ó
    ˆ*£h2”  ã                   óä  • S SK JrJr  \S 5       r\S 5       r\S3S j5       r\S3S j5       r\S3S j5       r\S 5       r\S 5       r	\S	 5       r
\S
 5       r\S 5       r\S 5       r\S 5       r\S 5       rS r\S 5       r\S 5       r\S 5       r\S 5       r\S 5       r\S 5       r\S 5       r\S 5       rS r\S 5       r\S 5       r\S 5       r\S 5       rS rS r\S3S j5       r \S3S  j5       r!S4S! jr"\S3S" j5       r#\S5S# j5       r$S$ r%\S% 5       r&\S& 5       r'\0 4S' j5       r(\S6S( j5       r)\0 4S) j5       r*\S6S* j5       r+S+ r,S, r-S- r.S.0 4S/ jr/\S3S0 j5       r0\S3S1 j5       r1g2)7é   )ÚdefunÚdefun_wrappedc                 ó&   • U R                  SU5      $ )zCComputes the Bessel function `J_0(x)`. See :func:`~mpmath.besselj`.é    ©Úbesselj©ÚctxÚxs     ÚT/home/mande/repo/quber/.venv/lib/python3.13/site-packages/mpmath/functions/bessel.pyÚj0r      ó   € ð �;‰;�q˜!ÓÐó    c                 ó&   • U R                  SU5      $ )zDComputes the Bessel function `J_1(x)`.  See :func:`~mpmath.besselj`.r   r   r	   s     r   Új1r      r   r   c                 óR  ^ ^^^	^
^• [        T5      [        L a  SnOCT R                  T5      mT R                  T5      nU(       a  [        T R	                  T5      5      mU(       a"  TS:  a  ST-  T R
                  " T* TU40 UD6-  $ T R                  T5      mT R                  T5      m	U(       aÆ  T R                  U5      m
T R                  T
5      (       a~  T
S:¼  ax  [        T
5      m
T R                  n T =R                  S-  sl        T R                  U U
UU4S j[        T
S-   5       5       5      nUT l        UT R                  S5      T
* -  -  nU$ U	U U4S jnT R                  " UTT
/40 UD6n U$ U(       d8  U(       a1  [        T	5      S	:  a"  [        T5      S
:  a   T R                  TT5      $ T(       dH  T(       d  T R                  T-   T-   nO¥T R!                  T5      S:”  a  TT-  nOŠT R"                  T-   T-   nOwT R                  n T =R                  [%        S[        T	5      -  T R                  5      -  sl        T R'                  TSSS9mU	U U4S jnT R                  " UT/40 UD6nUT l        U7nU$ ! UT l        f = f! [         a     Nãf = f! UT l        f = f)NTr   éÿÿÿÿé   c              3   óŒ   >#   • U  H9  nS U-  TR                  TU5      -  TR                  SU-  T-   T-
  T5      -  v •  M;     g7f)r   é   N)Úbinomialr   )Ú.0Úkr
   ÚdÚnÚzs     €€€€r   Ú	<genexpr>Úbesselj.<locals>.<genexpr>%   sH   øé € ð )Ú'˜ð ! 1™W s§|¡|°A°aÓ'8Ñ8¸3¿;¹;ÀqÈÁsÈ1ÁuÈQÁwÈqÓ;QÖQÚ'ùs   ƒAAr   r   c                 óô   >• TR                  TR                  TTTR                  T-   S9SSS9nSX-
  S-   -  SX-
  S-   -  /nSTR                  T/USU -  -
  SX-
  // X0S-   S-  U S-   S-  /X0S-   /-   U4/nU$ )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hÚbesselj.<locals>.h+   s¡   ø€ Ø—H‘H˜SŸX™X a¨°·±¸!±˜XÐ<¸eÈ4�HÐP�Ø˜!™#˜a™%‘[ # q¡s¨1¡u¡+Ð.�Ø˜Ÿ™ �l A a¨¡c¡E¨#¨a©c ?°2°a¸A¹#¸s¹ÀAÀaÁCÈÁ9Ð8MÈaÐSTÑQTÐPUÉgÐVWÐXÐY�Ø�r   é
   é   é   r%   r#   c                 ó˜   >• TR                  TR                  TT[        STR                  T-   5      S9SS9nT/U // U S-   // U S-   /U4/$ )Nr   r    Tr#   r   )Úfnegr'   Úmaxr!   )r   r*   r-   r
   Úws     €€€r   r.   r/   G   s[   ø€ ØŸ™ §¡¨!¨Q´S¸¸3¿8¹8ÀA¹:Ó5F Ð!GÈt˜ÐT�AØ˜S 1 # r¨A¨a©C¨5°"°q¸±s°e¸QÐ?Ð@Ð@r   )ÚtypeÚintÚconvertÚisintÚ_rer   Úmagr!   ÚfsumÚrangeÚmpfÚ	hypercombÚabsÚ_besseljÚNotImplementedErrorÚoneÚreÚinfÚminr'   )r
   r   r   Ú
derivativeÚkwargsÚn_isintÚorigÚvr.   r-   r   r6   s   ```      @@@r   r   r      s[  ý€ äˆAƒw”#‚~Ø‰à�K‰K˜‹NˆØ—)‘)˜A“,ˆÞÜ�C—G‘G˜A“J“ˆAÞ�1�q“5Ø�Q‰w˜Ÿš a R¨¨JÑA¸&ÑAÑAÐAØ�‰�A‹€AØ�‰�‹
€AÞØ�K‰K˜
Ó#ˆð
 �9‰9�Q�<‰<˜A ›FÜ�A“ˆAØ—8‘8ˆDð Ø—’˜B‘•Ø—H‘H÷ )Ü" 1 Q¡3œZó)ó )�ð  �”Ø�—‘˜“˜q˜bÑ!Ñ!ˆAðJ €H÷Gð
 —’˜a ! A Ñ1¨&Ñ1‰Að< €Hö7 ¦¬C°«F°R«K¼CÀ»FÀR»KðØ—|‘| A qÓ)Ð)ö ÞØ—G‘G˜a‘K ‘M‘Ø—‘˜“˜Q“Ø�a‘C‘à—G‘G˜a‘K !‘O‘ð —8‘8ˆDð
 ð —’œC ¤# a£&¡¨#¯(©(Ó3Ñ3•Ø—H‘H˜Q ¨4�HÐ0�÷Að —M’M ! a SÑ3¨FÑ3�à�”ØˆBˆØ€HøðM  �•ûô 'ó Ùðûð,  �•ús+   Ã6?J Æ$J ÈA#J Ê	J
Ê
JÊJÊ	J&c                 ó,  ^ ^^	• T R                  U5      nT R                  T5      mT(       dƒ  U(       a  [        eU(       d  SU-   T-   $ T R                  U5      (       a  SUT-   -  $ T R                  U5      nUS:X  a  T R                  UT-   -  $ US:”  a  SUT-   -  $ T R
                  UT-   -   $ T R                  T5      m	U(       a0  T R                  U5      nU	U U4S jnT R                  " XqU/40 UD6nU$ U	U U4S jnT R                  " Xq/40 UD6nU$ )Nr   r   c                 óö   >• TR                  TR                  TTTR                  T-   S9SSS9nSX-
  S-   -  SX-
  S-   -  U S-   /nSTR                  T/USU -  -
  SX-
  /U S-   /X0S-   S-  U S-   S-  /X24/nU$ )Nr    ç      Ð?Tr#   r%   r   r   r&   r)   s        €€€r   r.   Úbesseli.<locals>.hf   sŸ   ø€ Ø—‘˜Ÿ™ ! Q¨S¯X©X°a©Z˜Ð8¸$Àd�ÐKˆAØ�a‘c˜!‘e‘˜c 1¡3 q¡5™k¨1¨Q©3Ð/ˆAØ�S—V‘V˜A�,  ! A¡#¡ c¨!©#˜°°!±¨u°Q¸1¹¸c¹	À1ÀQÁ3ÈÁ)Ð7LÈQÐQÐRˆAØˆHr   c           	      óœ   >• TR                  TSSS9nTR                  X[        STR                  T-   5      S9nU/U // U S-   // U S-   /U4/$ )Nr%   Tr#   r   r    r   )r'   r5   r!   )r   r6   r*   r-   r
