ-ˆBˆBˆBð	.øøøàˆuˆs   ƒBB
 Â
B"Â!B"r   )r>   rS   rC   ©	r
   rE  rF  r   r4   ÚchoprG   r-   rJ   s	   `  ``    r   ÚcoulombfrS  Ê  s«   øøø€ ð
ð 
ð 
ð 
ð 
ð 
ð 
ð 	ˆŒ�a˜!˜C˜Ð+Ð+ FÐ+Ð+€AØð �S—V’V˜A‘Y”Yð ¨¯ª°©¬ð ¸s¿vºvÀa¹y¼yð Ø	�Š�‰Œ�aŠˆØ�FŠF�1‰IŒIˆØ€Hr   c                 óÀ   ‡ ‡‡— ‰‰f|v r)|‰‰f         d         ‰ j         k    r|‰‰f         d         S ˆ ˆˆfd„}‰                      |d¦  «        }‰ j         |f|‰‰f<   |S )Nr   r   c                  ó*  •— ‰ dz
  } ‰j         ‰z  }‰                     d‰z   |z   ¦  «        dz  ‰                     d‰z   |z
  ¦  «        dz  ‰                     d| z   |z   ¦  «        dz  ‰                     d| z   |z
  ¦  «        dz  ‰dz    ‰j        z  gS )Nr   y       €      à¿y              à?r$   )rk   rD  r'   )Úl2Újetar
   rF  rE  s     €€€r   Útermsz_coulomb_chi.<locals>.termså  s¢   ø€ ØˆR�‰TˆØŒu�S‰yˆØ—’˜Q˜q™S ™XÑ&Ô&¨%Ñ0Ø�LŠL˜˜1™˜T™Ñ"Ô" dÑ+Ø�LŠL˜˜2™˜d™Ñ#Ô# tÑ,Ø�LŠL˜˜2™˜d™Ñ#Ô# uÑ-Ø�‰eˆH�S”V‰Oð	ð 	r   )r    Úsum_accurately)r
   rE  rF  rG  rX  rJ   s   ```   r   Ú_coulomb_chirZ  á  s’   øøø€ à	ˆ3€x�6ÐÐ˜f Q s Uœm¨AÔ.°#´(Ò:Ð:Ø�a˜�eŒ}˜QÔÐðð ð ð ð ð ð ð 	×Ò˜5 !Ñ$Ô$€AØ”X˜q�M€Fˆ1ˆSˆ5�MØ€Hr   c                 ón  ‡ ‡‡— ‰                       |¦  «        s‰                      |¦  «        }ˆ ˆˆfd„} ‰ j        |||gfi |¤Ž}|rm‰                       |¦  «        sX‰                       |¦  «        sC‰                       ‰¦  «        s.‰                      ‰¦  «        dk    r‰                      |¦  «        }|S )Nc                 óh  •— ‰                      | dz  ¦  «        rdgdgg g g g df}|fS |  dz
  }	 ‰                     | |¦  «        }‰j        ‰z  }‰                     |¦  «        }‰                     |¦  «        }‰                     | |¦  «        }‰                     ||¦  «        }	‰                     |‰z  ¦  «        }
d|z  ‰z  }||‰|
|gdd| dz   ddgg g d| z   ||z  z   gd| z  dz   g|f}| |	‰|
