+
    ‹°³j5H  ã                   ó    € ^ RI HtHtHtHtHt ^ RIHtHt ^ RI	H
t
 ^ RIHt ^ RIHtHtHt ^ RIHtHt ^ RIHtHt R tR	 t ! R
 R]4      tR# )é    )ÚFunctionÚSÚMulÚPowÚAdd)ÚorderedÚdefault_sort_key)Úexpand_func)ÚDummy)ÚgammaÚsqrtÚsin)ÚfactorÚcancel)ÚsiftÚuniqc                óÒ  € V P                  \        4      p V P                  \        4      pV Uu0 uF  p\	        V\        4      '       g   K  VkK  	  ppV'       g   V # W,          pWP                  4       P                  \        4      ,          pV'       d¶   \        \        V4       UUu. uFB  p\        4       YDP                  ! VP                   Uu. uF  p\        VRR7      NK  	  up!  3NKD  	  upp!  w  rgpV P                  \        \        Wv4      4      4      p	\        V	RR7      P                  \        \        Wh4      4      4      # \        V RR7      # u upi u upi u uppi )a}  
Simplify expressions with gamma functions.

Explanation
===========

This function takes as input an expression containing gamma
functions or functions that can be rewritten in terms of gamma
functions and tries to minimize the number of those functions and
reduce the size of their arguments.

The algorithm works by rewriting all gamma functions as expressions
involving rising factorials (Pochhammer symbols) and applies
recurrence relations and other transformations applicable to rising
factorials, to reduce their arguments, possibly letting the resulting
rising factorial to cancel. Rising factorials with the second argument
being an integer are expanded into polynomial forms and finally all
other rising factorial are rewritten in terms of gamma functions.

Then the following two steps are performed.

1. Reduce the number of gammas by applying the reflection theorem
   gamma(x)*gamma(1-x) == pi/sin(pi*x).
2. Reduce the number of gammas by applying the multiplication theorem
   gamma(x)*gamma(x+1/n)*...*gamma(x+(n-1)/n) == C*gamma(n*x).

It then reduces the number of prefactors by absorbing them into gammas
where possible and expands gammas with rational argument.

All transformation rules can be found (or were derived from) here:

.. [1] https://functions.wolfram.com/GammaBetaErf/Pochhammer/17/01/02/
.. [2] https://functions.wolfram.com/GammaBetaErf/Pochhammer/27/01/0005/

Examples
========

>>> from sympy.simplify import gammasimp
>>> from sympy import gamma, Symbol
>>> from sympy.abc import x
>>> n = Symbol('n', integer = True)

>>> gammasimp(gamma(x)/gamma(x - 3))
(x - 3)*(x - 2)*(x - 1)
>>> gammasimp(gamma(n + 3))
gamma(n + 3)

F)Úas_comb)Úrewriter   Úatomsr   Ú
isinstanceÚas_dummyÚzipr   r   ÚfuncÚargsÚ
_gammasimpÚxreplaceÚdict)
ÚexprÚfÚiÚgammasÚfiÚaÚdumÚfunÚsimpÚds
   &         ÚY/home/luca/screen-checking/.venv/lib/python3.14/site-packages/sympy/simplify/gammasimp.pyÚ	gammasimpr*   
   s*  € ðd �<‰<œÓ€Dð
 	�
‰
”8Ó€AáÓ3™�Aœj¨¬E×2�aˆa™€FÐ3ßØˆØ…K€Aà	�M‰M‹O×!Ñ!¤(Ó+Õ+€AßÜô ˜a”jô"ñ !�ô ‹W�bŸ'š'Ø68·g²gó$?Ù6=°”
˜1 e×,±gñ$?ñ @ó Aá ò"ñ #‰ˆ�$ð �M‰Mœ$œs 3›}Ó-Ó.ˆÜ˜! UÔ+×4Ñ4´T¼#¸c».Ó5IÓJÐJä�d EÔ*Ð*ùò 4ùò$?ùó"s#   ¯EÁEÂ*(E#
ÃEÃ(
E#
ÅE#
c                ó.  aa€ V P                  \        R 4      p S'       d   V P                  \        R 4      p MV P                  \        R 4      p RVV3R llo\        V 4      pS! V4      p W8w  d   \        V 4      p V P                  \        R 4      p V # )a  
Helper function for gammasimp and combsimp.