   r   s      €€€r   r.   rP   m   s`   ø€ Ø—‘˜˜C t�Ð,ˆAØ—‘˜¤C¨¨#¯(©(°1©*Ó$5�Ð6ˆAØ�S˜1˜#˜r A a¡C 5¨"¨q°©s¨e°QÐ7Ð8Ð8r   )r9   Ú
ValueErrorr:   rE   ÚnanrF   r<   r@   )
r
   r   r   rH   rI   r*   r   r.   rL   r-   s
   ` `      @r   ÚbesselirT   P   s  ú€ à�‰�A‹€AØ�‰�A‹€AÞÞÜÐÞà�Q‘3�q‘5ˆLØ�9‰9�Q�<‰<Ø�a˜‘c‘7ˆNØ�F‰F�1‹IˆØ�‹6Ø—7‘7˜A˜a™C‘=Ð Ø�‹UØ�a˜‘c‘7ˆNà—7‘7˜A˜a™C‘=Ð Ø�‰�‹
€AÞØ�K‰K˜
Ó#ˆ÷	ð
 �MŠM˜! ˜UÑ- fÑ-ˆð €H÷	9ð �MŠM˜!˜SÑ+ FÑ+ˆØ€Hr   c                 óN  • U(       dâ  U(       a  [         eU(       d  U R                  * X-   -   $ U R                  U5      (       a  U R                  X-   -  $ U R	                  U5      nUS-   nU R                  U5      (       a  US:”  a  U R                  * X-   -   $ SX-   -  $ US:  a3  [        U R                  U5      5      S-  (       a  U R                  X-   -   $ U R                  X-   -   $ U =R                  S-  sl	        U R                  U5      u  pxX€R                  * :  a'  U R                  7n	U =R                  S-  sl	        X-  nOUS:  a  U =R                  U-  sl	        U R                  U5      u  p«U R                  " XU40 UD6U
-  U R                  " U* X#40 UD6-
  U-  $ )Nr%   r   r   r0   )rR   rF   ÚimrS   rE   r:   r8   ÚfloorÚninfr!   Únint_distanceÚepsÚcospi_sinpir   )r
   r   r   rH   rI   r*   ÚqÚmr   r.   ÚcosÚsins               r   Úbesselyr`   t   sr  € æÞäÐÞà—G‘G�8˜q™sÑ#Ð#Ø�6‰6�!�9‰9Ø—7‘7˜a™c‘?Ð"Ø�F‰F�1‹IˆØˆc‰EˆØ�9‰9�Q�<‰<Ø�1‹uØŸ™�x 1¡3Ñ'Ð'à˜A™C‘yÐ Øˆq‹5”S˜Ÿ™ 1›Ó&¨×*Ø—7‘7˜a™c‘?Ð"à—8‘8˜q™sÑ#Ð#à‡H‚H��N…HØ×Ñ˜QÓ�D€AØ�H‰Hˆ9ƒ}Ø�W‰WˆHˆØ�Š�A‰�Ø	‰‰Ø	
ˆQ‹Ø�Š�A‰�à�‰˜qÓ!�H€CØ�KŠK˜˜JÑ0¨Ñ0°Ñ4Ø�Š�Q�B�qÑ- fÑ-ñ.Ø/2ñ3ð 3r   c                 óÈ   ^ ^• T(       d  T R                   $ T R                  T5      nUS:  a  U4S jnOT =R                  U-  sl        U U4S jnT R                  " XQ/40 UD6$ )Nr   c                 ór   >• TS-  S-  nTS/U * U S-
  /U // / SU -
  /U4nTS/X * S-
  /U * // / SU -   /U4nX#4$ )Nr   r   © )r   r*   ÚT1ÚT2r   s       €r   r.   Úbesselk.<locals>.hŸ   sk   ø€ Ø�1‘�q‘ˆAØ�Q�˜1˜"˜a ™c˜ Q C¨¨R°!°A±#°¸Ð9ˆBØ�Q�˜!˜R ™T˜ a R D¨"¨b°1°Q±3°%¸Ð:ˆBØ�6ˆMr   c           	      óx   >• TR                   S-  TTR                  T* 5      // SQ/ / U S-   SU -
  // SST-  -  4/$ )Nr   )r%   ç      à¿r   r%   r   )r(   Úexp©r   r
   r   s    €€r   r.   rf   ª   sO   ø€ Ø—f‘f˜Q‘h  3§7¡7¨A¨2£;Ð/²¸rÀ2Ø�3‘˜˜A™�  B¨¨!©¡Hð.ð /ð /r   )rF   r<   r!   r@   )r
   r   r   rI   r-   r.   s   ` `   r   Úbesselkrk   ˜   sT   ù€ æØ�w‰wˆØ�‰�‹
€AØˆ1ƒuö	ð 	�Š�A‰�ö	/ð �=Š=˜˜CÑ* 6Ñ*Ð*r   c                 ój   • U R                   " X40 UD6U R                  U R                  " X40 UD6-  -   $ ©N©r   Újr`   ©r
   r   r   rI   s       r   Úhankel1rq   ¯   ó2   € à�;Š;�qÑ$˜VÑ$ s§u¡u¨S¯[ª[¸Ñ-F¸vÑ-FÑ'FÑFÐFr   c                 ój   • U R                   " X40 UD6U R                  U R                  " X40 UD6-  -
  $ rm   rn   rp   s       r   Úhankel2rt   ³   rr   r   c                 ó6  • US:X  aJ  U R                  U5      S:”  a  U$ U R                  U5      S:  a  U R                  U-   $ U R                  U-  $ U R                  SUSS9nSU-   nU R	                  U5      X6-  -  U R
                  " Xa-
  SSU-  -   U40 UD6-  $ )Nr   rh   Tr#   r%   r   r   )rE   rF   rS   r'   ri   Úhyp1f1)r
   r   r]   r   rI   r   Úys          r   Úwhitmrx   ·   s�   € àˆAƒvà�6‰6�!‹9�tÓØˆHØ�V‰V�A‹Y˜ÓØ—7‘7˜Q‘;Ðà—7‘7˜Q‘;ÐØ�‰��q ˆÐ%€AØˆA‰€AØ�7‰7�1‹:˜™Ñ˜sŸzšz¨!©#¨q°°1±©u°aÑB¸6ÑBÑBÐBr   c                 ó.  • US:X  aF  [        U R                  U5      5      nUS:  a  U$ US:”  a  U R                  U-   $ U R                  U-  $ U R	                  SUSS9nSU-   nU R                  U5      X7-  -  U R                  " Xq-
  SSU-  -   U40 UD6-  $ )Nr   r%   rh   Tr#   r   r   )rA   rE   rF   rS   r'   ri   Úhyperu)r
   r   r]   r   rI   Úgr   rw   s           r   Úwhitwr|   Å   sœ   € àˆAƒvÜ�—‘�q“	‹NˆØˆs‹7ØˆHØ�‹WØ—7‘7˜Q‘;Ðà—7‘7˜Q‘;ÐØ�‰��q ˆÐ%€AØˆA‰€AØ�7‰7�1‹:˜™Ñ˜sŸzšz¨!©#¨q°°1±©u°aÑB¸6ÑBÑBÐBr   c           	      ó^  ^ ^• T R                  U5      u  pT R                  U5      u  p&T R                  T5      mT(       d@  T R                  U5      S::  a  T R                  SU-
  /X-
  S-   /5      $ T R                  T-   $ SU-   U-
  nT R                  U5      u  px T R
                  n	 T =R
                  S-  sl        T R                  SSXX4X/ST-  T R
                  S9n
U
TU-  -  U	T l        $ ! U	T l        f = f! T R                   a     Of = fU U4S jnT R                  " X±U/40 UD6$ )Nr   r0   r   r   r   )Úmaxtermsc                 óÄ   >• TR                  U5      nTR                  U/SS// X-
  S-   U/U /U/T4nTR                  * UT/SSSU-
  // U SU-
  /X-
  S-   /SU-
  /T4nX44$ )Nr   r   r   )Úsinpir(   )ÚaÚbr6   rd   re   r
   r   s        €€r   r.   Úhyperu.<locals>.hé   s…   ø€ Ø�I‰I�a‹LˆØ�v‰v�aˆj˜!˜B˜  A¡C¨¡E¨! 9¨a¨S°!°°QÐ7ˆØ—‘ˆw�q˜ˆm˜Q˜r ! A¡#˜J r¨1¨Q¨q©S¨'°1±3°q±5°'¸1¸Q¹3¸%ÀÐBˆØˆvˆr   )	Ú_convert_paramr9   rE   Ú	gammaprodrF   r!   ÚhypsumÚNoConvergencer@   )r
   r�   r‚   r   rI   ÚatypeÚbtypeÚbbÚbbtyperK   rL   r.   s   `  `        r   rz   rz   Ó   s(  ù€ à×!Ñ! !Ó$�H€AØ×!Ñ! !Ó$�H€AØ�‰�A‹€AÞØ�6‰6�!‹9˜‹>Ø—=‘= ! A¡# ¨©¨A© wÓ/Ð/à—7‘7˜Q‘;ÐØ	
ˆ1‰ˆQ‰€BØ×#Ñ# BÓ'�J€Bð	Ø�x‰xˆð	Ø�HŠH˜‰N�HØ—
‘
˜1˜a % °1°'¸2¸a¹4È#Ï(É(�
