+
    ‰°³j´F  ã                   ó  € ^RI Ht  ! R R]4      tR tR 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 4       t]R 4       t]R 4       t]R 4       t]R 4       t]R5R l4       t]R 4       t]R 4       t]R 4       t]R 4       t]R 4       t]R 4       t]R6R l4       t]R7R l4       t]R 4       t]R 4       t]R 4       t]R  4       t ]R! 4       t!]R" 4       t"]R# 4       t#]R8R% l4       t$]R& 4       t%]R' 4       t&]R( 4       t']R) 4       t(R* t)^ R$I*t*^ R$I+t+R+ t,R, t-]R6R- l4       t.]R5R. l4       t/R7R/ lt0]R0 4       t1]R1 4       t2]R2 4       t3]R7R3 l4       t4]R7R4 l4       t5R$# )9é   )Úxrangec                   óŠ   a € ] tR t^t o Rt/ tRtR t]R 4       t	R t
R tR tR tR tR	 tR
 tR tR tR tR tR tRtV tR# )ÚSpecialFunctionszï
This class implements special functions using high-level code.

Elementary and some other functions (e.g. gamma function, basecase
hypergeometric series) are assumed to be predefined by the context as
"builtins" or "low-level" functions.
c                ó  € V P                   pVP                   F*  pVP                  V,          w  r4VP                  W#V4       K,  	  V P                  R4      V n        V P                  R4      V n        V P                  R4      V n        V P                  R4      V n        V P                  R4      V n        V P                  R4      V n	        V P                  R4      V n
        V P                  R4      V n        V P                  R4      V n        V P                  R4      V n        V P                  R4      V n        V P                  R4      V n        V P                  R4      V n        V P                  R4      V n        V P                  R4      V n        V P                  R4      V n        V P                  R 4      V n        / V n        V P,                  P/                  RRRRRRRRR	R
RRRR/4       V P1                  V P2                  4      V n        R# )!é   ÚphaseÚargÚ	conjugateÚconjÚnthrootÚrootÚ	polygammaÚpsiÚhurwitzÚzetaÚ	fibonacciÚfibÚ	factorialÚfacN)r   r   )é    r   )r   r   )é   r   )r   é   )r   é   )r   r   )é   r   )r   r   )é   r   )r   r   )r   r   )r   r   )r   r   )r   é   )r   r   )r   r   )Ú	__class__Údefined_functionsÚ_wrap_specfunÚ_mpqÚmpq_1Úmpq_0Úmpq_1_2Úmpq_3_2Úmpq_1_4Úmpq_1_16Úmpq_3_16Úmpq_5_2Úmpq_3_4Úmpq_7_4Úmpq_5_4Úmpq_1_3Úmpq_2_3Úmpq_4_3Úmpq_1_6Úmpq_5_6Úmpq_5_3Ú_misc_const_cacheÚ_aliasesÚupdateÚmemoizeÚzetazeroÚzetazero_memoized)ÚselfÚclsÚnameÚfÚwraps   &    Ú[/home/luca/screen-checking/.venv/lib/python3.14/site-packages/mpmath/functions/functions.pyÚ__init__ÚSpecialFunctions.__init__   sª  € Ø�n‰nˆØ×)Ô)ˆDØ×+Ñ+¨DÕ1‰GˆAØ×Ñ˜d tÖ,ñ *ð —Y‘Y˜uÓ%ˆŒ
Ø—Y‘Y˜uÓ%ˆŒ
Ø—y‘y Ó'ˆŒØ—y‘y Ó'ˆŒØ—y‘y Ó'ˆŒØŸ	™	 &Ó)ˆŒØŸ	™	 &Ó)ˆŒØ—y‘y Ó'ˆŒØ—y‘y Ó'ˆŒØ—y‘y Ó'ˆŒØ—y‘y Ó'ˆŒØ—y‘y Ó'ˆŒØ—y‘y Ó'ˆŒØ—y‘y Ó'ˆŒØ—y‘y Ó'ˆŒØ—y‘y Ó'ˆŒØ—y‘y Ó'ˆŒà!#ˆÔà�‰×ÑØ�eØ˜&Ø˜Ø˜%Ø˜ð
 ˜%Ø˜%ð
ô 	ð "&§¡¨d¯m©mÓ!<ˆÖó    c                ó   € \        WV4       R # ©N)Úsetattr)r9   r:   r;   r<   s   &&&&r=   r   ÚSpecialFunctions._wrap_specfun=   s   € ä�˜1Ör@   c                ó   € \         hrB   ©ÚNotImplementedError©ÚctxÚnÚzs   &&&r=   Ú_besseljÚSpecialFunctions._besseljD   s   € Ô#6Ð6r@   c                ó   € \         hrB   rF   ©rI   rK   s   &&r=   Ú_erfÚSpecialFunctions._erfE   s   € Ô/Ð/r@   c                ó   € \         hrB   rF   rO   s   &&r=   Ú_erfcÚSpecialFunctions._erfcF   ó   € Ô0Ð0r@   c                ó   € \         hrB   rF   )rI   rK   Úas   &&&r=   Ú_gamma_upper_intÚ!SpecialFunctions._gamma_upper_intG   s   € Ô+>Ð%>r@   c                ó   € \         hrB   rF   rH   s   &&&r=   Ú_expint_intÚSpecialFunctions._expint_intH   s   € Ô&9Ð 9r@   c                ó   € \         hrB   rF   ©rI   Úss   &&r=   Ú_zetaÚSpecialFunctions._zetaI   rU   r@   c                ó   € \         hrB   rF   )rI   r_   rW   rJ   ÚderivativesÚreflects   &&&&&&r=   Ú_zetasum_fastÚSpecialFunctions._zetasum_fastJ   s   € ÔATÐ;Tr@   c                ó   € \         hrB   rF   rO   s   &&r=   Ú_eiÚSpecialFunctions._eiK   ó   € Ô.