Explanation
===========

Simplifies expressions written in terms of gamma function. If
as_comb is True, it tries to preserve integer arguments. See
docstring of gammasimp for more information. This was part of
combsimp() in combsimp.py.
c                 óD   € \        ^V ^,
          P                  4       4      # ©é   )Ú_rfÚexpand©Úns   &r)   Ú<lambda>Ú_gammasimp.<locals>.<lambda>a   s   € ”#�a˜!˜a�%Ÿ™Ó)Ô*ó    c                 ó&   € \        V^,           4      # r-   ©r   ©r$   Úbs   &&r)   r3   r4   e   s   € œ˜q 1�uœr5   c                 óD   € \        W,           4      \        V 4      ,          # ©Nr7   r8   s   &&r)   r3   r4   h   s   € œ˜q�u›¤e¨A£hÖ.r5   c                ó.  <a(a)a*a+a,a-€ V P                   '       d   V # R pV+3R lo+V^ 8X  dA   V P                  ! V P                   Uu. uF  pS/! W1^,           4      NK  	  up!  p V^,          pV P                  '       g   V # V^8X  dm   V P	                  4       w  rEV'       g   V # V'       d@   S/! \
        P                  ! V4      V^,           4      \
        P                  ! V4      ,          # V^,          pV^8X  Edy   \        V P                  S+RR7      w  rg\        V!  p\        V!  p	V	P                  4       w  r«\        ^4       Fã  p\        \        \
        P                  ! V
4      4      4      p\        V4       F‡  w  rÞVP                  '       g   K  \        VP                   Uu. uF  pS/! V! Wû,          4      V^,           4      NK!  	  up!  P                  4       w  rëWäV&   VP!                  \"        4      '       d   K‡   M	  \        V!  p
V^ 8X  d   S+! V
4      '       g    MYºrºKå  	  WŠ,          V,          p V P                  '       d   S+! V4      '       g   S+! V
4      '       g   V # V^,          pV^8X  d    T pS/! V ^4      p V V8X  g   K  V # . p. p. p. pR p\        \        V P                  4      4      pV'       dŸ   VP%                  4       P                  4       w  pp	V! V4      w  ppV'       d   VP'                  V4       MVRJ d   VP'                  V4       V! V	4      w  ppV'       d   VP'                  V4       K‹  VRJ g   K“  VP'                  V4       K¦  S.'       EgÎ   VVV3VVV33 EFS  w  ppp. pV'       Ed<   VP%                  4       o*S*P(                  '       d   VP+                  S*4       K>  \        V4       Fã  w  ppS*V,           ^,
          pVP,                  '       g   K*  VP+                  \.        P0                  4       VP+                  \3        \.        P0                  S*,          4      4       VP%                  V4       V^ 8”  d&   VP'                  V*3R l\        V4       4       4       M,V^ 8  d&   VP'                  V*3R l\        V) 4       4       4        EK.  	  VP+                  S*4       EKD  VVR	&   EKV  	  VVVV3VVVV33 EF2  w  pp p!p" V F2  pV  F'  o-V^S-,          ,