ÐSˆAØ�q˜!‘t‘8àˆC�Hø�tˆC�HûØ×Ñó Ùðúöð
 �=Š=˜˜q˜EÑ, VÑ,Ð,s*   ÂC= Â*?C1 Ã)C= Ã1	C:Ã:C= Ã=DÄDc                 ó€   ^ ^• T R                  U5      nT R                  T5      mU U4S jnT R                  " XA/40 UD6$ )Nc                 ó†   >• TS-  STR                  TR                  5      -  /U S-   S// U S-   /S/SU S-   /TS-  S-  * 4/$ ©Nr   r%   r   r   ç      ø?©Úsqrtr(   rj   s    €€r   r.   Ústruveh.<locals>.hõ   sa   ø€ Ø�A‘#�s˜3Ÿ8™8 C§F¡FÓ+Ñ+Ð,¨q°©s°B¨i¸¸aÀ¹e¸WÀqÀcÈCÐQRÐSVÑQVÈ<Ð[\Ð]^Ñ[^ÐabÑZbÐYbÐcÐdÐdr   ©r9   r@   ©r
   r   r   rI   r.   s   ` `  r   Ústruvehr•   ð   s9   ù€ à�‰�A‹€AØ�‰�A‹€Aöeà�=Š=˜˜CÑ* 6Ñ*Ð*r   c                 ó€   ^ ^• T R                  U5      nT R                  T5      mU U4S jnT R                  " XA/40 UD6$ )Nc                 ó„   >• TS-  STR                  TR                  5      -  /U S-   S// U S-   /S/SU S-   /TS-  S-  4/$ rŽ   r�   rj   s    €€r   r.   Ústruvel.<locals>.hþ   s^   ø€ Ø�A‘#�s˜3Ÿ8™8 C§F¡FÓ+Ñ+Ð,¨q°©s°B¨i¸¸aÀ¹e¸WÀqÀcÈCÐQRÐSVÑQVÈ<ÐZ[Ð\]ÑZ]Ð`aÑYaÐbÐcÐcr   r“   r”   s   ` `  r   Ústruvelr™   ù   s9   ù€ à�‰�A‹€AØ�‰�A‹€Aödà�=Š=˜˜CÑ* 6Ñ*Ð*r   c                 óŠ   ^ ^^• T R                  U5      S   nT R                  T5      mU UU4S jnT R                  " XR/40 UD6$ )Nr   c                 ó  >• TR                   nX-  nUS-  nX2-
  X2-   SU-
  SU-   4u  pEpgTR                  U5      u  p‰TS:X  a	  UT-  U	/U/pºTS:X  a
  UT-  U* /U	/pºTR                  TSS9nW
SS// XE/S/XE/U4nWS// Xg/S/Xg/U4nXÞ4$ )Nr2   r   r   r"   ©Úmult)Úmpq_1_2r[   Úsquare_exp_arg)rL   r‚   Úur]   Úa1Úa2Úb1Úb2ÚcÚsÚAr+   r6   rd   re   r
   Úwhichr   s                  €€€r   r.   Ú_anger.<locals>.h  sÒ   ø€ Ø�K‰KˆØ‰CˆØˆa‰CˆØ‘c˜1™3  !¡ Q q¡SÐ(‰ˆˆbØ�‰˜qÓ!‰ˆØ�A‹:Ø�a‘C˜�8˜a˜SˆqØ�A‹:Ø�a‘C˜!˜�9˜q˜cˆqØ×Ñ˜q uÐÐ-ˆØ��A�˜˜R˜G a S¨2¨'°1Ð4ˆØ���R˜"˜ 1 #¨ w°Ð1ˆØˆvˆr   ©r„   r9   r@   )r
   r¨   rL   r   rI   r.   s   `` `  r   Ú_angerr«     sB   ú€ Ø×Ñ˜1Ó˜aÑ €AØ�‰�A‹€A÷ð �=Š=˜˜CÑ* 6Ñ*Ð*r   c                 ó   • [        U SX40 UD6$ ©Nr   ©r«   ©r
   rL   r   rI   s       r   Úangerjr°     ó   € ä�#�q˜!Ñ) &Ñ)Ð)r   c                 ó   • [        U SX40 UD6$ ©Nr   r®   r¯   s       r   Úweberer´     r±   r   c                 ó°   ^ ^• T R                  U5      S   nT R                  U5      S   nT R                  T5      mU U4S jnT R                  " XQU/40 UD6$ )Nr   c           	      óœ   >• TR                   nTR                  TSS9nX-
  S-   X-   S-   T/SSU S-   // / S/X U-
  S-   -  X U-   S-   -  /U44$ )Nr"   rœ   r   r   r2   ©rž   rŸ   )r    rL   r‚   r6   r
   r   s       €€r   r.   Úlommels1.<locals>.h"  sx   ø€ Ø�K‰KˆØ×Ñ˜q uÐÐ-ˆØ‘�Q‘˜™˜A™˜qÐ! B¨¨A¨a©C =°"°b¸1¸#Ø�!‘�A‘‰Y�q˜A™#˜a™%‘yÐ! 1ð&ð 'ð 	'r   rª   ©r
   r    rL   r   rI   r.   s   `  `  r   Úlommels1rº     sY   ù€ à×Ñ˜1Ó˜aÑ €AØ×Ñ˜1Ó˜aÑ €AØ�‰�A‹€Aö'ð
 �=Š=˜˜q˜EÑ, VÑ,Ð,r   c                 ó°   ^ ^• T R                  U5      S   nT R                  U5      S   nT R                  T5      mU U4S jnT R                  " XQU/40 UD6$ )Nr   c           	      óJ  >• TR                   nTR                  TSS9nX-
  S-   X-   S-   T/SSU S-   // / S/X U-
  S-   -  X U-   S-   -  /U4nST/X-   S-
  U* /XX-   S-   -  /X!U -
  S-   -  // SU-
  /U4nST/X-
  S-
  U/U* X U-
  S-   -  /USU -
  U-
  -  // SU-   /U4nXEU4$ )Nr"   rœ   r   r   r2   r   r·   )	r    rL   r‚   r6   rd   re   ÚT3r
   r   s	          €€r   r.   Úlommels2.<locals>.h4  s  ø€ Ø�K‰KˆØ×Ñ˜q uÐÐ-ˆØ‰c�!‰e�Q‘S˜‘U˜AÐ  R¨¨1© ¨r°2¸°s¸QÀ!ÁÀAÁ¹YÀqÈAÉ#ÈaÉ%ÁyÐ<QÐSTÐTˆØ�ˆV�a‘c˜!‘e˜a˜R�[ 1¨©¨Q©¡i .°1¸±c¸!±e±9°+¸rÀAÀaÁCÀ5È!ÐKˆØ�ˆV�a‘c˜!‘e˜Q�Z 1 " a¨1©¨Q©¡i °1°a¸±c¸!±e±9°+¸rÀAÀaÁCÀ5È!ÐKˆð �rˆzÐr   rª   r¹   s   `  `  r   Úlommels2r¿   )  sY   ù€ à×Ñ˜1Ó˜aÑ €AØ×Ñ˜1Ó˜aÑ €AØ�‰�A‹€Aöð  �=Š=˜˜q˜EÑ, VÑ,Ð,r   c                 ó€   ^ ^• T R                  U5      nT R                  T5      mU U4S jnT R                  " XA/40 UD6$ )Nc           	      óÚ   >• TS-  S-  * nTR                  SU -  5      u  p#UTS-  /SU // U S-   // SSU S-   -  SU -  S-   /U4nUTS-  /SU S-   // U S-   // SSU S-   -  SU -  S-   /U4nXE4$ )Né   ç      è¿r   r   r%   r�   r2   ©r[   ©r   r*   r^   r_   rd   re   r
   r   s         €€r   r.   Úber.<locals>.hK  s«   ø€ Ø�‰c�A‰XˆIˆØ—?‘? 5¨¡7Ó+‰ˆØ�1�Q‘3ˆZ˜!˜Q˜  a¨¡c U¨B°°c¸1¸Q¹3±iÀÀQÁÀqÁÐ0IÈ1ÐLˆØ�1�Q‘3ˆZ˜!˜Q˜q™S˜ 2¨¨!© u¨b°3¸¸Q¸q¹S¹	À3ÀqÁ5ÈÁ7Ð2KÈQÐNˆØˆvˆr   r“   r”   s   ` `  r   ÚberrÇ   F  ó:   ù€ à�‰�A‹€AØ�‰�A‹€Aöð �=Š=˜˜CÑ* 6Ñ*Ð*r   c                 ó€   ^ ^• T R                  U5      nT R                  T5      mU U4S jnT R                  " XA/40 UD6$ )Nc           	      óÚ   >• TS-  S-  * nTR                  SU -  5      u  p#UTS-  /SU S-   // U S-   // SSU S-   -  SU -  S-   /U4nUTS-  /SU // U S-   // SSU S-   -  SU -  S-   /U4nXE4$ )NrÂ   ç      è?r   r   r�   r%   r2   rÄ   rÅ   s         €€r   r.   Úbei.<locals>.hX  s«   ø€ Ø�‰c�A‰XˆIˆØ—?‘? 4¨¡6Ó*‰ˆØ�1�Q‘3ˆZ˜!˜Q˜q™S˜ 2¨¨!© u¨b°3¸¸Q¸q¹S¹	À3ÀqÁ5ÈÁ7Ð2KÈQÐNˆØ�1�Q‘3ˆZ˜!˜Q˜  a¨¡c U¨B°°c¸1¸Q¹3±iÀÀQÁÀqÁÐ0IÈ1ÐLˆØˆvˆr   r“   r”   s   ` `  r   ÚbeirÍ   S  rÈ   r   c                 ó€   ^ ^• T R                  U5      nT R                  T5      mU U4S jnT R                  " XA/40 UD6$ )Nc           
      ó¸  >• TS-  S-  * nT
R                  SU -  5      u  p#T
R                  SU -  5      u  pESTSU-  /U * S-
