+
    ‰°³j2”  ã                   óæ  € ^RI HtHt ]R 4       t]R 4       t]R3R l4       t]R3R l4       t]R3R l4       t]R 4       t]R 4       t	]R	 4       t
]R
 4       t]R 4       t]R 4       t]R 4       t]R 4       tR t]R 4       t]R 4       t]R 4       t]R 4       t]R 4       t]R 4       t]R 4       t]R 4       tR t]R 4       t]R 4       t]R 4       t]R 4       tR tR t]R3R l4       t ]R3R  l4       t!R4R! lt"]R3R" l4       t#]R5R# l4       t$R$ t%]R% 4       t&]R& 4       t']/ 3R' l4       t(]R6R( l4       t)]/ 3R) l4       t*]R6R* l4       t+R+ t,R, t-R- t.R./ 3R/ lt/]R3R0 l4       t0]R3R1 l4       t1R2# )7é   )ÚdefunÚdefun_wrappedc                ó&   € V P                  ^ V4      # )zCComputes the Bessel function `J_0(x)`. See :func:`~mpmath.besselj`.©Úbesselj©ÚctxÚxs   &&ÚX/home/luca/screen-checking/.venv/lib/python3.14/site-packages/mpmath/functions/bessel.pyÚj0r      ó   € ð �;‰;�q˜!ÓÐó    c                ó&   € V P                  ^V4      # )zDComputes the Bessel function `J_1(x)`.  See :func:`~mpmath.besselj`.r   r   s   &&r   Új1r      r   r   c                 óà  a aaa	a
a€ \        S4      \        J d   R pMDS P                  S4      oS P                  S4      pV'       d   \        S P	                  S4      4      oV'       d,   S^ 8  d%   RS,          S P
                  ! S) SV3/ VB ,          # S P                  S4      oS P                  S4      o	V'       dÙ   S P                  V4      o
S P                  S
4      '       d�   S
^ 8¼  d‰   \        S
4      o
S P                  p S ;P                  ^,          un        S P                  V V
VV3R l\        S
^,           4       4       4      pVS n        VS P                  ^4      S
) ,          ,          pV# V	V V3R lpS P                  ! VSS
.3/ VB p V# V'       g<   V'       d4   \        S	4      ^
8  d$   \        S4      ^8  d    S P                  SS4      # S'       g_   S'       g   S P                  S,           S,           pM½S P!                  S4      ^ 8”  d   SS,          pM�S P"                  S,           S,           pM‚S P                  p S ;P                  \%        ^\        S	4      ,          S P                  4      ,          un        S P'                  SRR R7      oV	V V3R lpS P                  ! VS.3/ VB pVS n        V5pV#   TS n        i ; i  \         d     ELi ; i  TS n        i ; i)Tc              3   ó¼   <"  € T FQ  pRV,          SP                  SV4      ,          SP                  ^V,          S,           S,
          S4      ,          x € KS  	  R# 5i)r   Néÿÿÿÿ)Úbinomialr   )Ú.0Úkr	   ÚdÚnÚzs   & €€€€r   Ú	<genexpr>Úbesselj.<locals>.<genexpr>%   sK   øé € ð )Ù'˜ð ! 1�W s§|¡|°A°aÓ'8Õ8¸3¿;¹;ÀqÈÅsÈ1ÅuÈQÅwÈqÓ;Q×QÒQÛ'ùs   ƒAAc                 ó€  <€ SP                  SP                  SSSP                  S,           R 7      RRR7      pRW,
          ^,           ,          RW,
          ^,           ,          .p^SP                  S.V^V ,          ,
          RW,
          .. W0^,           R,          V ^,           R,          .W0^,           .,           V3.pV# )©ÚprecT©Úexactç      à?ç      Ð¿©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¨°·±¸!µ˜XÓ<¸eÈ4�HÓP�Ø˜!�#˜a�%•[ # q¥s¨1¥u¥+Ð.�Ø˜Ÿ™ �l A a¨¥c¥E¨#¨a­c ?°2°a¸A½#¸s½ÀAÀaÅCÈÅ9Ð8MÈaÐSTÕQTÐPUÍgÐVWÐXÐY�Ø�r   r!   r   c                 ó¼   <€ SP                  SP                  SS\        ^ SP                  S,           4      R7      RR7      pS.V .. V ^,           .. V ^,           .V3.# )é    r   Tr   )Úfnegr$   Úmaxr   )r   r'   r*   r	   Úws   & €€€r   r+   r,   G   s[   ø€ ØŸ™ §¡¨!¨Q´S¸¸3¿8¹8ÀA½:Ó5F Ó!GÈt˜ÓT�AØ˜S 1 # r¨A¨a­C¨5°"°q¸µs°e¸QÐ?Ð@Ð@r   r   )ÚtypeÚintÚconvertÚisintÚ_rer   Úmagr   ÚfsumÚrangeÚmpfÚ	hypercombÚabsÚ_besseljÚNotImplementedErrorÚoneÚreÚinfÚminr$   )r	   r   r   Ú
derivativeÚkwargsÚn_isintÚorigÚvr+   r*   r   r1   s   fff&,    @@@r   r   r      s[  ý€ äˆAƒw”#ƒ~Ø‰à�K‰K˜‹NˆØ—)‘)˜A“,ˆßÜ�C—G‘G˜A“J“ˆAß�1�q”5Ø�Q�w˜Ÿš a R¨¨JÑA¸&ÑAÕAÐAØ�‰�A‹€AØ�‰�‹
€AßØ�K‰K˜
Ó#ˆð
 �9‰9�Q�<Š<˜A œFÜ�A“ˆAØ—8‘8ˆDð Ø—’˜B••Ø—H‘H÷ )Ü" 1 Q¥3œZó)ó )�ð  �”Ø�—‘˜“˜q˜bÕ!Õ!ˆAðJ €H÷Gð
 —’˜a ! A Ñ1¨&Ñ1‰Að< €H÷7 §¬C°«F°R¬K¼CÀ»FÀR¼KðØ—|‘| A qÓ)Ð)÷ ßØ—G‘G˜a•K •M‘Ø—‘˜“˜Q”Ø�a•C‘à—G‘G˜a•K !•O‘ð —8‘8ˆDð
 ð —’œC ¤# a£&¥¨#¯(©(Ó3Õ3•Ø—H‘H˜Q ¨4�HÓ0�÷Að —M’M ! a SÑ3¨FÑ3�à�”ØˆBˆØ€HøðM  �•ûô 'ô Úðûð,  �•ús,   ÄAK ÇK ÉA.K$ Ë	KËK!Ë K!Ë$	K-c                 óŠ  a aa	€ S P                  V4      pS P                  S4      oS'       g±   V'       d   \        hV'       g   ^V,           S,           # S P                  V4      '       d   ^ VS,           ,          # S P                  V4      pV^ 8X  d   S P                  VS,           ,          # V^ 8”  d   ^ VS,           ,          # S P
                  VS,           ,           # S P                  S4      o	V'       d1   S P                  V4      pV	V V3R lpS P                  ! WqV.3/ VB pV# V	V V3R lpS P                  ! Wq.3/ VB pV# )r   c                 ó‚  <€ SP                  SP                  SSSP                  S,           R 7      RRR7      pRW,
          ^,           ,          RW,
          ^,           ,          V ^,           .p^SP                  S.V^V ,          ,
          RW,