          pVP,                  '       g   K'   M	  K2   M	  KD  VP5                  V4       V P5                  S-4       V^ 8”  d&   V!P'                  V-3R
 l\        V4       4       4       M,V^ 8  d&   V"P'                  V-3R l\        V) 4       4       4       VP+                  S-\.        P6                  ,           4       V!P+                  ^^S-,          ^,
          ,          4       V"P+                  \9        \.        P0                  4      4       EK,  	  R o(V(3R lp#VVV3VVV33 F  w  pppV#! VVV4       K  	  V^8¼  Eda   V)V,3R lp$/ o,V,3R lo)V)V,3R lp%VV,           V,           V,            F  p V%! V 4       K  	  VVV3VVV33 EF  w  ppp. pV'       dÿ   VP%                  4       p&Rp'V''       dÑ   Rp'V$! VV&4      o-S-eK   VP5                  S-4       S-V&8w  d(   VP+                  S-V&,          4       V%! S-V&,          4       V&^,          p&Rp'V$! VV&^,
          4      o-S-f   Kw  VP5                  S-4       S-V&^,
          8w  d6   VP+                  V&^,
          S-,          4       V%! V&^,
          S-,          4       V&^,          p&Rp'KØ  VP+                  V&4       EK  VVR	&   EK  	  \        V U&u. uF  p&\#        V&4      NK  	  up&!  \        V U&u. uF  p&\#        V&4      NK  	  up&!  ,          \        V!  ,          \        V!  ,          # u upi u upi u up&i u up&i )z.Simplify products of gamma functions further. c                 ó–   € V P                  \        4      pV P                  \        R  4      pVP                  \        4      V8  d   Tp V # )c                 ón   € \        ^V ^,
          P                  4       4      P                  \         R 4      # )r.   c                 óD   € \        W,           4      \        V 4      ,          # r;   r7   r8   s   &&r)   r3   ÚU_gammasimp.<locals>.rule_gamma.<locals>.gamma_rat.<locals>.<lambda>.<locals>.<lambda>t   s   € ¬E°!µ%«L¼¸q»Ö,Ar5   )r/   r0   Úreplacer1   s   &r)   r3   ÚC_gammasimp.<locals>.rule_gamma.<locals>.gamma_rat.<locals>.<lambda>s   s+   € ¬C°°A¸µE·>±>Ó3Có -ß‘'œ#ÑAÓBð-Cr5   )Úcountr   rA   )ÚxÚwasÚxxs   &  r)   Ú	gamma_ratÚ1_gammasimp.<locals>.rule_gamma.<locals>.gamma_ratp   s@   € à—'‘'œ%“.ˆCØ—‘œ5ñ #Có DˆBà�x‰xœ‹ Ô$Ø�ØˆHr5   c                 óÔ  <€ \        V \        4      '       d   R # V P                  '       g   V P                  '       dO   \        ;QJ d)    V3R lV P
                   4       F  '       g   K   R # 	  R# ! V3R lV P