Ð.r@   c                ó   € \         hrB   rF   rO   s   &&r=   Ú_e1ÚSpecialFunctions._e1L   rj   r@   c                ó   € \         hrB   rF   rO   s   &&r=   Ú_ciÚSpecialFunctions._ciM   rj   r@   c                ó   € \         hrB   rF   rO   s   &&r=   Ú_siÚSpecialFunctions._siN   rj   r@   c                ó   € \         hrB   rF   r^   s   &&r=   Ú_altzetaÚSpecialFunctions._altzetaO   s   € Ô 3Ð3r@   )r2   r"   r!   r&   r#   r,   r%   r/   r-   r'   r$   r)   r.   r(   r1   r+   r0   r*   r7   NgËPÊÿÿï?)Ú__name__Ú
__module__Ú__qualname__Ú__firstlineno__Ú__doc__r   ÚTHETA_Q_LIMr>   Úclassmethodr   rL   rP   rS   rX   r[   r`   re   rh   rl   ro   rr   ru   Ú__static_attributes__Ú__classdictcell__)Ú__classdict__s   @r=   r   r      s\   ø‡ € ñð Ðð
 €Kò(=ðV ñó ðò 7Ú/Ú0Ú>Ú9Ú0ÚTÚ.Ú.Ú.Ú.ß3Ð3r@   r   c                 óD   € V R 3\         P                  V P                  &   V # )T©r   r   rw   ©r;   s   &r=   Údefun_wrappedr„   Q   s    € Ø56¸°WÔ×&Ñ& q§z¡zÑ2Ø€Hr@   c                 óD   € V R 3\         P                  V P                  &   V # ©Fr‚   rƒ   s   &r=   Údefunr‡   U   s    € Ø56¸°XÔ×&Ñ& q§z¡zÑ2Ø€Hr@   c                 ó<   € \        \        V P                  V 4       V # rB   )rC   r   rw   rƒ   s   &r=   Údefun_staticr‰   Y   s   € ÜÔ˜aŸj™j¨!Ô,Ø€Hr@   c                 óF   € V P                   V P                  V4      ,          # rB   )ÚoneÚtanrO   s   &&r=   Úcotr�   ]   ó   € ØŸ™ #§'¡'¨!£*Õ,Ð,r@   c                 óF   € V P                   V P                  V4      ,          # rB   )r‹   ÚcosrO   s   &&r=   Úsecr‘   `   rŽ   r@   c                 óF   € V P                   V P                  V4      ,          # rB   )r‹   ÚsinrO   s   &&r=   Úcscr”   c   rŽ   r@   c                 óF   € V P                   V P                  V4      ,          # rB   )r‹   ÚtanhrO   s   &&r=   Úcothr—   f   ó   € ØŸ™ 3§8¡8¨A£;Õ.Ð.r@   c                 óF   € V P                   V P                  V4      ,          # rB   )r‹   ÚcoshrO   s   &&r=   Úsechr›   i   r˜   r@   c                 óF   € V P                   V P                  V4      ,          # rB   )r‹   ÚsinhrO   s   &&r=   Úcschrž   l   r˜   r@   c                 ó|   € V'       g   V P                   R ,          # V P                  V P                  V,          4      # )ç      à?)ÚpiÚatanr‹   rO   s   &&r=   Úacotr£   o   s*   € çØ�v‰v˜�|Ðà�x‰x˜Ÿ™ !�Ó$Ð$r@   c                 óF   € V P                  V P                  V,          4      # rB   )Úacosr‹   rO   s   &&r=   Úasecr¦   v   ó   € ØŸ™ #§'¡'¨A¥+Ó.Ð.r@   c                 óF   € V P                  V P                  V,          4      # rB   )Úasinr‹   rO   s   &&r=   Úacscrª   y   r§   r@   c                 ó|   € V'       g   V P                   R ,          # V P                  V P                  V,          4      # )y              à?)r¡   Úatanhr‹   rO   s   &&r=   Úacothr­   |   s*   € çØ�v‰v˜�}Ðà�y‰y˜Ÿ™ 1�Ó%Ð%r@   c                 óF   € V P                  V P                  V,          4      # rB   )Úacoshr‹   rO   s   &&r=   Úasechr°   „   ó   € ØŸ)™) C§G¡G¨a¥KÓ0Ð0r@   c                 óF   € V P                  V P                  V,          4      # rB   )Úasinhr‹   rO   s   &&r=   Úacschr´   ‡   r±   r@   c                 óø   € V P                  V4      pV'       d   V P                  V4      '       d   V# V P                  V4      '       d!   V^ 8”  d   V P                  # V P                  ) # V\	        V4      ,          # ©r   )ÚconvertÚisnanÚ_is_real_typer‹   Úabs©rI   Úxs   &&r=   Úsignr½   Š   s]   € à�‰�A‹€Aß�—	‘	˜!—’ØˆØ