          .V ^,           .W0^,           R,          V ^,           R,          .W23.pV# )r   ç      Ð?Tr   r!   r#   r&   s   &&   €€€r   r+   Úbesseli.<locals>.hf   sŸ   ø€ Ø—‘˜Ÿ™ ! Q¨S¯X©X°a­Z˜Ó8¸$Àd�ÓKˆAØ�a•c˜!•e•˜c 1¥3 q¥5�k¨1¨Q­3Ð/ˆAØ�S—V‘V˜A�,  ! A¥#¥ c¨!­#˜°°!µ¨u°Q¸1½¸c½	À1ÀQÅ3ÈÅ)Ð7LÈQÐQÐRˆAØˆHr   c           	      óÀ   <€ SP                  SR RR7      pSP                  W\        ^ SP                  S,           4      R7      pV.V .. V ^,           .. V ^,           .V3.# )r!   Tr   r   )r$   r0   r   )r   r1   r'   r*   r	   r   s   &  €€€r   r+   rK   m   s`   ø€ Ø—‘˜˜C t�Ó,ˆAØ—‘˜¤C¨¨#¯(©(°1­*Ó$5�Ó6ˆAØ�S˜1˜#˜r A a¥C 5¨"¨q°­s¨e°QÐ7Ð8Ð8r   )r4   Ú
ValueErrorr5   r@   ÚnanrA   r7   r;   )
r	   r   r   rC   rD   r'   r   r+   rG   r*   s
   f&f&,    @r   ÚbesselirO   P   s  ú€ à�‰�A‹€AØ�‰�A‹€AßßÜÐßà�Q•3�q•5ˆLØ�9‰9�Q�<Š<Ø�a˜•c•7ˆNØ�F‰F�1‹IˆØ�Œ6Ø—7‘7˜A˜a�C•=Ð Ø�ŒUØ�a˜•c•7ˆNà—7‘7˜A˜a�C•=Ð Ø�‰�‹
€AßØ�K‰K˜
Ó#ˆ÷	ð
 �MŠM˜! ˜UÑ- fÑ-ˆð €H÷	9ð �MŠM˜!˜SÑ+ FÑ+ˆØ€Hr   c                 ó  € V'       Eg"   V'       d   \         hV'       g   V P                  ) W,           ,           # V P                  V4      '       d   V P                  W,           ,          # V P	                  V4      pVR ,           pV P                  V4      '       d1   V^ 8”  d   V P                  ) W,           ,           # ^ W,           ,          # V^ 8  dA   \        V P                  V4      4      ^,          '       d   V P                  W,           ,           # V P                  W,           ,           # V ;P                  ^
,          un	        V P                  V4      w  rxW€P                  ) 8  d0   V P                  5p	V ;P                  ^,          un	        W,          pM V^ 8  d   V ;P                  V,          un	        V P                  V4      w  r«V P                  ! WV3/ VB V
,          V P                  ! V) W#3/ VB ,
          V,          # )r!   )rM   rA   ÚimrN   r@   r5   r3   ÚfloorÚninfr   Únint_distanceÚepsÚcospi_sinpir   )r	   r   r   rC   rD   r'   ÚqÚmr   r+   ÚcosÚsins   &&&&,       r   Úbesselyr[   t   sw  € çˆ1ßäÐßà—G‘G�8˜q�sÕ#Ð#Ø�6‰6�!�9Š9Ø—7‘7˜a�c•?Ð"Ø�F‰F�1‹IˆØˆc�EˆØ�9‰9�Q�<Š<Ø�1ŒuØŸ™�x 1¥3Õ'Ð'à˜A�C•yÐ ØˆqŒ5”S˜Ÿ™ 1›Ó&¨×*Ô*Ø—7‘7˜a�c•?Ð"à—8‘8˜q�sÕ#Ð#à‡H‚H�…N…HØ×Ñ˜QÓ�D€AØ�H‰Hˆ9„}Ø�W‰WˆHˆØ�Š�A��Ø	�‰Ø	
ˆQŒØ�Š�A��à�‰˜qÓ!�H€CØ�KŠK˜˜JÑ0¨Ñ0°Õ4Ø�Š�Q�B�qÑ- fÑ-õ.Ø/2õ3ð 3r   c                 óÔ   a a€ S'       g   S P                   # S P                  S4      pV^8  d   V3R lpM S ;P                  V,          un        V V3R lpS P                  ! WQ.3/ VB # )r   c                 ó¢   <€ S^,          ^,          pS^.V ) V ^,
          .V .. . ^V ,
          .V3pS^.W ) ^,
          .V ) .. . ^V ,           .V3pW#3# )é   © )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°!°Aµ#°¸Ð9ˆBØ�Q�˜!˜R �T˜ a R D¨"¨b°1°Qµ3°%¸Ð:ˆBØ�6ˆMr   c           	      ó    <€ SP                   ^,          SSP                  S) 4      .. RO. . V R,           RV ,
          .. R^S,          ,          3.# )r^   r!   )r!   ç      à¿r   r   )r%   Úexp©r   r	   r   s   &€€r   r+   rb   ª   sO   ø€ Ø—f‘f˜Q•h  3§7¡7¨A¨2£;Ð/²¸rÀ2Ø�3•˜˜A��  B¨¨!­¥Hð.ð /ð /r   )rA   r7   r   r;   )r	   r   r   rD   r*   r+   s   f&f,  r   Úbesselkrg   ˜   sT   ù€ çØ�w‰wˆØ�‰�‹
€AØˆ1„uö	ð 	�Š�A��ö	/ð �=Š=˜˜CÑ* 6Ñ*Ð*r   c                 óz   € V P                   ! W3/ VB V P                  V P                  ! W3/ VB ,          ,           # ©N©r   Újr[   ©r	   r   r
   rD   s   &&&,r   Úhankel1rm   ¯   ó2   € à�;Š;�qÑ$˜VÑ$ s§u¡u¨S¯[ª[¸Ñ-F¸vÑ-FÕ'FÕFÐFr   c                 óz   € V P                   ! W3/ VB V P                  V P                  ! W3/ VB ,          ,
          # ri   rj   rl   s   &&&,r   Úhankel2rp   ³   rn   r   c                 óŠ  € V^ 8X  dU   V P                  V4      R8”  d   V# V P                  V4      R8  d   V P                  V,           # V P                  V,          # V P                  RVRR7      pRV,           pV P	                  V4      W6,          ,          V P
                  ! Wa,
          ^^V,          ,           V3/ VB ,          # ©r.   r!   Tr   rd   )r@   rA   rN   r$   re   Úhyp1f1)r	   r   rX   r   rD   r
   Úys   &&&&,  r   Úwhitmru   ·   s�   € àˆA„và�6‰6�!‹9�tÔØˆHØ�V‰V�A‹Y˜ÔØ—7‘7˜Q•;Ðà—7‘7˜Q•;ÐØ�‰��q ˆÓ%€AØˆA�€AØ�7‰7�1‹:˜�Õ˜sŸzšz¨!­#¨q°°1µ­u°aÑB¸6ÑBÕBÐBr   c                 ó‚  € V^ 8X  dQ   \        V P                  V4      4      pVR8  d   V# VR8”  d   V P                  V,           # V P                  V,          # V P	                  RVRR7      pRV,           pV P                  V4      W7,          ,          V P                  ! Wq,
          ^^V,          ,           V3/ VB ,          # rr   )r<   r@   rA   rN   r$   re   Úhyperu)r	   r   rX   r   rD   Úgr
   rt   s   &&&&,   r   Úwhitwry   Å   sœ   € àˆA„vÜ�—‘�q“	‹NˆØˆsŒ7ØˆHØ�ŒWØ—7‘7˜Q•;Ðà—7‘7˜Q•;ÐØ�‰��q ˆÓ%€AØˆA�€AØ�7‰7�1‹:˜�Õ˜sŸzšz¨!­#¨q°°1µ­u°aÑB¸6ÑBÕBÐBr   c           	      óº  a a€ S P                  V4      w  rS P                  V4      w  r&S P                  S4      oS'       gR   S P                  V4      ^8:  d)   S P                  ^V,
          .W,
          ^,           .4      # S P                  S,           # ^V,           V,
          pS P                  V4      w  rx S P
                  p	 S ;P
                  ^
,          un        S P                  ^^ WX3W.RS,          S P
                  R7      p
V
SV,          ,          V	S n        #   T	S n        i ; i  S P                   d     Mi ; iT T3R lpS P                  ! Y±T.3/ TB # )r   )Úmaxtermsc                 óü   <€ SP                  V4      pSP                  V.^R.. W,
          ^,           V.V .V.S3pSP                  ) VS.^R^V,
          .. V ^V,
          .W,
          ^,           .^V,