  U S/U * // / SSSU -   -  SU S-   -  /U4nSTU* /U * S-
  SU -   S/U * S-
  // / SSSU -   -  SU S-   -  /U4nSTSU-  /U S-
  U * S/U // / SSSU -
  -  SSU -  -
  /U4nSTU* /U S-
  SU -
  S/U S-
  // / SSSU -
  -  SSU -  -
  /U4n	XgX‰4$ )	NrÂ   rO   rË   r   r2   r   r%   r�   rÄ   ©r   r*   Úcos1Úsin1Úcos2Úsin2rd   re   r½   ÚT4r
   r   s             €€r   r.   Úker.<locals>.he  sg  ø€ Ø�‰c�A‰XˆIˆØ—_‘_ T¨!¡VÓ,‰
ˆØ—_‘_ T¨!¡VÓ,‰
ˆØ��A�d‘Fˆ^˜q˜b ™d A q˜\¨Q¨B¨4°°R¸#¸sÀAÀaÁC¹yÈ#ÈqÐQRÉsÉ)Ð9TÐVWÐWˆØ��T�Eˆ]˜a˜R ™T 1 Q¡3¨˜N¨a¨R°©T¨F°B¸¸SÀ#ÀqÈÁsÁ)ÈSÐRSÐTUÑRUÉYÐ<WÐYZÐZˆØ��A�d‘Fˆ^˜a ™c A 2 q˜\¨A¨3°°B¸¸cÀ1ÀQÁ3¹iÈÈ3ÈqÉ5ÉÐ8QÐSTÐTˆØ��T�Eˆ]˜Q˜q™S ! A¡# q˜M¨A¨a©C¨5°"°b¸3ÀÀQÀqÁSÁ	È1ÈSÐQRÉUÉ7Ð:SÐUVÐVˆØ�rˆ~Ðr   r“   r”   s   ` `  r   Úkerr×   `  ó:   ù€ à�‰�A‹€AØ�‰�A‹€Aöð �=Š=˜˜CÑ* 6Ñ*Ð*r   c                 ó€   ^ ^• T R                  U5      nT R                  T5      mU U4S jnT R                  " XA/40 UD6$ )Nc           
      ó°  >• TS-  S-  * nT
R                  SU -  5      u  p#T
R                  SU -  5      u  pEU* ST/SU S-
  SU -
  /U S-
  // / SSSU -
  -  SSU -  -
  /U4nU* ST/SU S-
  U * /U // / SSSU -
  -  SSU -  -
  /U4nU* ST/SU * S-
  U /U * // / SSU S-   -  SU S-   -  /U4nU* ST/SU * S-
  U S-   /U * S-
  // / SSU S-   -  SU S-   -  /U4n	XgX‰4$ )	NrÂ   rË   rO   r   r   r2   r�   r%   rÄ   rÐ   s             €€r   r.   Úkei.<locals>.hu  sc  ø€ Ø�‰c�A‰XˆIˆØ—_‘_ T¨!¡VÓ,‰
ˆØ—_‘_ T¨!¡VÓ,‰
ˆØ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¨$°°B¸¸cÀ1ÀQÁ3¹iÈÈaÐPQÉcÉÐ8SÐUVÐVˆØˆe�Q˜ˆ]˜Q   1¡ a¨¡c˜N¨a¨R°©T¨F°B¸¸SÀ#ÀqÈÁsÁ)ÈSÐRSÐTUÑRUÉYÐ<WÐYZÐZˆØ�rˆ~Ðr   r“   r”   s   ` `  r   ÚkeirÜ   p  rØ   r   c                 ó0   ^ ^• T R                   mU U4S jnU$ )Nc                 ó–   >• U R                   nU R                  nUR                  TS5      u  p4X2:¼  a  U7$ UT" U 5      4UT'   UT   S   $ )N)r   r   r   )Ú_misc_const_cacher!   Úget)r
   Úcacher!   ÚprL   ÚfÚnames        €€r   Ú	f_wrappedÚc_memo.<locals>.f_wrappedƒ  sT   ø€ Ø×%Ñ%ˆØ�x‰xˆØ�i‰i˜˜fÓ%‰ˆØ‹9Ø�2ˆIà¡ 3£˜.ˆE�$‰KØ˜‘;˜q‘>Ð!r   )Ú__name__)rã   rå   rä   s   ` @r   Úc_memorè   �  s   ù€ Ø�:‰:€Dö"ð Ðr   c                 ór   • SU R                  S5      U R                  U R                  S5      S-  5      -  -  $ )Nr   é	   r   r2   ©ÚcbrtÚgammar?   ©r
   s    r   Ú
_airyai_C1rï   Ž  s/   € à�—‘˜“˜cŸi™i¨¯©°«
°1©Ó5Ñ5Ñ6Ð6r   c                 ór   • SU R                  S5      U R                  U R                  S5      S-  5      -  -  $ )Nr   r2   r   rë   rî   s    r   Ú
_airyai_C2rñ   ’  s/   € à�—‘˜!“˜sŸy™y¨¯©°«°A©Ó6Ñ6Ñ7Ð7r   c                 ót   • SU R                  SS5      U R                  U R                  S5      S-  5      -  -  $ )Nr   r2   é   r   ©Únthrootrí   r?   rî   s    r   Ú
_airybi_C1rö   –  s2   € à�—‘˜A˜aÓ  3§9¡9¨S¯W©W°Q«Z¸©\Ó#:Ñ:Ñ;Ð;r   c                 ón   • U R                  SS5      U R                  U R                  S5      S-  5      -  $ )Nr2   ró   r   rô   rî   s    r   Ú
_airybi_C2rø   š  s-   € à�;‰;�q˜Ó˜cŸi™i¨¯©°«
°1©Ó5Ñ5Ð5r   c                 ó¬   • U R                   n U R                  SS5      U R                  S5      -  SU R                  -  -  nXl         U7$ ! Xl         f = f)Nr2   ú2/3r   )r!   Úpowerrí   r(   )r
   r!   rL   s      r   Ú_airybi_n2_infrü   ž  sP   € Ø�8‰8€DðØ�I‰I�a˜Ó˜sŸy™y¨Ó/Ñ/°°3·6±6±Ñ:ˆàŒØˆ2€Iøð �ús   Ž4A ÁAc                 ó  • US:X  aì  US:  a  U$ U R                   nU R                  n U =R                  S-  sl        U R                  US-   U-  5      U R                  SX%-  5      -  U R                  -  nUS:X  a1  XpR                  SUS-   -  U-  5      -  nXpR                  SS5      -  nO:U[        U R                  SUS-   -  U-  5      5      -  nXpR                  SS5      -  nX`l        U7U-   $ [        e! X`l        f = f)	NÚZr   r0   r   r2   r   rú   z1/6)Úmpq_1_3r!   rí   rû   r(   r€   rA   rC   )r
   r   r   Úntyper¨   r*   r!   rL   s           r   Ú_airyderiv_0r  ¨  sù   € Ø�ƒ|Øˆq‹5ØˆHØ�K‰KˆØ�x‰xˆð
	Ø�HŠH˜‰N�HØ—	‘	˜1˜Q™3 ™'Ó" S§Y¡Y¨q°±Ó%5Ñ5¸¿¹Ñ>ˆAØ˜‹zØ—Y‘Y˜q ! A¡#™w q™yÓ)Ñ)�Ø—Y‘Y˜q Ó'Ñ'‘à”S˜Ÿ™ 1 a¨¡c¡7¨1¡9Ó-Ó.Ñ.�Ø—Y‘Y˜q Ó'Ñ'�àŒHØˆr�A‰vˆô "Ð!øð	 �Hús   ¨B?C9 Ã9Dc           	      ó  ^ ^^• T R                  T5      mU(       a  T R                  U5      u  pEOSnT R                  T5      (       dÖ  T(       aÏ  U(       a‘  WS:X  a‹  US:X  aT  TT R                  :X  a  T R	                  S5      S-  ST-  -   $ TT R
                  :X  a  T R	                  S5      S-  ST-  -   $ US:  a+  TT R                  :X  a  T$ TT R
                  :X  a	  SU-  T* -  $ U(       d  TT R                  :X  d  TT R
                  :X  a  ST-  $ [        S5      eT(       a(  [        S[        ST R                  T5      -  5      5      mOSmU(       a’  US:X  a  U UU4S	 jnT R                  " U/ 40 UD6$ TS:X  a  [        T TUWS5      $ U UU4S
 jnT R                  " Xd/40 UD6nT R                  T5      (       a'  T R                  U5      (       a  T R                  U5      nU$ U UU4S jnT R                  " U/ 40 UD6$ )Nr   rþ   r   r   r2   éþÿÿÿzessential singularity of Ai(z)r�   c                  óB  >• TR                  T
5      S:”  aŽ  T=R                  T	-  sl        T