×Ñ˜×ÒØˆqŒ5Ø—7‘7ˆNà—G‘G�8ˆOØŒs�1‹v�:Ðr@   c                 ó˜   € V^8X  d   V P                  V4      # V P                  V4      pV P                  V4      pV P                  W4      # ©r   )Úagm1r·   Ú_agm)rI   rW   Úbs   &&&r=   ÚagmrÃ   –   s?   € àˆA„vØ�x‰x˜‹{ÐØ�‰�A‹€AØ�‰�A‹€AØ�8‰8�A‹>Ðr@   c                 ó”   € V P                  V4      '       d
   ^V,          # V'       g
   V^,           # V P                  V4      V,          # r¿   )Úisinfr“   r»   s   &&r=   ÚsincrÆ   ž   s6   € à
‡y�y�‡|‚|Ø��sˆ
ßØ��sˆ
Ø�7‰7�1‹:�a�<Ðr@   c                 ó¶   € V P                  V4      '       d
   ^V,          # V'       g
   V^,           # V P                  V4      V P                  V,          ,          # r¿   )rÅ   Úsinpir¡   r»   s   &&r=   ÚsincpirÉ   ¦   s?   € à
‡y�y�‡|‚|Ø��sˆ
ßØ��sˆ
Ø�9‰9�Q‹<˜Ÿ™ �Õ"Ð"r@   c                 óÌ   a a€ S'       g   S P                   # S P                  S4      S P                  ) 8  d   SR S^,          ,          ,           # S P                  V V3R l^4      # )r    c                  ó<   <€ \        S P                  S4      R.4      # ©r   éÿÿÿÿ)ÚiterÚexpr»   s   €€r=   Ú<lambda>Úexpm1.<locals>.<lambda>·   s   ø€ ¤d¨C¯G©G°A«J°r¨?Ô&;r@   )ÚzeroÚmagÚprecÚsum_accuratelyr»   s   ffr=   Úexpm1rÖ   ¯   sK   ù€ çØ�x‰xˆà
‡w�wˆqƒz�S—X‘X�IÔØ�3�q˜!•t•8�|Ðà×ÑÕ;¸AÓ>Ð>r@   c                 ó  € V'       g   V P                   # V P                  V4      V P                  ) 8  d   VR V^,          ,          ,
          # V P                  V P	                  ^V^V P                  ,          R7      4      # )r    ©rÔ   )rÒ   rÓ   rÔ   ÚlogÚfaddr»   s   &&r=   Úlog1prÛ   ¹   s\   € çØ�x‰xˆØ
‡w�wˆqƒz�S—X‘X�IÔØ�3�q˜!•t•8�|ÐØ�7‰7�3—8‘8˜A˜q q¨¯©¥z�8Ó2Ó3Ð3r@   c                 óÐ  aa€ V P                   pV P                  pSS,          V,
          pV! V4      pVR8”  d   V# V'       g)   S'       d   SR9   d   V P                  S4      '       d   V# SV,
          pV! S4      pV P                  S4      p	Wƒ! V	4      ,           V P                  ) 8  d&   V	S,          V	S,          ^,          ^,          ,           # V P                  VV3R l^4      # )é   c                  ó,   <€ \        S S,          R.4      # rÌ   )rÎ   )r¼   Úys   €€r=   rÐ   Úpowm1.<locals>.<lambda>Õ   s   ø€ ¤d¨A¨q­D°"¨:Ô&6r@   iøÿÿÿ)r   rÍ   y              ð?y       €      ð¿)rÓ   r‹   ÚisintÚlnrÔ   rÕ   )
rI   r¼   rß   rÓ   r‹   ÚwÚMÚx1ÚmagyÚlnxs
   &ff       r=   Úpowm1rè   Á   sº   ù€ à
�'‰'€CØ
�'‰'€CØ	ˆ1�ˆs�
€AÙˆA‹€Aàˆ2„vØˆçß�qÐ,Ô,°·±¸1·²ØˆHØ	
ˆS�€BÙˆq‹6€DØ
�&‰&�‹)€Càˆc�#‹h…˜#Ÿ(™(˜Ô"Ø�1�u˜˜A� •z !•|Õ#Ð#à×ÑÕ6¸Ó:Ð:r@   c                 óp  € \        V4      p\        V4      pW,          pV'       g   V P                  # ^V,          V8X  d   V P                  ) # ^V,          V8X  d   V P                  # ^V,          ^V,          8X  d   V P                  ) # V P                  ^V P	                  V4      ,          V,          4      # )r   )Úintr‹   ÚjÚexpjpiÚmpf)rI   ÚkrJ   s   &&&r=   Ú_rootof1rï   ×   sŠ   € äˆA‹€AÜˆA‹€AØ…F€AßØ�w‰wˆØ	
ˆ1��ŒØ—‘ˆxˆØ	
ˆ1��ŒØ�u‰uˆØ	
ˆ1���!•ŒØ—‘ˆvˆØ�:‰:�a˜Ÿ™ ›