          .S3pW43# ©r   r   )Úsinpir%   )ÚaÚbr1   r`   ra   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¨q­S¨'°1µ3°qµ5°'¸1¸Q½3¸%ÀÐBˆØˆvˆr   r   )	Ú_convert_paramr4   r@   Ú	gammaprodrA   r   ÚhypsumÚNoConvergencer;   )r	   r   r€   r   rD   ÚatypeÚbtypeÚbbÚbbtyperF   rG   r+   s   f&&f,       r   rw   rw   Ó   s(  ù€ à×!Ñ! !Ó$�H€AØ×!Ñ! !Ó$�H€AØ�‰�A‹€AßØ�6‰6�!‹9˜Œ>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—
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ÓSˆAØ�q˜!•t•8àˆC�Hø�tˆC�HûØ×Ñô Ùðúöð
 �=Š=˜˜q˜EÑ, VÑ,Ð,s+   Â7D* ÃAD ÄD* Ä	D'Ä'D* Ä*D>Ä=D>c                 ó€   a a€ S P                  V4      pS P                  S4      oV V3R  lpS P                  ! WA.3/ VB # )c                 ó¾   <€ S^,          RSP                  SP                  4      ,          .V ^,           R.. V R,           .^.RV R,           .S^,          ^,          ) 3.# ©r^   r!   ç      ø?r   ©Úsqrtr%   rf   s   &€€r   r+   Ústruveh.<locals>.hõ   sa   ø€ Ø�A•#�s˜3Ÿ8™8 C§F¡FÓ+Õ+Ð,¨q°­s°B¨i¸¸aÀ½e¸WÀqÀcÈCÐQRÐSVÕQVÈ<Ð[\Ð]^Õ[^ÐabÕZbÐYbÐcÐdÐdr   ©r4   r;   ©r	   r   r   rD   r+   s   f&f, r   Ústruvehr“   ð   s9   ù€ à�‰�A‹€AØ�‰�A‹€Aöeà�=Š=˜˜CÑ* 6Ñ*Ð*r   c                 ó€   a a€ S P                  V4      pS P                  S4      oV V3R  lpS P                  ! WA.3/ VB # )c                 ó¼   <€ S^,          RSP                  SP                  4      ,          .V ^,           R.. V R,           .^.RV R,           .S^,          ^,          3.# rŒ   rŽ   rf   s   &€€r   r+   Ústruvel.<locals>.hþ   s^   ø€ Ø�A•#�s˜3Ÿ8™8 C§F¡FÓ+Õ+Ð,¨q°­s°B¨i¸¸aÀ½e¸WÀqÀcÈCÐQRÐSVÕQVÈ<ÐZ[Ð\]ÕZ]Ð`aÕYaÐbÐcÐcr   r‘   r’   s   f&f, r   Ústruvelr—   ù   s9   ù€ à�‰�A‹€AØ�‰�A‹€Aödà�=Š=˜˜CÑ* 6Ñ*Ð*r   c                 ó’   a aa€ S P                  V4      ^ ,          pS P                  S4      oV VV3R lpS P                  ! WR.3/ VB # )r.   c                 ó^  <€ SP                   pW,          pV^,          pW2,
          W2,           ^V,
          ^V,           3w  rErgSP                  V4      w  r‰S^ 8X  d   VS,          V	.V.rºS^8X  d   VS,          V) .V	.rºSP                  SRR7      pX
^^.. WE.^.WE.V3pX^.. Wg.^.Wg.V3pWÞ3# )é   ©Úmultr"   )Úmpq_1_2rV   Úsquare_exp_arg)rG   r€   ÚurX   Úa1Úa2Úb1Úb2ÚcÚsÚAr(   r1   r`   ra   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�AŒ:Ø�a•C˜�8˜a˜SˆqØ�AŒ:Ø�a•C˜!˜�9˜q˜cˆqØ×Ñ˜q uÐÓ-ˆØ��A�˜˜R˜G a S¨2¨'°1Ð4ˆØ���R˜"˜ 1 #¨ w°Ð1ˆØˆvˆr   ©r‚   r4   r;   )r	   r§   rG   r   rD   r+   s   ff&f, r   Ú_angerrª     sB   ú€ Ø×Ñ˜1Ó˜aÕ €AØ�‰�A‹€A÷ð �=Š=˜˜CÑ* 6Ñ*Ð*r   c                 ó   € \        V ^ W3/ VB # ©r.   ©rª   ©r	   rG   r   rD   s   &&&,r   Úangerjr¯     ó   € ä�#�q˜!Ñ) &Ñ)Ð)r   c                 ó   € \        V ^W3/ VB # ©r   r­   r®   s   &&&,r   Úweberer³     r°   r   c                 óÀ   a a€ S P                  V4      ^ ,          pS P                  V4      ^ ,          pS P                  S4      oV V3R lpS P                  ! WQV.3/ VB # )r.   c           	      óú   <€ SP                   pSP                  SRR7      pW,
          ^,           W,           ^,           S.RRV ^,           .. . ^.W V,
          ^,           ,          W V,           ^,           ,          .V33# ©rJ   r›   r"   r   ©r�   rž   )rŸ   rG   r€   r1   r	   r   s   &&  €€r   r+   Úlommels1.<locals>.h"  sx   ø€ Ø�K‰KˆØ×Ñ˜q uÐÓ-ˆØ•�Q•˜�˜A�˜qÐ! B¨¨A¨a­C =°"°b¸1¸#Ø�!•�A•�Y�q˜A�#˜a�%•yÐ! 1ð&ð 'ð 	'r   r©   ©r	   rŸ   rG   r   rD   r+   s   f&&f, r   Úlommels1rº     sY   ù€ à×Ñ˜1Ó˜aÕ €AØ×Ñ˜1Ó˜aÕ €AØ�‰�A‹€Aö'ð
 �=Š=˜˜q˜EÑ, VÑ,Ð,r   c                 óÀ   a a€ S P                  V4      ^ ,          pS P                  V4      ^ ,          pS P                  S4      oV V3R lpS P                  ! WQV.3/ VB # )r.   c           	      ó8  <€ SP                   pSP                  SRR7      pW,
          ^,           W,           ^,           S.RRV ^,           .. . ^.W V,
          ^,           ,          W V,           ^,           ,          .V3p^S.W,           ^,
          V) .WW,           ^,           ,          .W!V ,
          ^,           ,          .. ^V,
          .V3p^S.W,
          ^,
          V.V) W V,
          ^,           ,          .V^V ,
          V,
          ,          .. ^V,           .V3pWEV3# r¶   r·   )	rŸ   rG   r€   r1   r`   ra   ÚT3r	   r   s	   &&     €€r   r+   Úlommels2.<locals>.h4  s  ø€ Ø�K‰KˆØ×Ñ˜q uÐÓ-ˆØ�c�!�e�Q•S˜•U˜AÐ  R¨¨1­ ¨r°2¸°s¸QÀ!ÅÀAÅ½YÀqÈAÍ#ÈaÍ%ÅyÐ<QÐSTÐTˆØ�ˆV�a•c˜!•e˜a˜R�[ 1¨­¨Q­¥i .°1¸µc¸!µeµ9°+¸rÀAÀaÅCÀ5È!ÐKˆØ�ˆV�a•c˜!•e˜Q�Z 1 " a¨1­¨Q­¥i °1°a¸µc¸!µeµ9°+¸rÀAÀaÅCÀ5È!ÐKˆð �rˆzÐr   r©   r¹   s   f&&f, r   Úlommels2r¿   )  sY   ù€ à×Ñ˜1Ó˜aÕ €AØ×Ñ˜1Ó˜aÕ €AØ�‰�A‹€Aöð  �=Š=˜˜q˜EÑ, VÑ,Ð,r   c                 ó€   a a€ S P                  V4      pS P                  S4      oV V3R  lpS P                  ! WA.3/ VB # )c           	      óZ  <€ S^,          ^,          ) pSP                  RV ,          4      w  r#VS^,          .^V .. V ^,           .. RRV ^,           ,          RV ,          ^,           .V3pVS^,          .^V ^,           .. V ^,           .. RRV ^,           ,          RV ,          ^,           .V3pWE3# )é   r!   r�   ç      è¿©rV   ©r   r'   rY   rZ   r`   ra   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¸Q½3µiÀÀQÅÀqÅÐ0IÈ1ÐLˆØ�1�Q•3ˆZ˜!˜Q˜q�S˜ 2¨¨!­ u¨b°3¸¸Q¸q½S½	À3ÀqÅ5ÈÅ7Ð2KÈQÐNˆØˆvˆr   r‘   r’   s   f&f, r   ÚberrÇ   F  ó:   ù€ à�‰�A‹€AØ�‰�A‹€Aöð �=Š=˜˜CÑ* 6Ñ*Ð*r   c                 ó€   a a€ S P                  V4      pS P                  S4      oV V3R  lpS P                  ! WA.3/ VB # )c           	      óZ  <€ S^,          ^,          ) pSP                  RV ,          4      w  r#VS^,          .^V ^,           .. V ^,           .. RRV ^,           ,          RV ,          ^,           .V3pVS^,          .^V .. V ^,           .. RRV ^,           ,          RV ,          ^,           .V3pWE3# )rÂ   ç      è?r�   r!   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¸q½S½	À3ÀqÅ5ÈÅ7Ð2KÈQÐNˆØ�1�Q•3ˆZ˜!˜Q˜  a¨¥c U¨B°°c¸1¸Q½3µiÀÀQÅÀqÅÐ0IÈ1ÐLˆØˆvˆr   r‘   r’   s   f&f, r   ÚbeirÍ   S  rÈ   r   c                 ó€   a a€ S P                  V4      pS P                  S4      oV V3R  lpS P                  ! WA.3/ VB # )c           