S-  n SU -  nSU -  S-  nT=R                  T	-  sl        TR                  U5      * STR                  TR                  5      -  -  TR                  T
S5      -  nU/S// / SS	// U44$ T=R                  T	-  sl        T
S-  S
-  n T=R                  T	-  sl        [        T5      S-  n[        T5      nUT
/SS// / / TR                  /U 4nU/S// / / TR                  /U 4nXg4$ )NrÂ   r�   rÃ   r  r2   r   r   )r   ró   )é   ró   rê   r%   )
r;   r!   ri   r‘   r(   rõ   rï   rñ   Úmpq_5_3rÿ   ©r6   r*   r    ÚCÚC1ÚC2rd   re   r
   Ú	extraprecr   s           €€€r   r.   Úairyai.<locals>.hÝ  s*  ø€ à—7‘7˜1“: “>Ø—H’H 	Ñ)•HØ˜3™�A E¨!¡G °°A±°a±¨QØ—H’H 	Ñ)•HØŸ™ ›˜ Q s§x¡x°·±Ó'7Ñ%7Ñ8¸¿¹ÀQÀqÓ9IÑI�AØ˜C   B r¨6°%¨.¸¸AÐ>Ð?Ð?ð —H’H 	Ñ)•HØ˜1™˜q™�AØ—H’H 	Ñ)•HÜ# C›¨3Ñ.�BÜ# C›�BØ˜Q˜  1  b¨¨B°·±¨}¸QÐ>�BØ˜˜q˜c " R¨¨C¯K©K¨=¸Ð:�BØ˜6�Mr   c                 óz  >• T=R                   T-  sl         TS-  S-  nT=R                   T-  sl         TR                  TR                  TR                  pCnUnSnSU -
  U-  nSU -
  U-  nSX-  -
  n	ST/X-
  U * /U/XxU	/XV/XxU	/U4n
UnSU -
  U-  nSX-  -
  nSU -
  U-  n	STT* /X-
  U * S/U/XxU	/XV/XxU	/U4nX«4$ ©Nr2   rê   r   r   rÂ   )r!   rÿ   Úmpq_2_3Úmpq_4_3)r   r6   Úq13Úq23Úq43r¡   r¢   r£   r¤   Úb3rd   re   r
   r  r   s               €€€r   r.   r  ô  s   ø€ Ø—’˜IÑ%•Ø�q‘D˜‘F�Ø—’˜IÑ%•Ø!Ÿk™k¨3¯;©;¸¿¹˜�Ø�˜1˜ ! A¡# s¡˜b°°!±°S©y¨B¸Q¸q¹u¹W¸"Ø˜�V˜a™e a R˜[¨2¨$°°r°
Ø�G˜b B˜Z¨ð+�à�˜A˜a™C ™9˜¨¨1©5© b°a¸±c¸3±Y°"Ø˜˜Q˜B�Z !¡%¨!¨¨Q °"°¸¸b°zØ�G˜b B˜Z¨ð+�à�v�r   c                  ó<  >• TR                  T
5      S:”  a�  T=R                  T	-  sl        T
S-  n SU -  nSU -  S-  nT=R                  T	-  sl        TR                  U5      STR                  TR                  5      -  TR                  T
S5      -  -  nU/S// / SS	// U44$ T=R                  T	-  sl        T
S-  S
-  n T=R                  T	-  sl        [        T5      n[        T5      nU/S// / / TR                  /U 4nT
U-  /S// / / TR                  /U 4nXg4$ )NrÂ   r�   rÃ   r  r2   r   r   )r   ró   )é   ró   rê   )
r;   r!   ri   r‘   r(   rõ   rï   rñ   r  r  r  s           €€€r   r.   r    s#  ø€ Ø�w‰w�q‹z˜A‹~ð —’˜IÑ%•Ø�s‘F�  a¡˜A¨R°©T°!©V¨Ø—’˜IÑ%•Ø—G‘G˜A“J  #§(¡(¨3¯6©6Ó"2Ñ 2°3·;±;¸qÀÓ3CÑ CÑD�Ø˜˜Q˜C  2 u¨U m°B°qÐ9Ð:Ð:à—’˜IÑ%•Ø�q‘D˜1‘H�Ø—’˜IÑ%•Ü “_�Ü “_�Ø�T˜1˜#˜b  B¨¯© }°QÐ6�Ø˜‘d�V˜Q˜C  2 b¨#¯+©+¨°qÐ8�Ø�v�r   )r9   r„   ÚisnormalrF   r?   rX   rR   r5   r8   r<   r@   r  Ú_is_real_typer:   r;   ©	r
   r   rH   rI   r   r   r.   rL   r  s	   ``      @r   Úairyair  ¾  sÃ  ú€ à�‰�A‹€AÞØ×%Ñ% jÓ1‰ˆˆ5àˆà�<‰<˜�?‰?žqÞ�˜#“Ø�B‹wØ˜Ÿ™“<ØŸ7™7 1›: a™<¨!¨A©#Ñ-Ð-Ø˜Ÿ™“=ØŸ7™7 2›; q™=¨1¨Q©3Ñ.Ð.Ø�2‹vØ˜Ÿ™“<Ø�HØ˜Ÿ™“=Ø ™7 q b™>Ð)Þ�q˜CŸG™G“| q¨C¯H©H£}Ø�Q‘3ˆJäÐ9Ó:Ð:æÜ˜œ3˜s 3§7¡7¨1£:™~Ó.Ó/‰	àˆ	ÞØ�‹6÷"ð$ —=’=  BÑ1¨&Ñ1Ð1à�A‹vÜ# C¨¨A¨u°aÓ8Ð8÷ð —’˜a Ñ/¨Ñ/ˆAØ× Ñ  ×#Ñ#¨¯	©	°!¯©Ø—G‘G˜A“J�ØˆH÷	ð& �}Š}˜Q Ñ- fÑ-Ð-r   c           	      ó�  ^ ^^• T R                  T5      mU(       a  T R                  U5      u  pEOSnT R                  T5      (       dš  T(       a“  U(       aS  WS:X  aM  TT R                  :X  a  T$ TT R                  :X  a+  US:X  a  ST-  $ US:X  a  [        T 5      $ US:  a	  SU-  T* -  $ U(       d'  TT R                  :X  a  T$ TT R                  :X  a  ST-  $ [        S5      eT(       a(  [        S[        ST R                  T5      -  5      5      mOSmU(       a’  US:X  a  U UU4S jnT R                  " U/ 40 UD6$ TS:X  a  [        T TUWS5      $ U UU4S	 jnT R                  " Xd/40 UD6nT R                  T5      (       a'  T R                  U5      (       a  T R                  U5      nU$ U UU4S
 jnT R                  " U/ 40 UD6$ )Nr   rþ   r   r   r  zessential singularity of Bi(z)r�   c                  óü   >• T=R                   T-  sl         TS-  S-  n T=R                   T-  sl         [        T5      S-  n[        T5      nUT/SS// / / TR                  /U 4nU/S// / / TR                  /U 4nX44$ )Nr2   rê   r%   r   r   )r!   rö   rø   r  rÿ   ©r6   r	  r
  rd   re   r
   r  r   s        €€€r   r.   Úairybi.<locals>.h;  sŠ   ø€ Ø—’˜IÑ%•Ø�q‘D˜1‘H�Ø—’˜IÑ%•Ü “_ SÑ(�Ü “_�Ø˜�V˜Q˜q˜E " R¨¨C¯K©K¨=¸Ð:�Ø�T˜1˜#˜b  B¨¯© }°QÐ6�Ø�v�r   c                 ó¨  >• T=R                   T-  sl         TS-  S-  nT=R                   T-  sl         TR                  TR                  TR                  pCnTR                  nTR
                  nUnSnSU -
  U-  n	SU -
  U-  n
SX-  -
  nST/X-
  U * /U/XšU/Xx/XšU/U4nUnSU -
  U-  n	SX-  -
  n
SU -
  U-  nST/X-
  SU -
  /U/XšU/Xx/XšU/U4nXÍ4$ r  )r!   rÿ   r  r  Úmpq_1_6Úmpq_5_6)r   r6   r  r  r  Úq16Úq56r¡   r¢   r£   r¤   r  rd   re   r
   r  r   s                 €€€r   r.   r  H  s  ø€ Ø—’˜IÑ%•Ø�q‘D˜‘F�Ø—’˜IÑ%•Ø!Ÿk™k¨3¯;©;¸¿¹˜�Ø—k‘k�Ø—k‘k�Ø�˜1˜ ! A¡# s¡˜b°°!±°S©y¨B¸Q¸q¹u¹W¸"Ø˜�V˜a™e a R˜[¨2¨$°°r°
Ø�G˜b B˜Z¨ð+�à�˜A˜a™C ™9˜¨¨1©5© b°a¸±c¸3±Y°"Ø˜�V˜a™e Q q¡S˜\¨B¨4°"¸°Ø�G˜b B˜Z¨ð+�à�v�r   c                  óø   >• T=R                   T-  sl         TS-  S-  n T=R                   T-  sl         [        T5      n[        T5      nU/S// / / TR                  /U 4nTU-  /S// / / TR                  /U 4nX44$ )Nr2   rê   r   )r!   rö   rø   r  r  r  s        €€€r   r.   r  [  s…   ø€ Ø�HŠH˜	Ñ!�HØ�1‘�q‘ˆAØ�HŠH˜	Ñ!�HÜ˜C“ˆBÜ˜C“ˆBØ��q�c˜"˜R  C§K¡K =°Ð2ˆBØ�B‘$�˜˜˜B˜r " c§k¡k ]°1Ð4ˆBØ�6ˆMr   )r9   r„   r  rF   rX   rü   rR   r5   r8   r<   r@   r  r  r:   r;   r  s	   ``      @r   Úairybir%    s™  ú€ à�‰�A‹€AÞØ×%Ñ% jÓ1‰ˆˆ5àˆà�<‰<˜�?‰?žqÞ�˜#“Ø�C—G‘G‹|Ø�Ø�C—H‘H‹}Ø˜“7Ø˜Q™3�JØ˜“7Ü)¨#Ó.Ð.Ø�r“6Ø ™7 q b™>Ð)ÞØ�C—G‘G‹|Ø�Ø�C—H‘H‹}Ø˜‘s�
äÐ9Ó:Ð:ÞÜ˜œ3˜s 3§7¡7¨1£:™~Ó.Ó/‰	àˆ	ÞØ�‹6÷ð —=’=  BÑ1¨&Ñ1Ð1à�A‹vÜ# C¨¨A¨u°aÓ8Ð8÷ð —’˜a Ñ/¨Ñ/ˆAØ× Ñ  ×#Ñ#¨¯	©	°!¯©Ø—G‘G˜A“J�ØˆH÷	ð �}Š}˜Q Ñ- fÑ-Ð-r   c                 ó\  ^ • S nS n[        U5      nUS:  a  [        S5      eUS;  a  [        S5      eUS:X  ay  U(       a6  T R                  U 4S jU" S	T R                  -  S
U-  S	-
  -  S-  5      * 5      $ T R                  T R                  U" S	T R                  -  S
U-  S-
  -  S-  5      * 5      $ US:X  a  US:X  ay  U(       a6  T R                  U 4S jU" S	T R                  -  S
U-  S-
  -  S-  5      * 5      $ T R                  T R
                  U" S	T R                  -  S
U-  S	-
  -  S-  5      * 5      $ US:X  aî  US:X  aç  U(       am  S	T R                  -  S
U-  S	-
  -  S-  ST R                  -  -   nT R                  T R                  S5      S	-  5      U" U5      -  nT R                  U 4S jU5      $ S	T R                  -  S
U-  S-
  -  S-  ST R                  -  -   nT R                  T R                  S5      S	-  5      U" U5      -  nT R                  T R
                  U5      $ g g )Nc                 ó*   • U S-  SSU S-  S-  -  -
  -  $ )NçUUUUUUå?r   r  r   é0   rc   ©Úts    r   ÚUÚ_airy_zero.<locals>.Uh  ó    € �Q˜‘Y  ! Q¨¡T¨"¡W¡+¡Ñ.Ð.r   c                 ó*   • U S-  SSU S-  S-  -  -   -  $ )Nr(  r   r  r   r)  rc   r*  s    r   r,   Ú_airy_zero.<locals>.Ti  r.  r   r   zk cannot be less than 1©r   r   z%Derivative should lie between 0 and 1r   c                 ó(   >• TR                  U S5      $ r³   )r  ©r   r
   s    €r   Ú<lambda>Ú_airy_zero.<locals>.<lambda>q  ó   ø€ ¨#¯*©*°Q°q¬/r   r2   rÂ   é   Fc                 ó(   >• TR                  U S5      $ r³   ©r%  r3  s    €r   r4  r5  v  r6  r   Ty              è?c                 ó(   >• TR                  U S5      $ r³   r9  r3  s    €r   r4  r5  }  r6  r   )	r8   rR   Úfindrootr(   r  r%  Úln2Úexpjpir?   )	r
   r¨   r   rH   Úcomplexr,  r,   r+  r¦   s	   `        r   Ú
_airy_zeror?  f  s  ø€ â.Ú.ÜˆA‹€AØˆ1ƒuÜÐ2Ó3Ð3Ø˜ÓÜÐ@ÓAÐAØ�ƒzÞØ—<‘<Ô 9Ù�1�S—V‘V‘8˜Q˜q™S ™UÑ# AÑ%Ó&Ð&ó(ð (à�|‰|˜CŸJ™J©¨1¨S¯V©V©8°Q°q±S¸±UÑ+;¸AÑ+=Ó)>Ð(>Ó?Ð?Ø�ƒz�g Ó&ÞØ—<‘<Ô 9Ù�1�S—V‘V‘8˜Q˜q™S ™UÑ# AÑ%Ó&Ð&ó(ð (à�|‰|˜CŸJ™J©¨1¨S¯V©V©8°Q°q±S¸±UÑ+;¸AÑ+=Ó)>Ð(>Ó?Ð?Ø�ƒz�g “oÞØ�#—&‘&‘˜!˜A™#˜a™%Ñ  Ñ" U¨3¯7©7¡]Ñ2ˆAØ—
‘
˜3Ÿ7™7 1›: a™<Ó(©1¨Q«4Ñ/ˆAØ—<‘<Ô 9¸1Ó=Ð=Øˆc�f‰f‰H�a˜‘c˜!‘eÑ˜QÑ  s§w¡w¡Ñ.ˆØ�J‰J�s—w‘w˜q“z !‘|Ó$¡q¨£tÑ+ˆØ�|‰|˜CŸJ™J¨Ó*Ð*ð &€zr   c                 ó   • [        U SXS5      $ )Nr   F©r?  )r
   r   rH   s      r   Ú
airyaizerorB  ‚  s   € ä�c˜1˜a¨UÓ3Ð3r   c                 ó   • [        U SXU5      $ r³   rA  )r
   r   rH   r>  s       r   Ú
airybizerorD  †  s   € ä�c˜1˜a¨WÓ5Ð5r   c           	      ór  ^ ^^^^• T R                  T5      mT R                  T5      (       aC  TT R                  :X  a  TS:X  a  ST-  $ TS:X  a  T$ TT R                  :X  a  ST-  $ [	        S5      eT(       a(  [        S[        ST R                  T5      -  5      5      mOSmTR                  S5      (       a  [        e T R                  T5      S:”  a²  TS:X  aQ  [        T R                  T5      5      T R                  S-  S-  :  a#  U U4S jnT R                  U/ T R                  S	S
9$ TS:X  aU  [        T R                  T* 5      5      ST R                  -  S-  S-  :  a#  U U4S jnT R                  U/ T R                  S	S
9$ U UUUU4S jnT R                  " U/ 40 TD6$ ! T R                   a     N0f = f)Nr   r   zessential singularityr�   rH   r2   g+‡ÙÎ÷ï?c            	      óD   >• T R                   T/SS// / / SQ/ STS-  -  44$ ©Nr   ))r   r2   )r   r2   r   rê   r2   ©r(   ©r
   r   s   €€r   r.   Ú_scorer.<locals>.hž  s/   ø€ Ø!Ÿf™f Q˜Z¨¨B¨°°2²oÀbÈÈ1ÈaÉ4ÉÐPÐRÐRr   T)r~   Úforce_seriesr   c            	      óF   >• T R                   * T/SS// / / SQ/ STS-  -  44$ rG  rH  rI  s   €€r   r.   rJ  ¢  s1   ø€ Ø"Ÿv™v˜g a˜[¨"¨R¨°°B²ÀrÈ!ÈAÈqÉDÉ&ÐQÐSÐSr   c                  ó6  >• TR                   " T	40 TD6S-  n STR                  -  nTS:X  a
  U S-  n US-  nT=R                  T-  sl        T	S-  S-  nT=R                  T-  sl        U /S// / / / S4nUT	/SS// / S/TR                  TR                  /U4nX44$ )Nr2   r  r   r   r   rê   r   )r%  r(   r!   r  r  )
r§   r+   r6   rd   re   r
   r  rI   r¨   r   s
        €€€€€r   r.   rJ  §  s´   ø€ Ø�JŠJ�qÑ#˜FÑ# AÑ%ˆØˆs�v‰v‰IˆØ�A‹:Ø�‰FˆAØ�‰GˆAØ�Š�IÑ�Øˆq‰D�‰FˆØ�Š�IÑ�ØˆS�1�#�r˜2˜r 2 qÐ(ˆØ�ˆU�R˜�F˜B  Q C¨#¯+©+°c·k±kÐ)BÀAÐEˆØˆvˆr   )r9   ÚisinfrF   rX   rR   r5   r8   r<   rà   rC   rA   Úargr(   r@   r!   r‡   )r
   r   r¨   rI   r.   r  s   ```` @r   Ú_scorerrP  Š  sƒ  ü€ Ø�‰�A‹€AØ
‡y�y�‡|�|Ø�—‘‹<Ø˜‹z ! A¡#˜:Ø˜‹z !˜8Ø�—‘‹=Ø�Q‘3ˆJÜÐ0Ó1Ð1ÞÜ˜œ3˜s 3§7¡7¨1£:™~Ó.Ó/‰	àˆ	Ø‡z�z�,×ÑÜ!Ð!ðØ�7‰7�1‹:˜‹>Ø˜‹zœc #§'¡'¨!£*›o°·±°q±¸5Ñ0@Ó@öSà—}‘} Q¨°S·X±XÈD�}ÐQÐQØ˜‹zœc #§'¡'¨1¨"£+Ó.°°3·6±6±¸!±¸eÑ1CÓCöTà—}‘} Q¨°S·X±XÈD�}ÐQÐQ÷ñ ð �=Š=˜˜BÑ) &Ñ)Ð)øð ×Ñó Ùðús   Â>A+F# Ä*AF# Æ#F6Æ5F6c                 ó   • [        XSU5      $ r­   ©rP  ©r