•l 1•nÓ%Ð%r@   c                 óú  € \        V4      pV P                  V4      pV'       d½   V^,          '       dW   ^V,          V^,
          8X  dB   V P                  V4      '       g+   V P                  V4      ^ 8  d   V P	                  V) V4      ) # V P
                  p V ;P
                  ^
,          un        V P	                  W^ 4      V P                  W24      ,          pW@n        V5# V P                  W4      #   Y@n        i ; ir¿   )rê   r·   ÚimÚrer   rÔ   rï   Ú_nthroot)rI   r¼   rJ   rî   rÔ   Úvs   &&&&  r=   r   r   æ   s¾   € äˆA‹€AØ�‰�A‹€Aßà��EŒE�a˜•c˜Q˜q�S”j¨3¯6©6°!¯9ª9¸3¿6¹6À!»9Àq¼=Ø—H‘H˜a˜R “OÐ#Ð#à�x‰xˆð	Ø�HŠH˜�N�HØ—‘˜˜qÓ! C§L¡L°Ó$6Õ6ˆAàŒHØˆrˆ	Ø�<‰<˜ÓÐøð �Hús   ÂAC2 Ã2C:c                 óš  € V P                   pV P                  p V ;P                  ^
,          un        V'       d9   \        V4       Uu. uF"  qS! WQ4      ^8X  g   K  V P                  WQ4      NK$  	  ppM(\        V4       Uu. uF  qPP                  WQ4      NK  	  ppW@n        V Uu. uF  qw5NK  	  up# u upi u upi   Y@n        i ; iu upi ©é
   )Ú_gcdrÔ   Úrangerï   )rI   rJ   Ú	primitiveÚgcdrÔ   rî   rô   r¼   s   &&&     r=   Ú	unitrootsrü   ø   s¦   € à
�(‰(€CØ�8‰8€DðØ�Š�B��ßÜ,1°!¬HÓF©H q¸¸A»ÀA¹Ô"�—‘˜aÖ"©HˆAÐFˆAô -2°!¬HÓ5©H q—‘˜aÖ"©HˆAÐ5àŒÙ‹?™�1‹B™‰?Ðùò Gùò 6øà�üÚs:   š B= »B= ÁB3ÁB3Á0B= Â B8ÂB= Â%CÂ3
B= Â=Cc                 óŠ   € V P                  V4      pV P                  V4      pV P                  V4      pV P                  W24      # rB   )r·   Ú_reÚ_imÚatan2)rI   r¼   rò   rñ   s   &&  r=   r	   r	     s8   € à�‰�A‹€AØ	�‰�‹€BØ	�‰�‹€BØ�9‰9�RÓÐr@   c                 ó6   € \        V P                  V4      4      # rB   )rº   r·   r»   s   &&r=   Úfabsr    s   € äˆs�{‰{˜1‹~ÓÐr@   c                 ód   € V P                  V4      p\        VR 4      '       d   VP                  # V# )Úreal)r·   Úhasattrr  r»   s   &&r=   rò   rò     s*   € à�‰�A‹€AÜˆq�&×ÒØ�v‰vˆØ€Hr@   c                 óx   € V P                  V4      p\        VR 4      '       d   VP                  # V P                  # )Úimag)r·   r  r  rÒ   r»   s   &&r=   rñ   rñ     s.   € à�‰�A‹€AÜˆq�&×ÒØ�v‰vˆØ�8‰8€Or@   c                 ól   € V P                  V4      p VP                  4       #   \         d    Tu # i ; irB   )r·   r
   ÚAttributeErrorr»   s   &&r=   r   r      s4   € à�‰�A‹€AðØ�{‰{‹}ÐøÜô ØŠðús   “# £3²3c                 óF   € V P                  V4      V P                  V4      3# rB   )r  r	   rO   s   &&r=   Úpolarr  (  s   € à�H‰H�Q‹K˜Ÿ™ ›Ð$Ð$r@   c                 óL   € WP                   ! V P                  V4      !  ,          # rB   )ÚmpcÚcos_sin)rI   ÚrÚphis   &&&r=   Úrectr  ,  s   € à�wŠw˜Ÿ™ CÓ(Ñ)Õ)Ð)r@   Nc                 ó¤   € Vf   V P                  V4      # V P                  ^,           pV P                  WR7      V P                  W#R7      ,          # )NrØ   )râ   rÔ   )rI   r¼   rÂ   Úwps   &&& r=   rÙ   rÙ   0  sC   € à‚yØ�v‰v�a‹yÐØ	�‰�B�€BØ�6‰6�!ˆ6Ó §¡ q Ó 2Õ2Ð2r@   c                 ó&   € V P                  V^
4      # rö   )rÙ   r»   s   &&r=   Úlog10r  7  s   € à�7‰7�1�b‹>Ðr@   c                 óP   € V P                  V4      V P                  V4      ,          # rB   )r·   )rI   r¼   rß   s   &&&r=   Úfmodr  ;  s   € à�;‰;�q‹>˜CŸK™K¨›NÕ*Ð*r@   c                 ó&   € WP                   ,          # rB   ©Údegreer»   s   &&r=   Údegreesr  ?  