                   4       4      # V P                  '       dK   V P                  P                  '       g   V P                  P                  '       d   S! V P                  4      # R# )Tc              3   ó4   <"  € T F  pS! V4      x € K  	  R # 5ir;   © )Ú.0ÚxiÚgamma_factors   & €r)   Ú	<genexpr>ÚG_gammasimp.<locals>.rule_gamma.<locals>.gamma_factor.<locals>.<genexpr>~   s   øé € Ð=±f°™<¨×+Ð+³fùs   ƒF)r   r   Úis_AddÚis_MulÚanyr   Úis_PowÚexpÚ
is_integerÚbaseÚis_positive)rD   rN   s   &€r)   rN   Ú4_gammasimp.<locals>.rule_gamma.<locals>.gamma_factory   sŽ   ø€ ä˜!œU×#Ò#ÙØ�x�xˆx˜1Ÿ8Ÿ8˜8ß“sÔ=°a·f²fÓ=—s”sÐ=’sÐ=�sÔ=°a·f²fÓ=Ó=Ð=Ø�x�xˆx˜QŸU™U×-×-Ð-°·±×1C×1CÐ1CÙ# A§F¡FÓ+Ð+Ùr5   T)Úbinaryc                 ó  € V \         P                  J d   R . 3# V P                  4       w  rVP                  '       d@   \	        V\
        4      '       d   RVP                  ^ ,          .V,          3# RV.V,          3# RV .3# )NTF)r   ÚOneÚas_base_expÚ
is_Integerr   r   r   )Úpr9   Úes   &  r)   Ú	explicateÚ1_gammasimp.<locals>.rule_gamma.<locals>.explicateº   sn   € Ø”A—E‘E‹zØ˜R�x�Ø—=‘=“?‰DˆAØ�|�|ˆ|Ü˜a¤×'Ò'Ø !§&¡&¨¥) ¨Q¥Ð.Ð.à  1 # a¥%˜<Ð'à˜q˜c�zÐ!r5   Fc              3   óD   <"  € T F  p^S,
          V,           x € K  	  R# 5i)r.   NrK   ©rL   ÚkÚg1s   & €r)   rO   Ú1_gammasimp.<locals>.rule_gamma.<locals>.<genexpr>ê   s   øé € Ð(F¹X¸¨¨R­°!¯ª»Xùó   ƒ c              3   ó8   <"  € T F  pS) V,
          x € K  	  R # 5ir;   rK   rd   s   & €r)   rO   rg   ì   s   øé € Ð(D¹)°Q¨"¨¨q¯ª»)ùs   ƒºNNNc              3   óD   <"  € T F  p^S,          V,           x € K  	  R# 5i©é   NrK   ©rL   re   Úys   & €r)   rO   rg     s   øé € Ð!<±8¨a ! A¥#¨§'¢'³8ùrh   c              3   óR   <"  € T F  p^S,          ^,
          V,
          x € K  	  R# 5irl   rK   rn   s   & €r)   rO   rg     s   øé € Ð!A±y°! ! A¥#¨¥'¨A§+¢+³yùs   ƒ$'c                 ó  € \        \        V 4      4      p\        \        V4      4       EFM  p\        V^,           \        V4      4       Uu. uF!  q1V,          W,          ,
          ^,          V3NK#  	  ppV Fü  w  rSVP                  ^8X  g   K  VP
                  ^8w  g   K+  VP
                  pV.p\        \        ^V4      4      pV FU  w  r“Wi,          p
V
P                  '       g   K!  W¨9   g   K)  VP                  V
4       VP                  V4       V'       d   KU   M	  K¬  \        V4       F"  w  r#W,          pV P                  V4       W·V&   K$  	  VP
                  V^ ,          VR,          3u u # 	  EKP  	  R# u upi )r.   :r.   NNN)
Úlistr   ÚrangeÚlenr_   Úqr^   ÚremoveÚappendÚ	enumerate)ÚcoeffsÚur!   ÚjÚdjÚoner2   ÚgotÚgetr(   ÚmÚcs   &           r)   Ú_runÚ,_gammasimp.<locals>.rule_gamma.<locals>._run"  s'  € ô œ˜f›Ó&�Üœs 1›vŸ�AÜ;@ÀÀQÅÌÈAËÔ;OÓPÑ;O°a˜q�T A¥D�[¨AÕ-¨qÓ1Ñ;O�BÐPÛ"$™˜ØŸ5™5 Až:¨#¯%©%°1®*Ø #§¡˜AØ#$ #˜CÜ"&¤u¨Q°£{Ó"3˜CÛ(*¡ Ø$%¥C Ø#$§<§<¢<°A¶HØ$'§J¡J¨q¤MØ$'§J¡J¨q¤Mß+.©3Ù(-ñ )+ñ !)Ü(1°#®¡ Ø$%¥D Ø &§¡¨aÔ 0Ø)* A£ñ )7ð $'§5¡5¨#¨a­&°#°bµ'Ð#9Ô9ô% #%ó 'ùÚPs   Á'E=c                 óD  <€ / pV  F6  pVP                  4       w  rVVP                  V. 4      P                  V4       K8  	  \        V\        R 7      pV EF%  p\        W6,          4      p. p	 S! V4      p