      ó¨  <€ S^,          ^,          ) pS
P                  RV ,          4      w  r#S
P                  RV ,          4      w  rE^S^V,          .V ) ^,
          V ^.V ) .. . RR^V ,           ,          RV ^,           ,          .V3p^SV) .V ) ^,
          ^V ,           ^.V ) ^,
          .. . RR^V ,           ,          RV ^,           ,          .V3p^S^V,          .V ^,
          V ) ^.V .. . RR^V ,
          ,          ^RV ,          ,
          .V3p^SV) .V ^,
          ^V ,
          ^.V ^,
          .. . RR^V ,
          ,          ^RV ,          ,
          .V3p	WgW‰3# )rÂ   rJ   rË   r!   r�   rÄ   ©r   r'   Úcos1Úsin1Úcos2Úsin2r`   ra   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¸#¸sÀAÀaÅC½yÈ#ÈqÐQRÍsÍ)Ð9TÐVWÐWˆØ��T�Eˆ]˜a˜R �T 1 Q¥3¨˜N¨a¨R°­T¨F°B¸¸SÀ#ÀqÈÅsÅ)ÈSÐRSÐTUÕRUÍYÐ<WÐYZÐZˆØ��A�d•Fˆ^˜a �c A 2 q˜\¨A¨3°°B¸¸cÀ1ÀQÅ3½iÈÈ3ÈqÍ5ÍÐ8QÐSTÐTˆØ��T�Eˆ]˜Q˜q�S ! A¥# q˜M¨A¨a­C¨5°"°b¸3ÀÀQÀqÅSÅ	È1ÈSÐQRÍUÍ7Ð:SÐUVÐVˆØ�rˆ~Ðr   r‘   r’   s   f&f, r   Úkerr×   `  ó:   ù€ à�‰�A‹€AØ�‰�A‹€Aöð �=Š=˜˜CÑ* 6Ñ*Ð*r   c                 ó€   a a€ S P                  V4      pS P                  S4      oV V3R  lpS P                  ! WA.3/ VB # )c           
      ó�  <€ S^,          ^,          ) pS
P                  RV ,          4      w  r#S
P                  RV ,          4      w  rEV) ^S.^V ^,
          ^V ,
          .V ^,
          .. . RR^V ,
          ,          ^RV ,          ,
          .V3pV) ^S.^V ^,
          V ) .V .. . RR^V ,
          ,          ^RV ,          ,
          .V3pV) ^S.^V ) ^,
          V .V ) .. . RRV ^,           ,          RV ^,           ,          .V3pV) ^S.^V ) ^,
          V ^,           .V ) ^,
          .. . RRV ^,           ,          RV ^,           ,          .V3p	WgW‰3# )rÂ   rË   rJ   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¨a­C¨5°"°b¸3ÀÀQÀqÅSÅ	È1ÈSÐQRÍUÍ7Ð:SÐUVÐVˆØˆe�Q˜ˆ]˜Q  !¥ a R˜L¨1¨#¨r°2¸¸SÀ!ÀAÅ#½YÈÈ#ÈaÍ%ÍÐ7PÐRSÐSˆØˆe�Q˜ˆ]˜Q   1¥ a˜L¨A¨2¨$°°B¸¸cÀ1ÀQÅ3½iÈÈaÐPQÍcÍÐ8SÐUVÐVˆØˆe�Q˜ˆ]˜Q   1¥ a¨¥c˜N¨a¨R°­T¨F°B¸¸SÀ#ÀqÈÅsÅ)ÈSÐRSÐTUÕRUÍYÐ<WÐYZÐZˆØ�rˆ~Ðr   r‘   r’   s   f&f, r   ÚkeirÜ   p  rØ   r   c                 ó0   a a€ S P                   oV V3R  lpV# )c                 ó¨   <€ V P                   pV P                  pVP                  SR4      w  r4W28¼  d   V5# VS! V 4      3VS&   VS,          ^,          # )r   )r   r.   )Ú_misc_const_cacher   Úget)r	   Úcacher   ÚprG   ÚfÚnames   &    €€r   Ú	f_wrappedÚc_memo.<locals>.f_wrappedƒ  sT   ø€ Ø×%Ñ%ˆØ�x‰xˆØ�i‰i˜˜fÓ%‰ˆØŒ9Ø�2ˆIà¡ 3£˜.ˆE�$‰KØ˜•;˜q•>Ð!r   )Ú__name__)rã   rå   rä   s   f @r   Úc_memorè   �  s   ù€ Ø�:‰:€Dö"ð Ðr   c                 óŠ   € ^V P                  ^	4      V P                  V P                  ^4      ^,          4      ,          ,          # r²   ©ÚcbrtÚgammar:   ©r	   s   &r   Ú
_airyai_C1rî   Ž  s/   € à�—‘˜“˜cŸi™i¨¯©°«
°1­Ó5Õ5Õ6Ð6r   c                 óŠ   € RV P                  ^4      V P                  V P                  ^4      ^,          4      ,          ,          # r}   rê   rí   s   &r   Ú
_airyai_C2rð   ’  s/   € à�—‘˜!“˜sŸy™y¨¯©°«°A­Ó6Õ6Õ7Ð7r   c                 óŒ   € ^V P                  ^^4      V P                  V P                  ^4      ^,          4      ,          ,          # r²   ©Únthrootrì   r:   rí   s   &r   Ú
_airybi_C1rô   –  s2   € à�—‘˜A˜aÓ  3§9¡9¨S¯W©W°Q«Z¸­\Ó#:Õ:Õ;Ð;r   c                 ó~   € V P                  ^^4      V P                  V P                  ^4      ^,          4      ,          # ©rš   rò   rí   s   &r   Ú
_airybi_C2r÷   š  s-   € à�;‰;�q˜Ó˜cŸi™i¨¯©°«
°1­Ó5Õ5Ð5r   c                 óÄ   € V P                   p V P                  ^R4      V P                  R4      ,          ^V P                  ,          ,          pWn         V5#   Yn         i ; i)rš   ú2/3)r   Úpowerrì   r%   )r	   r   rG   s   &  r   Ú_airybi_n2_infrû   ž  sP   € Ø�8‰8€DðØ�I‰I�a˜Ó˜sŸy™y¨Ó/Õ/°°3·6±6µÕ:ˆàŒØˆ2€Iøð �ús   ŽA A ÁAc                 ó˜  € VR 8X  Ed3   V^ 8  d   V# V P                   pV P                  p V ;P                  ^
,          un        V P                  V^,           V,          4      V P                  ^W%,          4      ,          V P                  ,          pV^ 8X  dF   WpP                  ^V^,           ,          V,          4      ,          pWpP                  ^R4      ,          pMNV\        V P                  ^V^,           ,          V,          4      4      ,          pWpP                  ^R4      ,          pW`n        V5V,           # \        h  Y`n        i ; i)ÚZrù   z1/6)Úmpq_1_3r   rì   rú   r%   r~   r<   r>   )r	   r   r   Úntyper§   r'   r   rG   s   &&&&&   r   Ú_airyderiv_0r   ¨  sù   € Ø�…|ØˆqŒ5ØˆHØ�K‰KˆØ�x‰xˆð