gdd|dz   dgg g d|z   ||z  z   gd|z  dz   g|f}||fS # t          $ r dgdgg g g g df}|fcY S w xY w)Nr   r   r   r   rô   )r8   rZ  rk   r\   r[   rK  re   rO   )rE  rF  ra   rV  ÚchirN  rŸ   rž   rú   rû   r™   r   rb   r
   r4   r   s                €€€r   r-   zcoulombg.<locals>.hø  s¡  ø€ à�9Š9�Q�q‘S‰>Œ>ð 	Ø��r�d˜B  B¨¨AÐ-ˆBØ�5ˆLØˆR�‰Tˆð	Ø×"Ò" 1 cÑ*Ô*ˆCØ”�q‘ˆBØ—’˜‘”ˆA #§'¢'¨#¡,¤,˜aØ—’˜a Ñ$Ô$ˆBØ—’˜b Ñ%Ô%ˆBØ—’˜˜1™‘”ˆAØ�2‘�a‘ˆAØ�R˜˜A˜qÐ! B¨¨1¨Q©3°°1Ð#5°r¸2Ø�1‘�R˜‘V‘�˜q ™s 1™u˜g qð)ˆBà�"�b˜!˜Q� B¨¨2¨a©4°Ð#3¸¸BØ�2‘�b˜‘f‘�  "¡ Q¡˜x¨ð+ˆBà�r�6ˆMøÝð 	ð 	ð 	Ø��r�d˜B  B¨¨AÐ-ˆBØ�5ˆLˆLˆLð	øøøs   ¯C&D ÄD1Ä0D1r   )Ú_imr9   r>   rQ  s	   `  ``    r   Úcoulombgr_  ñ  sÎ   øøø€ ð
 �7Š7�1‰:Œ:ð Ø�GŠG�A‰JŒJˆðð ð ð ð ð ð ð, 	ˆŒ�a˜!˜C˜Ð+Ð+ FÐ+Ð+€AØð �S—W’W˜Q‘Z”Zð ¨#¯'ª'°#©,¬,ð ÀÇÂÈÁÄð Ø	�Š�‰Œ�qŠˆØ�GŠG�A‰JŒJˆØ€Hr   c                 óˆ  — d|dz  z  }|dk    r|sd|z  d|z  z   dz
  | j         z  dz  }|dk    r|sd|z  d|z  z   dz
  | j         z  dz  }|dk    r|rd|z  d|z  z   dz
  | j         z  dz  }|dk    r|rd|z  d|z  z   dz
  | j         z  dz  }|s€|}|dz
   d|z  z  }d|dz
  z  d|z  dz
  z  dd|z  dz  z  z  }	d	|dz
  z  d
|dz  z  d|z  z
  dz   z  dd|z  dz  z  z  }
d|dz
  z  d|dz  z  d|dz  z  z
  d|z  z   dz
  z  dd|z  dz  z  z  }|r‰|}|dz    d|z  z  }dd|dz  z  d|z  z   dz
  z  dd|z  dz  z  z  }	d	d
|dz  z  d|dz  z  z   d|z  z
  dz   z  dd|z  dz  z  z  }
dd|dz  z  d|dz  z  z   d|dz  z  z
  d|z  z   dz
  z  dd|z  dz  z  z  }|||	|
|g}|}d}t          dt          |¦  «        ¦  «        D ]R}t          ||         ¦  «        t          ||dz
           ¦  «        k     r|||         z  }Œ=t          ||         ¦  «        }ŒS|t          |¦  «        dz
  k    rt          |d         ¦  «        }||fS ) aj  
    Computes an estimate for the location of the Bessel function zero
    j_{v,m}, y_{v,m}, j'_{v,m} or y'_{v,m} using McMahon's asymptotic
    expansion (Abramowitz & Stegun 9.5.12-13, DLMF 20.21(vi)).