ó   € à�z‰z�>Ðr@   c                 ó&   € WP                   ,          # rB   r  r»   s   &&r=   Úradiansr  C  r  r@   c                 ó˜  € V'       g   V'       g   V# V P                   V,           # WP                  8X  d<   V^ 8X  d   V# V^V,          V P                  ,          V P                  ,          ,           # WP                   8X  d;   V) ^V,          ^,           V P                  ,          V P                  ,          ,           # V P	                  V4      # r¶   )ÚninfÚinfr¡   rë   râ   )rI   rK   rî   s   &&&r=   Ú_lambertw_specialr"  G  s�   € çßØˆHØ�x‰x˜!�|ÐØ�G‰G„|Ø�Œ6ØˆHà�q˜•s˜3Ÿ6™6•z #§%¡%Õ'Õ'Ð'Ø�H‰H„}Ø��q˜•s˜1•u˜cŸf™f•n S§U¡UÕ*Õ*Ð*à�6‰6�!‹9Ðr@   c                 ó>  € ^ p\        V R4      '       dB   \        V P                  4      pV P                  pV'       d   RV^ 8  ,          p\        V4      pM\        V 4      pRp^ pV'       g   Rp\	        W44      p V^ 8X  Ed³   RTu;8  d	   R8  Ed8   M EM3RTu;8  d	   R8  Ed$   M EMV'       dy   VR8”  d   RRV R,
          ,          ,           # VR8”  d   RRV R,
          ,          ,           # VR8  d   RRV R,
          ,          ,           # VR8  d   RRV R,
          ,          ,           # VR8  d6   V^ 8¼  d   R R!V ^,           ,          ,           # R"R#V ^,           ,          ,           # R$pV'       g	   W58”  d   Tp VR%8  d2   RR	W,
          R,          ,          ,           R
W,
          ,          ,
          # VR8  d   V # RRV ,          ,           # V'       g6   VR8”  d/   \
        P                  ! V4      p\
        P                  ! V4      pEMZ\        P                  ! V 4      p\        P                  ! V4      pEM,VR8X  Ed%   R$pV'       g   YSu;8  d   R8  d   M MTp V^ 8¼  dK   VR8  dD   R&Tu;8  d   R%8  d5   M M1RR	W,
          R,          ,          ,
          R
W,
          ,          ,
          # V'       gH   R%Tu;8:  d   R8  d9   M M5\
        P                  ! V) 4      pV\
        P                  ! V) 4      ,
          # VR8X  d.   V'       g&   VR8  d   \        P                  ! V 4      R,
          pM\        P                  ! V 4      R,
          p\        P                  ! V4      pXX,
          Wv,          ,           Ww^,
          ,          ^V^,          ,          ,          ,           # )'r   r  g        g      @ç      ð?g      @g      Ð?r    gš™™™™™É?gµÿ¶õ4§@g}tp¹¸þü?g333333Ó?gš™™™™™¹?y        -DTû!	@y        -DTû!@rÍ   g      Àg      ð¿yÕxé&1ì?¤p=
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×£põ?yçû©ñÒMæ¿`åÐ"Ûùâ¿y'1¬ZÔ¿q=
×£põ¿yçû©ñÒMæ¿`åÐ"Ûùâ?g2ï,6V‹×¿gš™™™™™É¿g333333ã¿)r  Úfloatr  r  ÚcomplexÚmathrÙ   Úcmath)rK   rî   Ú	imag_signr¼   rß   r  ÚL1ÚL2s   &&      r=   Ú_lambertw_approx_hybridr,  Z  sª  € Ø€IÜˆq�&×ÒÜ�!—&‘&‹MˆØ�F‰FˆßØ  Q¡�ˆIÜ�!‹H‰ä�!‹HˆØˆØˆ	çØˆÜ�‹€AØˆA…vØ�!Ž>�c�>‰>˜d Qžn¨Ÿn™nßà�t”8 \°lÀQÈ	Å]Õ5SÕ$SÐSØ�t”8 \°lÀQÈ	Å]Õ5SÕ$SÐSØ�u”9 l°|ÀaÈÅmÕ6TÕ%TÐTØ�u”9 l°|ÀaÈÅmÕ6TÕ%TÐTà�4ŒxØ ”>Ø(¨]¸Q¸q½SÕ,AÕAÐAà(¨]¸Q¸q½SÕ,AÕAÐAà"ˆAß 1¤5Ø�à�4ŒxØÐ,¨a­c°C­ZÕ7Õ7Ð:JÈAÍCÕ:PÕPÐPà�3Œw˜q˜à˜˜Q�•;Ðß˜q 3œwÜ—’˜!“ˆB¤4§8¢8¨B£<šbä—’˜1“ˆB¤E§I¢I¨b£MšrØ	
ˆb�àˆß˜qž{ sŸ{ØˆAØ˜ŒN  C¤¨D°1®O°t¯OØÐ(¨!­#°­Õ3Õ3Ð6FÈÍÕ6LÕLÐLß˜t qž¨3ŸÜ—’˜1˜"“ˆBØœŸš " ›Õ%Ð%à˜BŒ¯¨q°3¬wÜ—Y’Y˜q“\Ð$7Õ7‘ä—Y’Y˜q“\Ð$7Õ7�Ü—’˜2“ˆBØ��7�R•U�?˜R A¥�Y¨¨"¨a­%­Õ0Õ0Ð0r@   c                ó,  a aaa€ S P                  S4      pR
Tu;8  d	   R8  EdÁ   M EM¼RTu;8  d	   R8  Ed­   M EM¨V^8  Ed…   \        SR,           4      R8  Edm   V^ 8X  g=   VR8X  d   S P                  S4      ^ 8¼  g    V^8X  EdA   S P                  S4      ^ 8  Ed*   S P                  V V3R l4      pS P                  V4      ) pS ;P                  V,          un        S P                  ^S P                  S,          ^,           ,          4      pS ;P                  V,          un        ^ S P                  R4      ^S P                  ^4      /o^ S P                  ^4      ^S P                  R4      /pV^ 8w  d   V) pS P                  p	\        \        ^V4      4       EF  oSS9  d´   S P                  VV3R l\        ^S4       4       4      VS&   S^,