	Ø�HŠH˜�N�HØ—	‘	˜1˜Q�3 �'Ó" S§Y¡Y¨q°µÓ%5Õ5¸¿¹Õ>ˆAØ˜ŒzØ—Y‘Y˜q ! A¥#�w q�yÓ)Õ)�Ø—Y‘Y˜q Ó'Õ'‘à”S˜Ÿ™ 1 a¨¥c¥7¨1¥9Ó-Ó.Õ.�Ø—Y‘Y˜q Ó'Õ'�àŒHØˆr�A�vˆô "Ð!øð	 �Hús   «D E ÅE	c           	      ó‚  a aa€ S P                  S4      oV'       d   S P                  V4      w  rEM^ pS P                  S4      '       Eg   S'       dÿ   V'       d¹   XR8X  d²   VR8X  do   SS P                  8X  d'   S P	                  ^4      ^,          ^S,          ,           # SS P
                  8X  d'   S P	                  R4      ^,          ^S,          ,           # VR8  d6   SS P                  8X  d   S# SS P
                  8X  d   RV,          S) ,          # V'       g   SS P                  8X  g   SS P
                  8X  d
   ^S,          # \        R4      hS'       d-   \        ^ \        RS P                  S4      ,          4      4      oM^ oV'       d—   V^8X  d   V VV3R lpS P                  ! V. 3/ VB # S^ 8X  d   \        S SVX^ 4      # V VV3R lpS P                  ! Wd.3/ VB pS P                  S4      '       d)   S P                  V4      '       d   S P                  V4      pV# V VV3R lpS P                  ! V. 3/ VB # )	r.   rý   zessential singularity of Ai(z)r�   c                  ó´  <€ SP                  S
4      ^8”  d³   S;P                  S	,          un        S
R,          p RV ,          pRV ,          ^,          pS;P                  S	,          un        SP                  V4      ) ^SP                  SP                  4      ,          ,          SP                  S
^4      ,          pV.^.. . RR.. V33# S;P                  S	,          un        S
^,          ^	,          p S;P                  S	,          un        \        S4      R,          p\        S4      pVS
.^^.. . . SP                  .V 3pV.^.. . . SP                  .V 3pWg3# )rÂ   r�   r!   rÃ   éþÿÿÿ)r   é   )é   r  )
r6   r   re   r�   r%   ró   rî   rð   Úmpq_5_3rþ   ©r1   r'   rŸ   ÚCÚC1ÚC2r`   ra   r	   Ú	extraprecr   s           €€€r   r+   Úairyai.<locals>.hÝ  s*  ø€ à—7‘7˜1“: ”>Ø—H’H 	Õ)•HØ˜3��A E¨!¥G °°Aµ°aµ¨QØ—H’H 	Õ)•HØŸ™ ›˜ Q s§x¡x°·±Ó'7Õ%7Õ8¸¿¹ÀQÀqÓ9IÕI�AØ˜C   B r¨6°%¨.¸¸AÐ>Ð?Ð?ð —H’H 	Õ)•HØ˜1�˜q��AØ—H’H 	Õ)•HÜ# C›¨3Õ.�BÜ# C›�BØ˜Q˜  1  b¨¨B°·±¨}¸QÐ>�BØ˜˜q˜c " R¨¨C¯K©K¨=¸Ð:�BØ˜6�Mr   c                 ó
  <€ S;P                   S,          un         S^,          ^	,          pS;P                   S,          un         SP                  SP                  SP                  rCpTp^p^V ,
          V,          p^V ,
          V,          p^W,          ,
          p	^S.W,
          V ) .V.WxV	.WV.WxV	.V3p
Tp^V ,
          V,          p^W,          ,
          p^V ,
          V,          p	^SS) .W,
          V ) ^.V.WxV	.WV.WxV	.V3pW«3# rö   )r   rþ   Úmpq_2_3Úmpq_4_3)r   r1   Úq13Úq23Úq43r    r¡   r¢   r£   Úb3r`   ra   r	   r  r   s   &           €€€r   r+   r  ô  s   ø€ Ø—’˜IÕ%•Ø�q•D˜•F�Ø—’˜IÕ%•Ø!Ÿk™k¨3¯;©;¸¿¹˜�Ø�˜1˜ ! A¥# s¥˜b°°!µ°S­y¨B¸Q¸q½u½W¸"Ø˜�V˜a�e a R˜[¨2¨$°°r°
Ø�G˜b B˜Z¨ð+�à�˜A˜a�C �9˜¨¨1­5­ b°a¸µc¸3µY°"Ø˜˜Q˜B�Z !¥%¨!¨¨Q °"°¸¸b°zØ�G˜b B˜Z¨ð+�à�v�r   c                  ó®  <€ SP                  S
4      ^8”  d²   S;P                  S	,          un        S
R,          p RV ,          pRV ,          ^,          pS;P                  S	,          un        SP                  V4      ^SP                  SP                  4      ,          SP                  S
^4      ,          ,          pV.^.. . RR.. V33# S;P                  S	,          un        S
^,          ^	,          p S;P                  S	,          un        \        S4      p\        S4      pV.^.. . . SP                  .V 3pS
V,          .^.. . . SP                  .V 3pWg3# )rÂ   r�   rÃ   r  )r   r  )é   r  )
r6   r   re   r�   r%   ró   rî   rð   r  r  r  s           €€€r   r+   r    s#  ø€ Ø�w‰w�q‹z˜AŒ~ð —’˜IÕ%•Ø�s•F�  a¥˜A¨R°­T°!­V¨Ø—’˜IÕ%•Ø—G‘G˜A“J  #§(¡(¨3¯6©6Ó"2Õ 2°3·;±;¸qÀÓ3CÕ CÕD�Ø˜˜Q˜C  2 u¨U m°B°qÐ9Ð:Ð:à—’˜IÕ%•Ø�q•D˜1•H�Ø—’˜IÕ%•Ü “_�Ü “_�Ø�T˜1˜#˜b  B¨¯© }°QÐ6�Ø˜•d�V˜Q˜C  2 b¨#¯+©+¨°qÐ8�Ø�v�r   r   r  )r4   r‚   ÚisnormalrA   r:   rS   rM   r0   r3   r7   r;   r   Ú_is_real_typer5   r6   ©	r	   r   rC   rD   r   rÿ   r+   rG   r  s	   ff&,    @r   Úairyair  ¾  sÃ  ú€ à�‰�A‹€AßØ×%Ñ% jÓ1‰ˆˆ5àˆà�<‰<˜�?‹?Ÿqß�˜#”Ø�BŒwØ˜Ÿ™”<ØŸ7™7 1›: a�<¨!¨A­#Õ-Ð-Ø˜Ÿ™”=ØŸ7™7 2›; q�=¨1¨Q­3Õ.Ð.Ø�2ŒvØ˜Ÿ™”<Ø�HØ˜Ÿ™”=Ø �7 q b�>Ð)ß�q˜CŸG™G”| q¨C¯H©H¤}Ø�Q•3ˆJäÐ9Ó:Ð:çÜ˜œ3˜s 3§7¡7¨1£:�~Ó.Ó/‰	àˆ	ßØ�Œ6÷"ð$ —=’=  BÑ1¨&Ñ1Ð1à�AŒvÜ# C¨¨A¨u°aÓ8Ð8÷ð —’˜a Ñ/¨Ñ/ˆAØ× Ñ  ×#Ò#¨¯	©	°!¯ªØ—G‘G˜A“J�ØˆH÷	ð& �}Š}˜Q Ñ- fÑ-Ð-r   c           	      óÞ  a aa€ S P                  S4      oV'       d   S P                  V4      w  rEM^ pS P                  S4      '       g¶   S'       d®   V'       df   XR8X  d_   SS P                  8X  d   S# SS P                  8X  d;   VR8X  d
   ^S,          # VR8X  d   \        S 4      # VR8  d   RV,          S) ,          # V'       g.   SS P                  8X  d   S# SS P                  8X  d
   ^S,          # \        R4      hS'       d-   \        ^ \        RS P                  S4      ,          4      4      oM^ oV'       d—   V^8X  d   V VV3R lpS P                  ! V. 3/ VB # S^ 8X  d   \        S SVX^4      # V VV3R lpS P                  ! Wd.3/ VB pS P                  S4      '       d)   S P                  V4      '       d   S P                  V4      pV# V VV3R lpS P                  ! V. 3/ VB # )	r.   rý   zessential singularity of Bi(z)r�   c                  ó$  <€ S;P                   S,          un         S^,          ^	,          p S;P                   S,          un         \        S4      R,          p\        S4      pVS.^^.. . . SP                  .V 3pV.^.. . . SP                  .V 3pW43# )rš   r!   )r   rô   r÷   r  rþ   ©r1   r	  r