    Returns (r,err) where r is the estimated location of the root
    and err is a positive number estimating the error of the
    asymptotic expansion.
    r¸   r   r   r0   r#  éüÿÿÿrö   é   iàÿÿÿéS   iÖ  iÃ  r   r  iÀÿÿÿi%  iÿX iO2 iuÈ_ éi   éR   rÛ   i  iß  iÑ  i,† i il"q i»QY g        r   )r'   r<   Úlenr?   )r
   ÚkindÚprimerJ   rZ   r™   r~   Ús1Ús2Ús3Ús4Ús5rX  rŸ   ÚerrÚis                   r   Úmcmahonrp    s  € ð 	
ˆ!ˆQ‰$‰€AØˆq‚y€y˜€y Q q¡S¨¨1©¡W¨Q¡Y°´Ñ$6°qÑ$8 Øˆq‚y€y˜€y Q q¡S¨¨1©¡W¨Q¡Y°´Ñ$6°qÑ$8 Øˆq‚y€y�U€y  1¡ Q q¡S¡¨¡¨C¬FÑ 2°1Ñ 4˜AØˆq‚y€y�U€y  1¡ Q q¡S¡¨¡¨C¬FÑ 2°1Ñ 4˜AØð PØˆØ�‰sˆV�Q�q‘S‰\ˆØ��1‘‰X�q˜‘s˜2‘vÑ  1 Q¡3¨¡(¡
Ñ+ˆØ�!�A‘#‰Y˜˜1˜a™4™  A¡™ dÑ*Ñ+¨R°°1±°q±©[Ñ9ˆØ�!�A‘#‰Y˜˜Q ™T™	 &¨¨A©¡+Ñ-¨g°a©iÑ7¸Ñ?Ñ@À#ÀqÈÁsÈQÁhÁ,ÑOˆØð WØˆØ�‰sˆV�Q�q‘S‰\ˆØ��1�a‘4‘˜˜1™‘˜Q‘Ñ  A a¡C¨!¡8¡Ñ,ˆØ�"�Q˜‘T‘'˜$˜q !™t™)Ñ# D¨¡FÑ*¨4Ñ/Ñ0°"°a¸±c¸A±X±+Ñ>ˆØ�$�q˜!‘t‘)˜F 1 a¡4™KÑ'¨°°1±©Ñ4°W¸Q±YÑ>¸wÑFÑGÈÈaÐPQÉcÐTUÉXÉÑVˆØ��2�b˜Ð€EØ
€AØ
€CÝ�1•S˜‘Z”ZÑ Ô ð  ð  ˆÝˆu�QŒx‰=Œ=�3˜u Q q¡Sœz™?œ?Ò*Ð*Ø��q”‰MˆAˆAå�e˜A”h‘-”-ˆCˆCØ�C�‰JŒJ�q‰LÒÐÝ�%˜”)‰nŒnˆØˆcˆ6€Mr   c                 ó  ‡ ‡‡‡— |dk     rt          d¦  «        ‚|dz   }g Šg Š	 ‰                      |||¦  «        Šˆ ˆfd„‰D ¦   «         Šˆˆfd„t          |dz
  ¦  «        D ¦   «         }t          |¦  «        |k    r|S |dz  }Œ`)zî
    Given f known to have exactly n simple roots within [a,b],
    return a list of n intervals isolating the roots
    and having opposite signs at the endpoints.