          SS^,
          ,          ^,          VS^,
          ,          ^,          ,           ,          S^,           ,          VS,          ^,          ,
          SS^,
          ,          S^,           ,          ,
          SS&   SS,          VS,          ,          p
Wš,          p	S P                  V
4      V) 8  d   V	R3u # S^,          oEK  	  S ;P                  V^,          ,          un        V	R3# V^ 8X  g   VR8X  d   \        SV4      R3# V^ 8X  d=   VR8  d   S^S,
          ,          R3# S P                  S4      pS P                  V4      pM­VR8X  df   S P                  S4      '       gO   RS P                  S4      u;8  d   ^ 8  d1   M M-S P                  S) 4      pVS P                  V) 4      ,
          R3# S P                  S4      R	S P                  ,          V,          ,           pS P                  V4      pW¼,
          WË,          ,           WÌ^,
          ,          ^V^,          ,          ,          ,           R3# )z“
Return rough approximation for W_k(z) from an asymptotic series,
sufficiently accurate for the Halley iteration to converge to
the correct value.
i„  iè  gï,6V‹×?gš™™™™™©?c                  ó*   <€ SS P                  R4      .# rÌ   )rÏ   rO   s   €€r=   rÐ   Ú"_lambertw_series.<locals>.<lambda>¥  s   ø€ °A°s·w±w¸r³{Ñ3Cr@   c              3   ón   <"  € T F*  pSV,          SS^,           V,
          ,          ,          x € K,  	  R# 5i)r   N© )Ú.0rë   ÚlÚus   & €€r=   Ú	<genexpr>Ú#_lambertw_series.<locals>.<genexpr>µ  s'   øé € Ð'K¹{¸!¨¨!­¨Q¨q°­s°1­u­X¯ª»{ùs   ƒ25TFy               @iöÿÿÿiüÿÿrÍ   gï,6V‹×¿)rÓ   rº   rÿ   rÕ   rÔ   ÚsqrtÚerí   rÒ   r   ÚmaxÚfsumr,  râ   rþ   r¡   )rI   rK   rî   ÚtolÚmagzÚdeltaÚcancellationÚprW   r_   Útermr*  r+  r3  r4  s   ff&&         @@r=   Ú_lambertw_seriesrA  ™  s  û€ ð �7‰7�1‹:€DØˆdÖ�S×Ñ˜u qÖ/¨4×/Ñ/à�!�8œ˜AÐ.Õ.Ó/°$Õ6Ø�AŒv˜!˜rœ' c§g¡g¨a£j°A¤oØ˜q�& c§g¡g¨a£j°1¥nØ×*Ñ*Õ+CÓD�Ø #§¡¨£˜�Ø—’˜LÕ(•à—H‘H˜Q §¡ a¥¨¥	�]Ó+�Ø—’˜LÕ(•Ø�s—w‘w˜r“{ A c§g¡g¨a£jÐ1�Ø�s—w‘w˜q“z 1 S§W¡W¨R£[Ð1�Ø˜”6Ø˜�AØ—H‘H�ô  ¤ A lÓ 3×4�AØ ”zØ"Ÿx™xÕ'K¼vÀaÈ¼{Ó'KÓK˜˜!™Ø ! !¥ a¨¨!­¥f¨Q¥h¨q°°1µ­v°a­xÕ&7Õ8¸!¸A½#Õ>¸qÀ½tÀA½vÕEÀaÈÈ!ÍÅfÈaÐPQÍcÅlÕR˜˜!™Ø˜Q�4 ! Q¥$�;�DØ•I�AØ—w‘w˜t“}¨ tÔ+Ø  $˜wšØ˜•F“Añ 5ð —’˜L¨!�OÕ+•Ø˜%�x�Ø�Œ6�Q˜"”WÜ*¨1¨aÓ0°%Ð7Ð7ØˆA„vØ�"Œ9Ø�a˜•c•7˜E�>Ð!Ø�V‰V�A‹YˆØ�V‰V�B‹Z‰Ø	
ˆbŒ˜#Ÿ'™' !Ÿ*š*Ð+<¸s¿w¹wÀq»zÖ+MÈA×+MØ�V‰V�Q�B‹ZˆØ�C—F‘F˜B˜3“KÕ Ð&Ð&ð �V‰V�A‹Y˜˜CŸF™F� 1�Õ$ˆØ�V‰V�B‹ZˆØ�7�R•U�?˜R A¥�Y¨¨"¨a­%­Õ0Õ0°%Ð7Ð7r@   c                 ó"  € V P                  V4      p\        V4      pV P                  V4      '       g   \        WV4      # V P                  pT ;P                  ^T P                  T;'       g    ^4      ,           ,          un        V P                  pV^,
          p\        WW%4      w  rgV'       gÙ   V P                  ^4      p\        ^d4       F™  p	V P                  V4      p
Wj,          pW±,
          pWlWº,           Wh,           V,          W†,          V,           ,          ,
          ,          ,
          pV P                  WÖ,
          4      V P                  V4      V,
          8:  d   Tp MTpK›  	  X	^d8X  d   V P                  RV,          4       W0n        V5# )é   z1Lambert W iteration failed to converge for z = %s)r·   rê   Úisnormalr"  rÔ   rÓ   rA  rí   r   rÏ   Úwarn)rI   rK   rî   rÔ   r  r;  rã   ÚdoneÚtwoÚiÚewÚwewÚwewzÚwns   &&&           r=   ÚlambertwrM  Ï  s-  € à�‰�A‹€AÜˆA‹€AØ�<‰<˜�?Š?Ü  ¨Ó+Ð+Ø�8‰8€DØ‡H‚H��S—W‘W˜QŸV˜V !“_Õ$Õ$…HØ	�‰€BØ