  r`   ra   r	   r  r   s        €€€r   r+   Úairybi.<locals>.h;  sŠ   ø€ Ø—’˜IÕ%•Ø�q•D˜1•H�Ø—’˜IÕ%•Ü “_ SÕ(�Ü “_�Ø˜�V˜Q˜q˜E " R¨¨C¯K©K¨=¸Ð:�Ø�T˜1˜#˜b  B¨¯© }°QÐ6�Ø�v�r   c                 ó@  <€ S;P                   S,          un         S^,          ^	,          pS;P                   S,          un         SP                  SP                  SP                  rCpSP                  pSP
                  pTp^p^V ,
          V,          p	^V ,
          V,          p
^W,          ,
          p^S.W,
          V ) .V.WšV.Wx.WšV.V3pTp^V ,
          V,          p	^W,          ,
          p
^V ,
          V,          p^S.W,
          ^V ,
          .V.WšV.Wx.WšV.V3pWÍ3# rö   )r   rþ   r  r  Úmpq_1_6Úmpq_5_6)r   r1   r  r  r  Úq16Úq56r    r¡   r¢   r£   r  r`   ra   r	   r  r   s   &             €€€r   r+   r  H  s  ø€ Ø—’˜IÕ%•Ø�q•D˜•F�Ø—’˜IÕ%•Ø!Ÿk™k¨3¯;©;¸¿¹˜�Ø—k‘k�Ø—k‘k�Ø�˜1˜ ! A¥# s¥˜b°°!µ°S­y¨B¸Q¸q½u½W¸"Ø˜�V˜a�e a R˜[¨2¨$°°r°
Ø�G˜b B˜Z¨ð+�à�˜A˜a�C �9˜¨¨1­5­ b°a¸µc¸3µY°"Ø˜�V˜a�e Q q¥S˜\¨B¨4°"¸°Ø�G˜b B˜Z¨ð+�à�v�r   c                  ó   <€ S;P                   S,          un         S^,          ^	,          p S;P                   S,          un         \        S4      p\        S4      pV.^.. . . SP                  .V 3pSV,          .^.. . . SP                  .V 3pW43# rö   )r   rô   r÷   r  r  r  s        €€€r   r+   r  [  s…   ø€ Ø�HŠH˜	Õ!�HØ�1•�q•ˆAØ�HŠH˜	Õ!�HÜ˜C“ˆBÜ˜C“ˆBØ��q�c˜"˜R  C§K¡K =°Ð2ˆBØ�B•$�˜˜˜B˜r " c§k¡k ]°1Ð4ˆBØ�6ˆMr   r   r  )r4   r‚   r  rA   rS   rû   rM   r0   r3   r7   r;   r   r  r5   r6   r  s	   ff&,    @r   Úairybir$    s™  ú€ à�‰�A‹€AßØ×%Ñ% jÓ1‰ˆˆ5àˆà�<‰<˜�?Š?Ÿqß�˜#”Ø�C—G‘GŒ|Ø�Ø�C—H‘HŒ}Ø˜”7Ø˜Q�3�JØ˜”7Ü)¨#Ó.Ð.Ø�r”6Ø �7 q b�>Ð)ßØ�C—G‘GŒ|Ø�Ø�C—H‘HŒ}Ø˜•s�
äÐ9Ó:Ð:ßÜ˜œ3˜s 3§7¡7¨1£:�~Ó.Ó/‰	àˆ	ßØ�Œ6÷ð —=’=  BÑ1¨&Ñ1Ð1à�AŒvÜ# C¨¨A¨u°aÓ8Ð8÷ð —’˜a Ñ/¨Ñ/ˆAØ× Ñ  ×#Ò#¨¯	©	°!¯ªØ—G‘G˜A“J�ØˆH÷	ð �}Š}˜Q Ñ- fÑ-Ð-r   c                 ó¨  a € R  pR p\        V4      pV^8  d   \        R4      hVR9   g   \        R4      hV^ 8X  d£   V'       dK   S P                  V 3R lV! ^S P                  ,          ^V,          ^,
          ,          ^,          4      ) 4      # S P                  S P                  V! ^S P                  ,          ^V,          ^,
          ,          ^,          4      ) 4      # V^8X  dª   VR8X  d£   V'       dK   S P                  V 3R lV! ^S P                  ,          ^V,          ^,
          ,          ^,          4      ) 4      # S P                  S P
                  V! ^S P                  ,          ^V,          ^,
          ,          ^,          4      ) 4      # V^8X  Ed;   VR8X  Ed1   V'       d’   ^S P                  ,          ^V,          ^,
          ,          ^,          RS P                  ,          ,           pS P                  S P                  ^4      ^,          4      V! V4      ,          pS P                  V 3R	 lV4      # ^S P                  ,          ^V,          ^,
          ,          ^,          RS P                  ,          ,           pS P                  S P                  ^4      ^,          4      V! V4      ,          pS P                  S P
                  V4      # R
# R
# )c                 óZ   € V R,          ^^V ^,          ^0,          ,          ,
          ,          # ©r^   gUUUUUUå?r_   ©Úts   &r   ÚUÚ_airy_zero.<locals>.Uh  ó    € �Q˜•Y  ! Q¨¥T¨"¥W¥+¥Õ.Ð.r   c                 óZ   € V R,          ^^V ^,          ^0,          ,          ,           ,          # r'  r_   r(  s   &r   r)   Ú_airy_zero.<locals>.Ti  r,  r   zk cannot be less than 1z%Derivative should lie between 0 and 1c                 ó(   <€ SP                  V ^4      # r²   )r  ©r   r	   s   &€r   Ú<lambda>Ú_airy_zero.<locals>.<lambda>q  ó   ø€ ¨#¯*©*°Q°q¬/r   Fc                 ó(   <€ SP                  V ^4      # r²   ©r$  r0  s   &€r   r1  r2  v  r3  r   Ty              è?c                 ó(   <€ SP                  V ^4      # r²   r5  r0  s   &€r   r1  r2  }  r3  r   N©r.   r   )	r3   rM   Úfindrootr%   r  r$  Úln2Úexpjpir:   )	r	   r§   r   rC   Úcomplexr*  r)   r)  r¥   s	   f&&&&    r   Ú
_airy_zeror<  f  s  ø€ â.Ú.ÜˆA‹€AØˆ1„uÜÐ2Ó3Ð3Ø˜ÔÜÐ@ÓAÐAØ�„zßØ—<‘<Ô 9Ù�1�S—V‘V•8˜Q˜q�S �UÕ# AÕ%Ó&Ð&ó(ð (à�|‰|˜CŸJ™J©¨1¨S¯V©V­8°Q°qµS¸µUÕ+;¸AÕ+=Ó)>Ð(>Ó?Ð?Ø�„z�g Ô&ßØ—<‘<Ô 9Ù�1�S—V‘V•8˜Q˜q�S �UÕ# AÕ%Ó&Ð&ó(ð (à�|‰|˜CŸJ™J©¨1¨S¯V©V­8°Q°qµS¸µUÕ+;¸AÕ+=Ó)>Ð(>Ó?Ð?Ø�…z�g •oßØ�#—&‘&•˜!˜A�#˜a�%Õ  Õ" U¨3¯7©7¥]Õ2ˆAØ—
‘
˜3Ÿ7™7 1›: a�<Ó(©1¨Q«4Õ/ˆAØ—<‘<Ô 9¸1Ó=Ð=Øˆc�f‰f�H�a˜•c˜!•eÕ˜QÕ  s§w¡w¥Õ.ˆØ�J‰J�s—w‘w˜q“z !•|Ó$¡q¨£tÕ+ˆØ�|‰|˜CŸJ™J¨Ó*Ð*ñ &�zr   c                 ó   € \        V ^ WR4      # ©r.   F©r<  )r	   r   rC   s   &&&r   Ú
airyaizeror@  ‚  s   € ä�c˜1˜a¨UÓ3Ð3r   c                 ó   € \        V ^WV4      # r²   r?  )r	   r   rC   r;  s   &&&&r   Ú
airybizerorB  †  s   € ä�c˜1˜a¨WÓ5Ð5r   c           	      óØ  a aaaa€ S P                  S4      oS P                  S4      '       dP   SS P                  8X  d   S^ 8X  d
   ^S,          # S^8X  d   S# SS P                  8X  d
   ^S,          # \	        R4      hS'       d-   \        ^ \        RS P                  S4      ,          4      4      oM^ oSP                  R4      '       d   \        h S P                  S4      ^8”  dÑ   S^ 8X  d^   \        S P                  S4      4      S P                  ^,          R,          8  d'   V V3R lpS P                  V. S P                  RR7      # S^8X  df   \        S P                  S) 4      4      ^S P                  ,          ^,          R,          8  d'   V V3R lpS P                  V. S P                  RR7      # T TTTT3R	 lpS P                  ! T. 3/ SB #   S P                   d     L1i ; i)