    TODO: this can be optimized, e.g. by reusing evaluation points.
    r   zn cannot be less than 1c                 óL   •— g | ] }‰                       ‰|¦  «        ¦  «        ‘Œ!S r`   )Úsign)r   r   r
   rÕ   s     €€r   ú
<listcomp>z)generalized_bisection.<locals>.<listcomp>J  s+   ø€ Ð0Ð0Ð0 A�—’˜!˜!˜A™$œ$‘”Ð0Ð0Ð0r   c                 óf   •— g | ]-}‰|         ‰|d z            z  dk    ¯‰|         ‰|d z            f‘Œ.S )r   r   r`   )r   ro  ÚpointsÚsignss     €€r   rt  z)generalized_bisection.<locals>.<listcomp>K  sO   ø€ ð *ð *ð *°AØ�QŒx˜˜a ™cœ
Ñ" bÒ(Ð(ð   œ 6¨!¨A©#¤;Ð/Ø(Ð(Ð(r   r   )rO   Úlinspacer<   rf  )	r
   rÕ   r}   r~   r   ÚNÚok_intervalsrv  rw  s	   ``     @@r   Úgeneralized_bisectionr{  ;  sÆ   øøøø€ ð 	ˆ1‚u€uÝÐ2Ñ3Ô3Ð3Ø	ˆ!‰€AØ€FØ€EðØ—’˜a  !Ñ$Ô$ˆØ0Ð0Ð0Ð0Ð0¨Ð0Ñ0Ô0ˆð*ð *ð *ð *ð *½¸qÀ¹s¹¼ð *ñ *ô *ˆåˆ|ÑÔ Ò!Ð!ØÐØˆa‰Cˆðr   c                 ó4   — |                       ||dd¬¦  «        S )NÚillinoisF)ÚsolverÚverify)r'  )r
   rÕ   Úabs      r   Úfind_in_intervalr�  Q  s   € Ø�<Š<˜˜2 j¸ˆ<Ñ?Ô?Ð?r   g{®Gáz„?c           	      ó  ‡ ‡— ‰ j         }t          |‰                      ‰¦  «        ‰                      |¦  «        ¦  «        dz   }	 |‰ _         ‰                      ‰¦  «        Št	          |¦  «        }t	          |¦  «        }‰dk     rt          d¦  «        ‚|dk     rt          d¦  «        ‚|dvrt          d¦  «        ‚|dk    r|rˆ ˆfd„}	nˆ ˆfd	„}	|d
k    r|rˆ ˆfd„}	nˆ ˆfd„}	|dk    rf|rd|dk    r^‰dk    r‰ j        |‰ _         S ‰dk    rDd
‰                      ‰d‰z   z  ‰d
z   z  ¦  «        z  }
t          ‰ |	|
dz  d
|
z  f¦  «        |‰ _         S ||‰|f|v r"t          ‰ |	|||‰|f         ¦  «        |‰ _         S t          ‰ ||‰|¦  «        \  }
}||k     r t          ‰ |	|
|z
  |
|z   f¦  «        |‰ _         S |dk    r|sd}|dk    r|rd}|d
k    r|sd}|d
k    r|rd}|dz   }	 t          ‰ ||‰|¦  «        \  }}||k     rtt          ‰ ||‰|dz   ¦  «        \  }}t          ‰ |	|d||z   z  |¦  «        }t          |¦  «        D ]\  }}||||‰|dz   f<   Œt          ‰ |	||dz
           ¦  «        |‰ _         S |d
z  }Œ–# |‰ _         w xY w)Nr.   r   zv cannot be negativer   zm cannot be less than 1r  z prime should lie between 0 and 1c                 ó4   •— ‰                      ‰| d¬¦  «        S ©Nr   )rF   r   ©r   r
   rJ   s    €€r   r!  zbessel_zero.<locals>.<lambda>c  ó   ø€  C§K¢K°°!¸q KÑ$AÔ$A€ r   c                 ó0   •— ‰                      ‰| ¦  «        S ri   r   r…  s    €€r   r!  zbessel_zero.<locals>.<lambda>d  ó   ø€  C§K¢K°°!Ñ$4Ô$4€ r   r   c                 ó4   •— ‰                      ‰| d¬¦  «        S r„  ©r]   r…  s    €€r   r!  zbessel_zero.<locals>.<lambda>f  r†  r   c                 ó0   •— ‰                      ‰| ¦  «        S ri   rŠ  r…  s    €€r   r!  zbessel_zero.<locals>.<lambda>g  rˆ  r   g333333@gÍÌÌÌÌÌü?gš™™™™™é?g       @r$   )r    r3   r:   r=   r6   rO   ÚzerorŒ   r�  rp  r{  Ú	enumerate)r
   rg  rh  rJ   rZ   ÚisoltolÚ_interval_cacher    ÚworkprecrÕ   r)   rn  Úlowr   Úr1Úr2Úerr2Ú	intervalsr   r€  s   `  `                r   Úbessel_zeror–  T  sP  øø€ ØŒ8€DÝ�4˜Ÿš ™œ S§W¢W¨Q¡Z¤ZÑ0Ô0°Ñ3€Hð0ØˆŒØ�GŠG�A‰JŒJˆÝ�‰FŒFˆÝ�E‘
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    For a real order `\nu \ge 0` and a positive integer `m`, returns
    `j_{\nu,m}`, the `m`-th positive zero of the Bessel function of the
    first kind `J_{\nu}(z)` (see :func:`~mpmath.besselj`). Alternatively,
    with *derivative=1*, gives the first nonnegative simple zero
    `j'_{\nu,m}` of `J'_{\nu}(z)`.