   r   rI   s      r   ÚscorergirT  µ  ó   € ä�3˜1˜fÓ%Ð%r   c                 ó   • [        XSU5      $ r³   rR  rS  s      r   ÚscorerhirW  ¹  rU  r   c                 ó  • X4U;   a!  X1U4   S   U R                   :¼  a
  X1U4   S   7$ U R                  SU-  S-   5      nU R                  SU-   U R                  U-  -   5      nU R                  SU-   U R                  U-  -
  5      nSU-  U R                  U R                  * U-  U-   U-   S-  U-
  5      -  nU R                  U5      (       d'  U R                  U5      (       d  U R                  U5      nU R                   U4X1U4'   U$ )Nr   r   r   )r!   Úloggammaro   ri   r(   rV   rE   )r
   ÚlÚetaÚ_cacheÚG3ÚG1ÚG2rL   s           r   Úcoulombcr`  ½  sú   € à	€x�6Ó˜f s U™m¨AÑ.°#·(±(Ó:Ø˜�u‘˜aÑ Ð Ð Ø	�‰�a˜‘c˜!‘eÓ	€BØ	�‰�a˜‘c˜#Ÿ%™% ™)‘mÓ	$€BØ	�‰�a˜‘c˜#Ÿ%™% ™)‘mÓ	$€BØ	ˆ1‰ˆs�w‰w˜Ÿ™˜ ™ B™ rÑ)¨1Ñ,¨rÑ1Ó2Ñ2€AØ�F‰F�1�I‰I˜Ÿ™ Ÿ™Ø�F‰F�1‹IˆØ—X‘X˜q�M€FˆSˆ5�MØ€Hr   c                 ó$  ^ ^^• U UU4S jnT R                   " XqU/40 UD6nU(       ah  T R                  U5      (       dR  T R                  U5      (       d<  T R                  T5      (       d&  T R                  T5      S:¼  a  T R                  U5      nU$ )Nc                 ó,  >•  TR                   T-  nTR                  UT	SS9nTR                  USSS9nTR                  X5      nUT	TR                  U5      /SU S-   S// / SU -   X!-  -   /SU -  S-   /U4nU4$ ! [         a    S/S// / / / S4n U4$ f = f)NTr#   r  r   r   r   r   )ro   r'   r`  ri   rR   )
rZ  r[  ÚjwÚjwzÚjwz2r  rd   r
   r6   r   s
          €€€r   r.   Úcoulombf.<locals>.hÐ  sÊ   ø€ ð	.Ø—‘�q‘ˆBØ—(‘(˜2˜q¨�(Ð-ˆCØ—8‘8˜C ¨4�8Ð0ˆDØ—‘˜QÓ$ˆAØ�Q˜Ÿ™ ›Ð%¨¨1¨Q©3° {°B¸¸Q¸q¹SÀÁ¹Z¸LØ�1‘�Q‘�˜ðˆBð ˆuˆøô ó 	.Ø��r�d˜B  B¨¨AÐ-‰BØˆuˆð	.ús   ƒA3A9 Á9BÂBr   )r@   rV   rE   ©	r
   rZ  r[  r   r6   ÚchoprI   r.   rL   s	   `  ``    r   Úcoulombfri  Ê  si   ú€ ÷
ð 	�Š�a˜C˜Ñ+ FÑ+€AÞ�S—V‘V˜A—Y‘Y¨¯©°¯©¸s¿v¹vÀa¿y¹yØ	�‰�‹�a‹Ø�F‰F�1‹IˆØ€Hr   c                 óº   ^ ^^• TT4U;   a"  UTT4   S   T R                   :¼  a
  UTT4   S   $ U UU4S jnT R                  US5      nT R                   U4UTT4'   U$ )Nr   r   c                  ó  >• T* S-
  n TR                   T-  nTR                  ST-   U-   5      S-  TR                  ST-   U-
  5      S-  TR                  SU -   U-   5      S-  TR                  SU -   U-
  5      S-  TS-   * TR                  -  /$ )Nr   y       €      à¿y              à?r%   )ro   rY  r(   )Úl2Újetar
   r[  rZ  s     €€€r   ÚtermsÚ_coulomb_chi.<locals>.termså  sš   ø€ ØˆR�‰TˆØ�u‰u�S‰yˆØ—‘˜Q˜q™S ™XÓ&¨%Ñ0Ø�L‰L˜˜1™˜T™Ó" dÑ+Ø�L‰L˜˜2™˜d™Ó# tÑ,Ø�L‰L˜˜2™˜d™Ó# uÑ-Ø�‰eˆH�S—V‘V‰Oð	ð 	r   )r!   Úsum_accurately)r
   rZ  r[  r\  rn  rL   s   ```   r   Ú_coulomb_chirq  á  sm   ú€ à	ˆ3€x�6Ó˜f Q s U™m¨AÑ.°#·(±(Ó:Ø�a˜�e‰}˜QÑÐ÷ð 	×Ñ˜5 !Ó$€AØ—X‘X˜q�M€Fˆ1ˆSˆ5�MØ€Hr   c                 ór  ^ ^^• T R                  U5      (       d  T R                  U5      nU UU4S jnT R                  " XqU/40 UD6nU(       ah  T R                  U5      (       dR  T R                  U5      (       d<  T R                  T5      (       d&  T R                  T5      S:¼  a  T R                  U5      nU$ )Nc                 ó4  >• TR                  U S-  5      (       a  S/S// / / / S4nU4$ U * S-
  n TR                  X5      nTR                  T-  nTR                  U5      nTR	                  U5      nTR                  X5      nTR                  X15      n	TR                  UT-  5      n
SU-  T-  nXhTX§/SSU S-   SS// / SU -   XQ-  -   /SU -  S-   /U4nU* U	TU
/SSUS-   S// / SU-   XQ-  -   /SU-  S-   /U4nX,4$ ! [         a    S/S// / / / S4nU4s $ f = f)Nr   r   r   r   r  )r:   rq  ro   r_   r^   r`  ri   rR   )rZ  r[  rd   rl  Úchirc  r¦   r¥   r	  r
  r    r   re   r
   r6   r   s                €€€r   r.   Úcoulombg.<locals>.hø  so  ø€ à�9‰9�Q�q‘S�>‰>Ø��r�d˜B  B¨¨AÐ-ˆBØ�5ˆLØˆR�‰Tˆð	Ø×"Ñ" 1Ó*ˆCØ—‘�q‘ˆBØ—‘˜“ˆA #§'¡'¨#£,˜aØ—‘˜aÓ$ˆBØ—‘˜bÓ%ˆBØ—‘˜˜1™“ˆAØ�2‘�a‘ˆAØ˜˜AÐ! B¨¨1¨Q©3°°1Ð#5°r¸2Ø�1‘�R‘V‘�˜q ™s 1™u˜g qð)ˆBà�"�b˜!˜Q� B¨¨2¨a©4°Ð#3¸¸BØ�2‘�b‘f‘�  "¡ Q¡˜x¨ð+ˆBà�6ˆMøÜó 	Ø��r�d˜B  B¨¨AÐ-ˆBØ�5ŠLð	ús   °CC< Ã<DÄDr   )Ú_imr;   r@   rg  s	   `  ``    r   Úcoulombgrw  ñ  s…   ú€ ð
 �7‰7�1�:‰:Ø�G‰G�A‹Jˆ÷ð, 	�Š�a˜C˜Ñ+ FÑ+€AÞ�S—W‘W˜Q—Z‘Z¨#¯'©'°#¯,©,ÀÇÁÈÇÁØ	�‰�‹�q‹Ø�G‰G�A‹JˆØ€Hr   c                 ó–  • SUS-  -  nUS:X  a%  U(       d  SU-  SU-  -   S-
  U R                   -  S-  nUS:X  a%  U(       d  SU-  SU-  -   S-
  U R                   -  S-  nUS:X  a%  U(       a  SU-  SU-  -   S-
  U R                   -  S-  nUS:X  a%  U(       a  SU-  SU-  -   S-
  U R                   -  S-  nU(       d€  WnUS-
  * SU-  -  nSUS-
  -  SU-  S-
  -  SSU-  S-  -  -  n	S	US-
  -  S
US-  -  SU-  -
  S-   -  SSU-  S-  -  -  n
SUS-
  -  SUS-  -  SUS-  -  -
  SU-  -   S-
  -  SSU-  S-  -  -  nU(       a‰  WnUS-   * SU-  -  nSSUS-  -  SU-  -   S-
  -  SSU-  S-  -  -  n	S	S
US-  -  SUS-  -  -   SU-  -
  S-   -  SSU-  S-  -  -  n
SSUS-  -  SUS-  -  -   SUS-  -  -
  SU-  -   S-
  -  SSU-  S-  -  -  nWWW	W
W/nUnSn[        S[        U5      5       H8  n[        XÏ   5      [        XÏS-
     5      :  a	  XÜU   -  nM+  [        XÏ   5      nM:     W[        U5      S-
  :X  a  [        US   5      nXÞ4$ ) aN  
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.