ˆq�&€CÜ˜s qÓ.�G€Aßà�g‰g�a‹jˆÜ˜–ˆAØ—‘˜“ˆBØ•$ˆCØ•5ˆDØ˜3�6 1¥5¨$¥,°µ°cµ	Õ":Õ:Õ;Õ;ˆBØ�w‰w�r•t‹} §¡¨£¨cÕ 1Ô1Ø�Ùà’ñ ð �Œ8Ø�H‰HÐHÈ1ÕLÔMØ„HØˆ2€Ir@   c                 ó  € V P                  V4      pV'       g+   V P                  V4      '       d   V# \        V4      ! ^4      # V P                  V4      '       gF   V P                  V4      '       g/   V P                  V4      '       g   V P                  V4      '       d	   W!,          # V^8X  d   V# V^8X  d   W"^,           ,          # V^ 8X  d   V P	                  V4      # \        WVR4      V P                  V4      ,          # )r   T)r·   r¸   ÚtyperÅ   rÉ   Ú_polyexprÏ   )rI   rJ   r¼   s   &&&r=   ÚbellrQ  ì  s¶   € à�‰�A‹€AßØ�9‰9�Q�<Š<ØˆHÜ�AŒw�q‹zÐØ
‡y�y�‡|‚|�s—y‘y —|’| s§y¡y°§|¢|°s·y±yÀ·|²|Ø�tˆØˆA„v�aˆxØˆA„v�a˜1�•gˆ~ØˆA„v�c—j‘j “mÐ#Ü�C˜A˜tÓ$ s§w¡w¨q£zÕ1Ð1r@   c                 óB   a aaa€ V VVV3R  lpS P                  V^R7      # )c               3   ó¦   <"  € S'       d   SP                  S4      x € Sp ^p VS,          V ,          x € V^,          pV S,          V,          p K.  5ir¿   )rÉ   )Útrî   rI   ÚextrarJ   r¼   s     €€€€r=   Ú_termsÚ_polyexp.<locals>._termsû  sG   øé € ßØ—*‘*˜Q“-ÒØˆØˆØØ�Q•$˜•(ŠNØ��FˆAØ�!•�A•ŠAùs   ƒAA)Ú
check_step)rÕ   )rI   rJ   r¼   rU  rV  s   ffff r=   rP  rP  ú  s%   û€ ÷ð ð ×Ñ˜f°ÐÓ3Ð3r@   c                 óÈ  € V P                  V4      '       gF   V P                  V4      '       g/   V P                  V4      '       g   V P                  V4      '       d	   W!,          # V^ 8X  d	   W!,          # V^ 8X  d   V P                  V4      # V^8X  d   V P                  V4      V,          # V^8X  d'   V P                  V4      V,          V^,           ,          # \	        WV4      # r¶   )rÅ   r¸   rÖ   rÏ   rP  )rI   r_   rK   s   &&&r=   ÚpolyexprZ    s¡   € à
‡y�y�‡|‚|�s—y‘y —|’| s§y¡y°§|¢|°s·y±yÀ·|²|Ø�tˆØˆA„v�a•cˆzØˆA„v�c—i‘i “lÐ"ØˆA„v�c—g‘g˜a“j •lÐ"ØˆA„v�c—g‘g˜a“j •l A a¥CÕ(Ð(Ü�C˜AÓÐr@   c                 óH  € \        V4      pV^ 8  d   \        R4      hV P                  pV^ 8X  d   V# V^8X  d	   W#,
          # V^8X  d	   W#,           # ^p^p^ p^ p\        ^V^,           4       F‰  pW,          '       d   K  V P	                  W,          4      p	V P                  W(4      ) p
V
'       d   W:V	,          ,          pKU  V	^8X  d   WH,          pV^,          pKo  V	R8X  g   Kx  WX,          pV^,          pK‹  	  V'       d"   Wg8”  d   V^ ,          pV# W4,          pW5,          pV# )r   zn cannot be negativerÍ   )rê   Ú
ValueErrorr‹   rù   Úmoebiusrè   )rI   rJ   rK   r?  Úa_prodÚb_prodÚ	num_zerosÚ	num_polesÚdrã   rÂ   s   &&&        r=   Ú
cyclotomicrc    s  € äˆA‹€AØˆ1„uÜÐ/Ó0Ð0Ø�‰€AØˆA„vØˆØˆA„vØ�uˆØˆA„vØ�uˆð €FØ€FØ€IØ€IÜ�1�Q�q•SŽ\ˆØ�uŽuØ—‘˜A�DÓ!ˆAð —‘˜1“Ð ˆAßØ˜•T•	’à˜”6Ø•K�FØ •N’IØ˜"–WØ•K�FØ •N’Iñ ÷  ØÔ Ø��FˆAð €Hð �KˆAØ�KˆAØ€Hr@   c                óÈ  € \        V4      pV^8  d   V P                  # V^,          ^ 8X  d.   W^,
          ,          ^ 8X  d   V P                  5# V P                  # R FY  pW,          '       d   K  W,          ^ rCV^8”  d&   \        W24      w  r4V'       g   K  V P                  u # V P	                  V4      u # 	  V P                  V4      '       d   V P	                  V4      # VR8”  d   \        h^p \        VRV,          ,          R,           4      pV^8  d   V P                  # W%,          V8X  d)   V P                  V4      '       d   V P	                  V4      # V^,          pKt  )aO  
Evaluates the von Mangoldt function `\Lambda(n) = \log p`
if `n = p^k` a power of a prime, and `\Lambda(n) = 0` otherwise.