r.   zessential singularityr�   rC   g+‡ÙÎ÷ï?c            	      óT   <€ S P                   S.RR.. . . RO. ^	S^,          ,          33# ©r   r   ))r   rš   )r^   rš   r   ©r%   ©r	   r   s   €€r   r+   Ú_scorer.<locals>.hž  s/   ø€ Ø!Ÿf™f Q˜Z¨¨B¨°°2²oÀbÈÈ1ÈaÍ4ÍÐPÐRÐRr   T)r{   Úforce_seriesc            	      óV   <€ S P                   ) S.RR.. . . RO. ^	S^,          ,          33# rE  rF  rG  s   €€r   r+   rH  ¢  s1   ø€ Ø"Ÿv™v˜g a˜[¨"¨R¨°°B²ÀrÈ!ÈAÈqÍDÍ&ÐQÐSÐSr   c                  óx  <€ SP                   ! S	3/ SB ^,          p RSP                  ,          pS^8X  d   V ^,          p VR,          pS;P                  S,          un        S	^,          ^	,          pS;P                  S,          un        V .^.. . . . ^ 3pVS	.R^.. . ^.SP                  SP                  .V3pW43# )rš   r  r   )r$  r%   r   r  r  )
r¦   r(   r1   r`   ra   r	   r  rD   r§   r   s
        €€€€€r   r+   rH  §  s´   ø€ Ø�JŠJ�qÑ#˜FÑ# AÕ%ˆØˆs�v‰v�IˆØ�AŒ:Ø��FˆAØ��GˆAØ�Š�IÕ�Øˆq�D��FˆØ�Š�IÕ�ØˆS�1�#�r˜2˜r 2 qÐ(ˆØ�ˆU�R˜�F˜B  Q C¨#¯+©+°c·k±kÐ)BÀAÐEˆØˆvˆr   )r4   ÚisinfrA   rS   rM   r0   r3   r7   rà   r>   r<   Úargr%   r;   r   r…   )r	   r   r§   rD   r+   r  s   ffff @r   Ú_scorerrN  Š  sƒ  ü€ Ø�‰�A‹€AØ
‡y�y�‡|‚|Ø�—‘Œ<Ø˜Œz ! A¥#˜:Ø˜Œz !˜8Ø�—‘Œ=Ø�Q•3ˆJÜÐ0Ó1Ð1ßÜ˜œ3˜s 3§7¡7¨1£:�~Ó.Ó/‰	àˆ	Ø‡z�z�,×ÒÜ!Ð!ðØ�7‰7�1‹:˜Œ>Ø˜Œzœc #§'¡'¨!£*›o°·±°qµ¸5Õ0@Ô@öSà—}‘} Q¨°S·X±XÈD�}ÓQÐQØ˜Œzœc #§'¡'¨1¨"£+Ó.°°3·6±6µ¸!µ¸eÕ1CÔCöTà—}‘} Q¨°S·X±XÈD�}ÓQÐQ÷ñ ð �=Š=˜˜BÑ) &Ñ)Ð)øð ×Ñô Ùðús   ÃA9G ÅA+G ÇG)Ç(G)c                 ó   € \        W^ V4      # r¬   ©rN  ©r	   r   rD   s   &&,r   ÚscorergirR  µ  ó   € ä�3˜1˜fÓ%Ð%r   c                 ó   € \        W^V4      # r²   rP  rQ  s   &&,r   ÚscorerhirU  ¹  rS  r   c                 ó´  € W3V9   d3   W1V3,          ^ ,          V P                   8¼  d   W1V3,          ^,          5# V P                  ^V,          ^,           4      pV P                  ^V,           V P                  V,          ,           4      pV P                  ^V,           V P                  V,          ,
          4      p^V,          V P                  V P                  ) V,          V,           V,           ^,          V,
          4      ,          pV P                  V4      '       g)   V P                  V4      '       g   V P                  V4      pV P                   V3W1V3&   V# r¬   )r   Úloggammark   re   r%   rQ   r@   )r	   ÚlÚetaÚ_cacheÚG3ÚG1ÚG2rG   s   &&&&    r   Úcoulombcr^  ½  sú   € à	€x�6Ô˜f s U�m¨AÕ.°#·(±(Ô:Ø˜�u•˜aÕ Ð Ð Ø	�‰�a˜•c˜!•eÓ	€BØ	�‰�a˜•c˜#Ÿ%™% �)•mÓ	$€BØ	�‰�a˜•c˜#Ÿ%™% �)•mÓ	$€BØ	ˆ1�ˆs�w‰w˜Ÿ™˜ � B� rÕ)¨1Õ,¨rÕ1Ó2Õ2€AØ�F‰F�1�IŠI˜Ÿ™ ŸšØ�F‰F�1‹IˆØ—X‘X˜q�M€FˆSˆ5�MØ€Hr   c                 ó.  a aa€ V VV3R  lpS P                   ! WqV.3/ VB pV'       dm   S P                  V4      '       gV   S P                  V4      '       g?   S P                  S4      '       g(   S P                  S4      ^ 8¼  d   S P                  V4      pV# )c                 ór  <€  SP                   S,          pSP                  VS	R R7      pSP                  VRR R7      pSP                  W4      pVS	SP                  V4      .^V ^,           ^.. . ^V ,           W!,          ,           .^V ,          ^,           .V3pV3#   \         d    ^ .R.. . . . ^ 3p T3# i ; i)Tr   r  r   )rk   r$   r^  re   rM   )
rX  rY  ÚjwÚjwzÚjwz2r  r`   r	   r1   r   s
   &&     €€€r   r+   Úcoulombf.<locals>.hÐ  sÊ   ø€ ð	.Ø—‘�q•ˆBØ—(‘(˜2˜q¨�(Ó-ˆCØ—8‘8˜C ¨4�8Ó0ˆDØ—‘˜QÓ$ˆAØ�Q˜Ÿ™ ›Ð%¨¨1¨Q­3° {°B¸¸Q¸q½SÀÅ½Z¸LØ�1•�Q•�˜ðˆBð ˆuˆøô ô 	.Ø��r�d˜B  B¨¨AÐ-‰BØˆuˆð	.ús   ƒBB ÂB6Â5B6)r;   rQ   r@   ©	r	   rX  rY  r   r1   ÚchoprD   r+   rG   s	   f&&ff&,  r   Úcoulombfrg  Ê  si   ú€ ÷
ð 	�Š�a˜C˜Ñ+ FÑ+€Aß�S—V‘V˜A—Y’Y¨¯©°¯ª¸s¿v¹vÀa¿yºyØ	�‰�‹�aŒØ�F‰F�1‹IˆØ€Hr   c                 óÞ   a aa€ SS3V9   d4   VSS3,          ^ ,          S P                   8¼  d   VSS3,          ^,          # V VV3R lpS P                  V^4      pS P                   V3VSS3&   V# )r.   c                  óž  <€ S) ^,
          p SP                   S,          pSP                  ^S,           V,           4      R,          SP                  ^S,           V,
          4      R,          SP                  ^V ,           V,           4      R,          SP                  ^V ,           V,
          4      R,          SR,           ) SP                  ,          .# )r   y              à?r!   y       €      à¿)rk   rW  r%   )Úl2Újetar	   rY  rX  s     €€€r   ÚtermsÚ_coulomb_chi.<locals>.termså  sš   ø€ ØˆR��TˆØ�u‰u�S�yˆØ—‘˜Q˜q�S �XÓ&¨%Õ0Ø�L‰L˜˜1�˜T�Ó" dÕ+Ø�L‰L˜˜2�˜d�Ó# tÕ,Ø�L‰L˜˜2�˜d�Ó# uÕ-Ø��eˆH�S—V‘V�Oð	ð 	r   )r   Úsum_accurately)r	   rX  rY  rZ  rl  rG   s   fff&  r   Ú_coulomb_chiro  á  sm   ú€ à	ˆ3€x�6Ô˜f Q s U�m¨AÕ.°#·(±(Ô:Ø�a˜�e�}˜QÕÐ÷ð 	×Ñ˜5 !Ó$€AØ—X‘X˜q�M€Fˆ1ˆSˆ5�MØ€Hr   c                 ó~  a aa€ S P                  V4      '       g   S P                  V4      pV VV3R  lpS P                  ! WqV.3/ VB pV'       dm   S P                  V4      '       gV   S P                  V4      '       g?   S P                  S4      '       g(   S P                  S4      ^ 8¼  d   S P                  V4      pV# )c                 óÈ  <€ SP                  V ^,          4      '       d   ^ .R.. . . . ^ 3pV3# V ) ^,
          p SP                  W4      pSP                  S,          pSP                  V4      pSP	                  V4      pSP                  W4      pSP                  W14      p	SP                  VS,          4      p