V
f   MßV
w  r¼pV FK  pWn,           ^,
          p\        \        Wì,
          4      4       F  pVP                  VV,
          4       K  	  KM  	  W¶V,           ,          pVP                  ^\        P                  ,          \        V^,
          4      ^,          ,          V\        P                  V,
          ,          ,          4       V	P                  V4       Kí  V Uu. uF  qVV,           NK  	  upV	,           W6&   EK(  	  . pV F  pWCV,          ,          pK  	  W@R&   R# u upi ))ÚkeyNrj   )
Úas_coeff_AddÚ
setdefaultrw   Úsortedr	   rs   Úintr   ÚPiÚHalf)r"   ÚnumerÚdenomÚratsÚgr�   ÚresidÚkeysry   ÚnewÚrunr2   ÚuiÚotherrz   Úconre   r‚   s   &&&              €r)   Ú	_mult_thmÚ1_gammasimp.<locals>.rule_gamma.<locals>._mult_thm<  se  ø€ ð
 �Û�AØ Ÿ~™~Ó/‘H�AØ—O‘O E¨2Ó.×5Ñ5°aÖ8ñ  ô
 ˜dÔ(8Ô9�Ü!�EÜ# D¥KÓ0�FØ�CØÙ" 6›l˜Øš;Ø!ð (+™˜˜uó "'˜AØ"'¥)¨a¥-˜CÜ%*¬3¨q­v«;Ö%7 Ø %§¡¨S°1­WÖ 5ó &8ñ "'ð
  ¨¥�n˜ð Ÿ™ a¬¯©¥f´°!°aµ%³¸µ
Õ%;Ø%&¬¯©°#­Õ%6õ&7ô 8ð Ÿ
™
 3žñ 7=Ó"=±f°¨1§9 9±fÑ"=ÀÕ"C�D”Kñ? "ðD �Û!�EØ˜e�Õ$’Añ "ð �q“	ùò #>s   ÅFc                 ó¤  <€ V '       g   R # S
! V4      w  r#V  F´  pSV,          w  rVW68w  g6   VP                  V4      '       g!   V\        4       8w  g   V\        4       8w  d   KK  \        \        W,          4      P                  4      p\        VP                  4      p\        VP                  4      p	V^ 8X  g   K¢  V^ 8”  g
   V	^ 8”  g   K²  Vu # 	  R # r;   )ÚintersectionÚsetrt   r   Úfree_symbols)ÚlrD   ÚS1ÚT1ro   ÚS2ÚT2r$   r9   r�   Ú
compute_STÚinvs   &&        €€r)   Ú
find_fuzzyÚ2_gammasimp.<locals>.rule_gamma.<locals>.find_fuzzy{  sœ   ø€ ßÙÙ# A›‘�Û�AØ  �V‘F�BØ”x¨¯©¸×(;Ò(;Ø%'¬3«5¤[°B¼#»%´KÙ ô œF 1¥3›K×4Ñ4Ó5�AÜ˜AŸN™NÓ+�AÜ˜AŸN™NÓ+�Aà˜A–v 1 q¤5¨A°®EØ šó r5   c                 óæ   <€ V S9   d
   SV ,          # V P                   V P                  \        4      P                  V P                  \        4       Uu0 uF  qP
                  kK  	  up4      3# u upi r;   )rœ   r   r   Úunionr   rU   )r   r`   r£   s   & €r)   r¢   Ú2_gammasimp.<locals>.rule_gamma.<locals>.compute_ST•  s\   ø€ Ø˜3”;Ø˜t�9Ð$Ø×)Ñ)¨4¯:©:´hÓ+?×+EÑ+EØ(,¯
©
´3¬Ó8© 1Ÿœ©Ñ8ó,:ð ;ð ;ùÚ8s   ÁA.
c                 ó   <€ S! V 4      SV &   R # r;   rK   )r   r¢   r£   s   &€€r)   Ú	update_STÚ1_gammasimp.<locals>.rule_gamma.<locals>.update_ST›  s   ø€ Ù& tÓ,��D“	r5   )Úis_Atomr   r   rR   Úargs_cncr   Ú