**Examples**

    >>> from mpmath import *
    >>> mp.dps = 25; mp.pretty = True
    >>> [mangoldt(n) for n in range(-2,3)]
    [0.0, 0.0, 0.0, 0.0, 0.6931471805599453094172321]
    >>> mangoldt(6)
    0.0
    >>> mangoldt(7)
    1.945910149055313305105353
    >>> mangoldt(8)
    0.6931471805599453094172321
    >>> fsum(mangoldt(n) for n in range(101))
    94.04531122935739224600493
    >>> fsum(mangoldt(n) for n in range(10001))
    10013.39669326311478372032

r$  r    )
r   r   r   é   é   é   é   é   é   é   l       ©7§3M“e')rê   rÒ   Úln2Údivmodrâ   ÚisprimerG   )rI   rJ   r?  Úqr  rî   s   &&    r=   Úmangoldtrp  ;  s  € ô0 	ˆA‹€AØˆ1„uØ�x‰xˆØˆ1…u�„zà�!•�9˜Œ>Ø—G‘G�8ˆOà—8‘8ˆOó
 *ˆØ�uŽuØ•6˜1ˆqØ�a”%Ü˜a“|‘�ß‘1ØŸ8™8’OØ—6‘6˜!“9Òñ *ð ‡{�{�1‡~‚~Ø�v‰v�a‹yÐàˆ6„zÜ!Ð!Ø	€AØ
Ü��B�q•D•	˜C•Ó ˆØˆqŒ5Ø—8‘8ˆOØ�6�QŒ;Ø�{‰{˜1�~Š~Ø—v‘v˜a“yÐ Ø	ˆQ�Šr@   c                 ó’   € V P                  \        V4      \        V4      4      pV'       d   \        V4      # V P                  V4      # rB   )Ú
_stirling1rê   rí   ©rI   rJ   rî   Úexactrô   s   &&&& r=   Ú	stirling1ru  w  ó4   € à�‰”s˜1“vœs 1›vÓ&€AßÜ�1‹vˆà�w‰w�q‹zÐr@   c                 ó’   € V P                  \        V4      \        V4      4      pV'       d   \        V4      # V P                  V4      # rB   )Ú
_stirling2rê   rí   rs  s   &&&& r=   Ú	stirling2ry    rv  r@   r¿   r¶   r†   rB   )6Úlibmp.backendr   Úobjectr   r„   r‡   r‰   r�   r‘   r”   r—   r›   rž   r£   r¦   rª   r­   r°   r´   r½   rÃ   rÆ   rÉ   rÖ   rÛ   rè   rï   r   rü   r	   r  rò   rñ   r   r  r  rÙ   r  r  r  r  r"  r'  r(  r,  rA  rM  rQ  rP  rZ  rc  rp  ru  ry  r1  r@   r=   Ú<module>r|     s>  ðÝ "ôL4�vô L4ò\òòð Ù ,ó Ø ,àÙ ,ó Ø ,àÙ ,ó Ø ,àÙ .ó Ø .àÙ .ó Ø .àÙ .ó Ø .àñ%ó ð%ð Ù .ó Ø .àÙ .ó Ø .àñ&ó ð&ð Ù 0ó Ø 0àÙ 0ó Ø 0àñ	ó ð	ð óó ðð ñó ðð ñ#ó ð#ð ñ?ó ð?ð ñ4ó ð4ð ñ;ó ð;ð* ñ&ó ð&ð óó ðð" óó ðð ñó ðð ñó ðð ñó ðð ñó ðð ñó ðð ñ%ó ð%ð ñ*ó ð*ð ó3ó ð3ð ñó ðð ñ+ó ð+ð ñó ðð ñó ðòó  Û ò=1ò~48ðl óó ðð8 ó2ó ð2ô
4ð ñó ðð ñ(ó ð(ðT ñ9ó ð9ðv óó ðð óó òr@   