    The indexing convention is that used by Abramowitz & Stegun
    and the DLMF. Note the special case `j'_{0,1} = 0`, while all other
    zeros are positive. In effect, only simple zeros are counted
    (all zeros of Bessel functions are simple except possibly `z = 0`)
    and `j_{\nu,m}` becomes a monotonic function of both `\nu`
    and `m`.

    The zeros are interlaced according to the inequalities

    .. math ::

        j'_{\nu,k} < j_{\nu,k} < j'_{\nu,k+1}

        j_{\nu,1} < j_{\nu+1,2} < j_{\nu,2} < j_{\nu+1,2} < j_{\nu,3} < \cdots

    **Examples**

    Initial zeros of the Bessel functions `J_0(z), J_1(z), J_2(z)`::

        >>> from mpmath import *
        >>> mp.dps = 25; mp.pretty = True
        >>> besseljzero(0,1); besseljzero(0,2); besseljzero(0,3)
        2.404825557695772768621632
        5.520078110286310649596604
        8.653727912911012216954199
        >>> besseljzero(1,1); besseljzero(1,2); besseljzero(1,3)
        3.831705970207512315614436
        7.01558666981561875353705
        10.17346813506272207718571
        >>> besseljzero(2,1); besseljzero(2,2); besseljzero(2,3)
        5.135622301840682556301402
        8.417244140399864857783614
        11.61984117214905942709415

    Initial zeros of `J'_0(z), J'_1(z), J'_2(z)`::

        0.0
        3.831705970207512315614436
        7.01558666981561875353705
        >>> besseljzero(1,1,1); besseljzero(1,2,1); besseljzero(1,3,1)
        1.84118378134065930264363
        5.331442773525032636884016
        8.536316366346285834358961
        >>> besseljzero(2,1,1); besseljzero(2,2,1); besseljzero(2,3,1)
        3.054236928227140322755932
        6.706133194158459146634394
        9.969467823087595793179143

    Zeros with large index::

        >>> besseljzero(0,100); besseljzero(0,1000); besseljzero(0,10000)
        313.3742660775278447196902
        3140.807295225078628895545
        31415.14114171350798533666
        >>> besseljzero(5,100); besseljzero(5,1000); besseljzero(5,10000)
        321.1893195676003157339222
        3148.657306813047523500494
        31422.9947255486291798943
        >>> besseljzero(0,100,1); besseljzero(0,1000,1); besseljzero(0,10000,1)
        311.8018681873704508125112
        3139.236339643802482833973
        31413.57032947022399485808

    Zeros of functions with large order::

        >>> besseljzero(50,1)
        57.11689916011917411936228
        >>> besseljzero(50,2)
        62.80769876483536093435393
        >>> besseljzero(50,100)
        388.6936600656058834640981
        >>> besseljzero(50,1,1)
        52.99764038731665010944037
        >>> besseljzero(50,2,1)
        60.02631933279942589882363
        >>> besseljzero(50,100,1)
        387.1083151608726181086283

    Zeros of functions with fractional order::

        >>> besseljzero(0.5,1); besseljzero(1.5,1); besseljzero(2.25,4)
        3.141592653589793238462643
        4.493409457909064175307881
        15.15657692957458622921634

    Both `J_{\nu}(z)` and `J'_{\nu}(z)` can be expressed as infinite
    products over their zeros::