RV,          S,          pWhSW§.R^V ^,           ^^.. . ^V ,           WQ,          ,           .^V ,          ^,           .V3pV) V	SV
.R^V^,           ^.. . ^V,           WQ,          ,           .^V,          ^,           .V3pW,3#   \         d    ^ .R.. . . . ^ 3pT3u # i ; i)r^   r   r  )r5   ro  rk   rZ   rY   r^  re   rM   )rX  rY  r`   rj  Úchira  r¥   r¤   r	  r
  rŸ   r
   ra   r	   r1   r   s   &&           €€€r   r+   Úcoulombg.<locals>.hø  so  ø€ à�9‰9�Q�q•S�>Š>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¨Q­3°°1Ð#5°r¸2Ø�1•�R•V•�˜q �s 1�u˜g qð)ˆBà�"�b˜!˜Q� B¨¨2¨a­4°Ð#3¸¸BØ�2•�b•f•�  "¥ Q¥˜x¨ð+ˆBà�6ˆMøÜô 	Ø��r�d˜B  B¨¨AÐ-ˆBØ�5ŠLð	ús   ¹DE ÅE!Å E!)Ú_imr6   r;   re  s	   f&&ff&,  r   Úcoulombgru  ñ  s…   ú€ ð
 �7‰7�1�:Š:Ø�G‰G�A‹Jˆ÷ð, 	�Š�a˜C˜Ñ+ FÑ+€Aß�S—W‘W˜Q—Z’Z¨#¯'©'°#¯,ª,ÀÇÁÈÇÂØ	�‰�‹�qŒØ�G‰G�A‹JˆØ€Hr   c                óJ  € ^V^,          ,          pV^8X  d?   V'       g7   ^V,          ^V,          ,           ^,
          V P                   ,          ^,          pV^8X  d?   V'       g7   ^V,          ^V,          ,           ^,
          V P                   ,          ^,          pV^8X  d?   V'       d7   ^V,          ^V,          ,           ^,
          V P                   ,          ^,          pV^8X  d?   V'       d7   ^V,          ^V,          ,           ^,
          V P                   ,          ^,          pV'       Eg   XpV^,
          ) ^V,          ,          pRV^,
          ,          ^V,          ^,
          ,          ^^V,          ^,          ,          ,          p	RV^,
          ,          ^SV^,          ,          RV,          ,
          R,           ,          ^^V,          ^,          ,          ,          p
RV^,
          ,          RV^,          ,          RV^,          ,          ,
          RV,          ,           R,
          ,          ^i^V,          ^,          ,          ,          pV'       Ed2   XpV^,           ) ^V,          ,          pR^V^,          ,          ^RV,          ,           ^	,
          ,          ^^V,          ^,          ,          ,          p	R^SV^,          ,          RV^,          ,          ,           RV,          ,
          R	,           ,          ^^V,          ^,          ,          ,          p
RRV^,          ,          R
V^,          ,          ,           RV^,          ,          ,
          RV,          ,           R,
          ,          ^i^V,          ^,          ,          ,          pXXX	X
X.pTpRp\        ^\        V4      4       FQ  p\        WÏ,          4      \        WÏ^,
          ,          4      8  d   WÜV,          ,          pK@  \        WÏ,          4      pKS  	  X\        V4      ^,
          8X  d   \        VR,          4      pWÞ3# )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.
iÖ  iÃ  i%  iÿX iO2 iuÈ_ i  iß  iÑ  i,† i il"q i»QY g        éüÿÿÿiàÿÿÿiÀÿÿÿr   )r%   r9   Úlenr<   )r	   ÚkindÚprimerG   rX   rŸ   r€   Ús1Ús2Ús3Ús4Ús5rl  r¥   ÚerrÚis   &&&&&           r   Úmcmahonr‚    sÝ  € ð 	
ˆ!ˆQ�$�€AØˆq„yŸ Q q¥S¨¨1­¥W¨Q¥Y°·±Õ$6°qÕ$8 Øˆq„yŸ Q q¥S¨¨1­¥W¨Q¥Y°·±Õ$6°qÕ$8 Øˆq„y—U  1¥ Q q¥S¥¨¥¨C¯F©FÕ 2°1Õ 4˜AØˆq„y—U  1¥ Q q¥S¥¨¥¨C¯F©FÕ 2°1Õ 4˜Aßˆ5ØˆØ��sˆV�Q�q•S�\ˆØ��1•�X�q˜•s˜2•vÕ  1 Q¥3¨¥(¥
Õ+ˆØ�!�A•#�Y˜˜1˜a�4�  A¥� dÕ*Õ+¨R°°1µ°qµ­[Õ9ˆØ�!�A•#�Y˜˜Q �T�	 &¨¨A­¥+Õ-¨g°a­iÕ7¸Õ?Õ@À#ÀqÈÅsÈQÅhÅ,ÕOˆß€uØˆØ��sˆV�Q�q•S�\ˆØ��1�a•4•˜˜1�•˜Q•Õ  A a¥C¨!¥8¥Õ,ˆØ�"�Q˜•T•'˜$˜q !�t�)Õ# D¨¥FÕ*¨4Õ/Õ0°"°a¸µc¸AµXµ+Õ>ˆØ�$�q˜!•t•)˜F 1 a¥4�KÕ'¨°°1µ­Õ4°W¸QµYÕ>¸wÕFÕGÈÈaÐPQÍcÐTUÍXÍÕVˆØ��2�b˜Ð€EØ
€AØ
€CÜ�1”S˜“ZÖ ˆÜˆu�x‹=œ3˜u q¥S�z›?Ô*Ø�q•�MŠAä�e•h“-ŠCñ	 !ð
 	ŒC�‹J�q�LÔÜ�%˜•)‹nˆØˆ6€Mr   c                ó°  € V^8  d   \        R4      hV^,           p. p. p V P                  W#V4      pV Uu. uF  q€P                  V! V4      4      NK  	  pp\        V^,
          4       U	u. uF=  p	Wy,          Wy^,           ,          ,          R8X  g   K&  Wi,          Wi^,           ,          3NK?  	  p
p	\	        V
4      V8X  d   V
# V^,          pK®  u upi u up	i )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.
zn cannot be less than 1r   )rM   ÚlinspaceÚsignr9   rx  )r	   rã   r   r€   r   ÚNÚpointsÚsignsr
   r�  Úok_intervalss   &&&&&      r   Úgeneralized_bisectionrŠ  ;  sÈ   € ð 	ˆ1„uÜÐ2Ó3Ð3Ø	ˆ!�€AØ€FØ€EØ
Ø—‘˜a !Ó$ˆÙ)/Ó0© A—‘™!˜A›$–©ˆÐ0Ü9>¸qÀ½s¼ó *¹°AØ�x˜ �c�
Õ" bÑ(ô 0˜� 6¨A­#¥;Ó/¹ˆð *äˆ|Ó Ô!ØÐØˆa�CŠùò 1ùò*s   ·CÁ,#CÂCc                 ó*   € V P                  WR RR7      # )ÚillinoisF)ÚsolverÚverify)r8  )r	   rã   Úabs   &&&r   Úfind_in_intervalr�  Q  s   € Ø�<‰<˜ j¸ˆ<Ó?Ð?r   g{®Gáz„?c           	      óŒ  a a€ S P                   p\        VS P                  S4      S P                  V4      4      ^
,           p VS n         S P                  S4      o\	        V4      p\	        V4      pS^ 8  d   \        R4      hV^8  d   \        R4      hVR9   g   \        R4      hV^8X  d   V'       d	   V V3R lp	MV V3R lp	V^8X  d   V'       d	   V V3R lp	MV V3R lp	V^8X  d‰   V'       d�   V^8X  dz   S^ 8X  d   S P                  VS n         # S^8:  dY   ^S P                  S^S,           ,          S^,           ,          4      ,          p
\        S X	V
^
,          ^V
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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	   rG   rX   rC   s   &&&&r   Úbesseljzeror«  ‰  s   € ô` ˜˜Q 
¨qÓ1Ð1Ð1r   c                ó    € \        V ^W1V4      5# )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

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+ó ð
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