rÂ   r   r   r2   r7  éüÿÿÿr  é   iàÿÿÿéS   iÖ  iÃ  r   r  iÀÿÿÿi%  iÿX iO2 iuÈ_ éi   éR   rê   i  iß  iÑ  i,† i il"q i»QY g        r   )r(   r>   ÚlenrA   )r
   ÚkindÚprimerL   r]   r    r‚   Ús1Ús2Ús3Ús4Ús5rn  r¦   ÚerrÚis                   r   Úmcmahonrˆ    sÙ  € ð 	
ˆ!ˆQ‰$‰€AØˆqƒyž Q q¡S¨¨1©¡W¨Q¡Y°·±Ñ$6°qÑ$8 Øˆqƒyž Q q¡S¨¨1©¡W¨Q¡Y°·±Ñ$6°qÑ$8 Øˆqƒy–U  1¡ Q q¡S¡¨¡¨C¯F©FÑ 2°1Ñ 4˜AØˆqƒy–U  1¡ Q q¡S¡¨¡¨C¯F©FÑ 2°1Ñ 4˜AÞØˆØ�‰sˆV�Q�q‘S‰\ˆØ��1‘‰X�q˜‘s˜2‘vÑ  1 Q¡3¨¡(¡
Ñ+ˆØ�!�A‘#‰Y˜˜1˜a™4™  A¡™ dÑ*Ñ+¨R°°1±°q±©[Ñ9ˆØ�!�A‘#‰Y˜˜Q ™T™	 &¨¨A©¡+Ñ-¨g°a©iÑ7¸Ñ?Ñ@À#ÀqÈÁsÈQÁhÁ,ÑOˆÞØˆØ�‰sˆV�Q�q‘S‰\ˆØ��1�a‘4‘˜˜1™‘˜Q‘Ñ  A a¡C¨!¡8¡Ñ,ˆØ�"�Q˜‘T‘'˜$˜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
€CÜ�1”S˜“ZÖ ˆÜˆu‰x‹=œ3˜u q¡S™z›?Ó*Ø�q‘‰MŠAä�e‘h“-ŠCñ	 !ð
 	ŒC�‹J�q‰LÓÜ�%˜‘)‹nˆØˆ6€Mr   c                 ó^  • US:  a  [        S5      eUS-   n/ n/ n U R                  X#U5      nU Vs/ s H  o€R                  U" U5      5      PM     nn[        US-
  5       V	s/ s H   n	Xy   XyS-      -  S:X  d  M  Xi   XiS-      4PM"     n
n	[	        U
5      U:X  a  U
$ US-  nM‰  s  snf s  sn	f )zÚ
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 1r   r   )rR   ÚlinspaceÚsignr>   r~  )r
   rã   r�   r‚   r   ÚNÚpointsÚsignsr   r‡  Úok_intervalss              r   Úgeneralized_bisectionr�  ;  sË   € ð 	ˆ1ƒuÜÐ2Ó3Ð3Ø	ˆ!‰€AØ€FØ€EØ
Ø—‘˜a !Ó$ˆÙ)/Ó0ª A—‘™!˜A›$–©ˆÐ0Ü9>¸qÀ¹s¼ó *º°AØ‰x˜ ™c™
Ñ" bÑ(ó 0˜™ 6¨A©#¡;Ó/¹ˆð *äˆ|Ó Ó!ØÐØˆa‰Cˆñ ùâ0ùò*s   ² B%Á$B*Á<B*c                 ó$   • U R                  XSSS9$ )NÚillinoisF)ÚsolverÚverify)r;  )r
   rã   Úabs      r   Úfind_in_intervalr–  Q  s   € Ø�<‰<˜ j¸ˆ<Ð?Ð?r   g{®Gáz„?c           	      óÆ  ^ ^• T R                   n[        UT R                  T5      T R                  U5      5      S-   n UT l         T R                  T5      m[	        U5      n[	        U5      nTS:  a  [        S5      eUS:  a  [        S5      eUS;  a  [        S5      eUS:X  a  U(       a  U U4S jn	OU U4S	 jn	US
:X  a  U(       a  U U4S jn	OU U4S jn	US:X  ah  U(       aa  US:X  a[  TS:X  a  T R                  UT l         $ TS::  a<  S
T R                  TST-   -  TS
-   -  5      -  n
[        T W	U
S-  S
U
-  45      UT l         $ XTU4U;   a  [        T W	XaUTU4   5      UT l         $ [        T XTU5      u  p«Xµ:  a  [        T W	X¥-
  X¥-   45      UT l         $ US:X  a	  U(       d  SnUS:X  a	  U(       a  SnUS
:X  a	  U(       d  SnUS
:X  a	  U(       a  SnUS-   n [        T XTU5      u  pëXµ:  ab  [        T XTUS-   5      u  nn[        T W	WSXï-   -  U5      n[        U5       H  u  nnUXaUTUS-   4'   M     [        T U	UUS-
     5      UT l         $ US
-  nM~  ! UT l         f = f)Nr0   r   zv cannot be negativer   zm cannot be less than 1r1  z prime should lie between 0 and 1c                 ó&   >• TR                  TU SS9$ ©Nr   )rH   r   ©r   r
   rL   s    €€r   r4  Úbessel_zero.<locals>.<lambda>c  ó   ø€  C§K¡K°°!¸q KÑ$Ar   c                 ó(   >• TR                  TU 5      $ rm   r   rš  s    €€r   r4  r›  d  ó   ø€  C§K¡K°°!Ô$4r   r   c                 ó&   >• TR                  TU SS9$ r™  ©r`   rš  s    €€r   r4  r›  f  rœ  r   c                 ó(   >• TR                  TU 5      $ rm   r   rš  s    €€r   r4  r›  g  rž  r   g333333@gÍÌÌÌÌÌü?gš™™™™™é?g       @r%   )r!   r5   r<   r?   r8   rR   Úzeror‘   r–  rˆ  r�  Ú	enumerate)r
   r  r€  rL   r]   ÚisoltolÚ_interval_cacher!   Úworkprecrã   r*   r†  Úlowr   Úr1Úr2Úerr2Ú	intervalsr   r•  s   `  `                r   Úbessel_zeror¬  T  s�  ù€ Ø�8‰8€DÜ�4˜Ÿ™ › S§W¡W¨Q£ZÓ0°Ñ3€Hð0ØˆŒØ�G‰G�A‹JˆÜ�‹FˆÜ�E“
ˆØˆq‹5ÜÐ3Ó4Ð4Øˆq‹5ÜÐ6Ó7Ð7Ø˜‹~ÜÐ?Ó@Ð@Ø�1‹9ÞÕA‘aÝ4�aØ�1‹9ÞÕA‘aÝ4�að �1‹9ž 1¨£6Ø�A‹vØ—x‘xð6 ˆ�ð5 �A‹và�c—h‘h˜q ! A¡#™w¨¨!©™}Ó-Ñ-�Ü'¨¨Q°°2±°q¸±s°Ó<ð. ˆ�ð- �q˜Ð˜Ó.Ü# C¨¨OÀÀqÈ¸NÑ,KÓLð* ˆ�ô) ˜˜d¨1¨aÓ0‰ˆØ‹=Ü# C¨¨Q©Y¸¹	Ð,BÓCð$ ˆ�ð! �1‹9žU¨# CØ�1‹9ž c Ø�1‹9žU¨# CØ�1‹9ž c Øˆa‰CˆØÜ˜c 4°°1Ó5‰GˆBØ‹}Ü" 3¨°Q¸¸!¹Ó<‘��DÜ1°#°q¸#¸sÀBÁE¹{ÈAÓN�	Ü& yÖ1‘E�A�rØ8:�O¨¨q°°1±Ð$4Ó5ñ 2ä'¨¨Q°	¸!¸A¹#±Ó?ð ˆ�ð �a‘C�ñ øð ˆ�ús+   ¾B=I Ä:I ÅI Å('I ÆB1I ÉI É	I c                 ó    • [        U SX1U5      7$ )a—  
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
   rL   r]   rH   s       r   Úbesseljzeror°  ‰  s   € ô` ˜˜Q 
¨qÓ1Ð1Ð1r   c                 ó    • [        U SX1U5      7$ )a®
  
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besselyzeror²  û  s   € ôr ˜˜Q 
¨qÓ1Ð1Ð1r   N)r   )F)r   F)r   T)2Ú	functionsr   r   r   r   r   rT   r`   rk   rq   rt   rx   r|   rz   r•   r™   r«   r°   r´   rº   r¿   rÇ   rÍ   r×   rÜ   rè   rï   rñ   rö   rø   rü   r  r  r%  r?  rB  rD  rP  rT  rW  r`  ri  rq  rw  rˆ  r�  r–  r¬  r°  r²  rc   r   r   Ú<module>r´     sF  ðß +àñó ðð ñó ðð ó@ó ð@ðD ó!ó ð!ðF ó!3ó ð!3ðF ñ+ó ð+ð, ñGó ðGð ñGó ðGð ñCó ðCð ñCó ðCð ñ-ó ð-ð8 ñ+ó ð+ð ñ+ó ð+ò+ð& ñ*ó ð*ð ñ*ó ð*ð ñ	-ó ð	-ð ñ-ó ð-ð8 ñ
+ó ð
+ð ñ
+ó ð
+ð ñ+ó ð+ð ñ+ó ð+ò ð ñ7ó ð7ð ñ8ó ð8ð ñ<ó ð<ð ñ6ó ð6òò"ð, óY.ó ðY.ðv óI.ó ðI.ôV+ð8 ó4ó ð4ð ó6ó ð6ò)*ðV ñ&ó ð&ð ñ&ó ð&ð Ø!#ó 
ó ð
ð óó ðð, Ø%'ó ó ðð ó ó ð òD%òNò,@ð 15Àbô 3ðj óo2ó ðo2ðb óX2ó ñX2r   