        >>> v,z = 2, mpf(1)
        >>> (z/2)**v/gamma(v+1) * \
        ...     nprod(lambda k: 1-(z/besseljzero(v,k))**2, [1,inf])
        ...
        0.1149034849319004804696469
        >>> besselj(v,z)
        0.1149034849319004804696469
        >>> (z/2)**(v-1)/2/gamma(v) * \
        ...     nprod(lambda k: 1-(z/besseljzero(v,k,1))**2, [1,inf])
        ...
        0.2102436158811325550203884
        >>> besselj(v,z,1)
        0.2102436158811325550203884

    r   ©r–  ©r
   rJ   rZ   rF   s       r   Úbesseljzerorš  ‰  s   € õ` ˜˜Q 
¨A¨qÑ1Ô1Ð1Ð1r   c                 ó*   — t          | d|||¦  «        
 S )aÆ  
    For a real order `\nu \ge 0` and a positive integer `m`, returns
    `y_{\nu,m}`, the `m`-th positive zero of the Bessel function of the
    second kind `Y_{\nu}(z)` (see :func:`~mpmath.bessely`). Alternatively,
    with *derivative=1*, gives the first positive zero `y'_{\nu,m}` of
    `Y'_{\nu}(z)`.

    The zeros are interlaced according to the inequalities

    .. math ::

        y_{\nu,k} < y'_{\nu,k} < y_{\nu,k+1}

        y_{\nu,1} < y_{\nu+1,2} < y_{\nu,2} < y_{\nu+1,2} < y_{\nu,3} < \cdots

    **Examples**

    Initial zeros of the Bessel functions `Y_0(z), Y_1(z), Y_2(z)`::

        >>> from mpmath import *
        >>> mp.dps = 25; mp.pretty = True
        >>> besselyzero(0,1); besselyzero(0,2); besselyzero(0,3)
        0.8935769662791675215848871
        3.957678419314857868375677
        7.086051060301772697623625
        >>> besselyzero(1,1); besselyzero(1,2); besselyzero(1,3)
        2.197141326031017035149034
        5.429681040794135132772005
        8.596005868331168926429606
        >>> besselyzero(2,1); besselyzero(2,2); besselyzero(2,3)
        3.384241767149593472701426
        6.793807513268267538291167
        10.02347797936003797850539

    Initial zeros of `Y'_0(z), Y'_1(z), Y'_2(z)`::

        >>> besselyzero(0,1,1); besselyzero(0,2,1); besselyzero(0,3,1)
        2.197141326031017035149034
        5.429681040794135132772005
        8.596005868331168926429606
        >>> besselyzero(1,1,1); besselyzero(1,2,1); besselyzero(1,3,1)
        3.683022856585177699898967
        6.941499953654175655751944
        10.12340465543661307978775
        >>> besselyzero(2,1,1); besselyzero(2,2,1); besselyzero(2,3,1)
        5.002582931446063945200176
        8.350724701413079526349714
        11.57419546521764654624265

    Zeros with large index::

        >>> besselyzero(0,100); besselyzero(0,1000); besselyzero(0,10000)
        311.8034717601871549333419
        3139.236498918198006794026
        31413.57034538691205229188
        >>> besselyzero(5,100); besselyzero(5,1000); besselyzero(5,10000)
        319.6183338562782156235062
        3147.086508524556404473186
        31421.42392920214673402828
        >>> besselyzero(0,100,1); besselyzero(0,1000,1); besselyzero(0,10000,1)
        313.3726705426359345050449
        3140.807136030340213610065
        31415.14112579761578220175

    Zeros of functions with large order::

        >>> besselyzero(50,1)
        53.50285882040036394680237
        >>> besselyzero(50,2)
        60.11244442774058114686022
        >>> besselyzero(50,100)
        387.1096509824943957706835
        >>> besselyzero(50,1,1)
        56.96290427516751320063605
        >>> besselyzero(50,2,1)
        62.74888166945933944036623
        >>> besselyzero(50,100,1)
        388.6923300548309258355475

    Zeros of functions with fractional order::

        >>> besselyzero(0.5,1); besselyzero(1.5,1); besselyzero(2.25,4)
        1.570796326794896619231322
        2.798386045783887136720249
        13.56721208770735123376018

    r   r˜  r™  s       r   Úbesselyzerorœ  û  s   € õr ˜˜Q 
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