_from_argsr   Úas_numer_denomrs   rr   r   Ú	make_argsrx   rQ   r   Úhasr   ÚpopÚextendrV   rw   r^   r   rŠ   r   rv   r‹   r   )0r   ÚlevelrG   rD   r   ÚncÚTÚFÚ	gamma_indr(   ÚndÚddÚipassr!   Únir$   rE   Únumer_gammasÚdenom_gammasÚnumer_othersÚdenom_othersra   Únewargsr2   Úisgr�   r"   rŒ   r�   r’   Úg2ÚngÚdgÚnoÚdor—   r¤   rª   r�   Úcontr‚   r¢   rf   rN   r£   ro   r   Ú
rule_gammas0   &&                                      @@@@@@€€r)   rÉ   Ú_gammasimp.<locals>.rule_gammaj   sè  þ€ ð �<�<ˆ<ØˆKò	õ	ð �AŒ:Ø—9’9ÀÇÂÓKÁ¸A™z¨!°Q­YÖ7ÁÑKÑLˆDØ�Q�JˆEà�{�{ˆ{ØˆKð �AŒ:Ø—}‘}“‰HˆDßØ�ßÙ!¤#§.¢.°Ó"6¸À½	ÓBÄ3Ç>Â>ÐRTÓCUÕUÐUØ�Q�JˆEð �A�:Ü˜Ÿ	™	 <¸Ô=‰DˆAÜ˜Q™ˆIÜ�Q‘ˆAà×%Ñ%Ó'‰FˆBÜ˜qž�ÜœG¤C§M¢M°"Ó$5Ó6Ó7�Ü& tž_‘E�AØ—y—y’yÜ!$ØLNÏGÊGó'UÙLSÀq™J¡y°µ£¸À½	ÖBÉGñ'Uñ "ç,™nÓ.ñ ˜ð #%˜Q™Ø!Ÿv™v¤eŸ}œ}Ù!ñ -ô ˜$‘Z�Ø˜Q”;¡|°B×'7Ò'7ÙØ’Bñ "ð •< •?ˆDØ—K—K�K¡\°"×%5Ò%5¹Àb×9IÒ9IØ�Ø�Q�JˆEð �AŒ:ØØ�Ù! $¨Ó*�Ø˜3–;Ø�KàˆØˆØˆØˆò
	"ô ”w˜tŸy™yÓ)Ó*ˆßØ—;‘;“=×/Ñ/Ó1‰DˆAˆqÙ˜q“\‰FˆC�ßØ×#Ñ# AÕ&Ø˜“Ø×#Ñ# AÔ&Ù˜q“\‰FˆC�ßØ×#Ñ# AÖ&Ø˜•Ø×#Ñ# AÖ&÷ ˆwð ˜l¨Lð*:à! <°Ð>ô)@Ñ$�˜˜uð �ß�fØŸ™›�BØ—}—}�}ØŸ
™
 2œÙ Ü!*¨6Ö!2™˜˜2Ø �G a�K˜Ø Ÿ|Ÿ|˜|Ù$ØŸ™¤Q§T¡TÔ*ØŸ™¤S¬¯©¨b­£\Ô2ØŸ
™
 1œØ˜qœ5Ø!ŸL™LÔ(F¼UÀ1¼XÓ(FÕFØ œUØ!ŸL™LÔ(D¼%ÀÀ¼)Ó(DÔDÛñ "3ð Ÿ
™
 2Ÿà��q”	ñ1)@ð> %1°,ÀØ$0ð$2à$0°,ÀØ$0ð$2ô#3‘��B˜˜Bð
 Û˜Û!#˜AØ ! A a¥C¥˜AØ Ÿ|Ÿ|š|Ù %ñ "$ñ
 %Ùñ  ñ Ø—I‘I˜a”LØ—I‘I˜a”LØ˜1”uØŸ	™	Ô!<´5¸´8Ó!<Õ<Ø˜QœØŸ	™	Ô!A´u¸a¸R´yÓ!AÔAØ—I‘I˜a¤!§&¡&�jÔ)Ø—I‘I˜a ! A¥#¨¥'�lÔ+Ø—I‘Iœd¤1§4¡4›j×)ñ1#3òT:õ42ðh &2°<ÀÐ$NØ%1°<ÀÐ$Nó$P‘��5˜%á˜!˜U EÖ*ñ$Pð �A�:ö
!ð0 ˆCõ;ö-à$ |Õ3°lÕBÀ\×QÐQ�Ù˜$–ñ Rð ˜l¨Lð*:à! <°Ð>ô)@Ñ$�˜˜uð �ßØŸ
™
›�AØ�DßØ$˜Ù& u¨aÓ0˜Øš=Ø!ŸL™L¨œOØ  AœvØ %§¡¨Q¨q­SÔ 1Ù )¨!¨A­#¤Ø �F˜AØ#'˜DÙ& u¨a°!­eÓ4˜Øœ=Ø!ŸL™L¨œOØ  A¨¥EœzØ %§¡¨a°!­e°Q­YÔ 7Ù )¨1¨q­5°!­)Ô 4Ø �F˜AØ#'šDØ—J‘J˜q—Mà��q”	ñ7)@ô> ¡|Ó4¡| !”U˜1–X¡|Ñ4Ñ5Ü¡lÓ3¡l ”E˜!–H¡lÑ3Ñ4õ5ä�<Ñ õ!ä#&¨Ñ#5õ6ð 	6ùòu	 Lùò4'Uùò@	 5ùÚ3s   Á`Æ&%`Þ'`ß`
c                 ód   € V P                   '       d   \        \        V 4      4      # \        V 4      # r;   )Úis_Rationalr
   r   r1   s   &r)   r3   r4   Ê  s!   € ¨1¯=¯=¨=”+œe A›hÓ'ÐF¼eÀA»hÐFr5   )r   )rA   r   r/   r   )r   r   rE   rÉ   s   &f @r)   r   r   T   s�   ù€ ð �<‰<œÙ*ó,€D÷ Ø�|‰|œCÙ%ó'‰ð �|‰|œCÙ.ó0ˆ÷W6ð W6ôr
 �‹,€Cá�c‹?€DØ„{Ü�d‹|ˆà�<‰<œÙFóH€Dð €Kr5   c                   ó0   a € ] tR tRt o ]R 4       tRtV tR# )r/   iÏ  c                óü  € VP                   '       d”   V'       g   \        P                  # \        V4      pV^ 8”  d)   \	        \        V4       Uu. uF  qAV,           NK  	  up!  # V^ 8  d9   ^\	        \        ^V) ^,           4       Uu. uF  qAV,
          NK  	  up!  ,          # R# VP                  '       d   VP                  4       w  rVVP                   '       d[   V^ 8”  d#   \        W4      \        W,           V4      ,          # V^ 8  d+   \        W4      \        W,           V,           V) 4      ,          # VP                  '       d«   VP                  4       w  rWVP                   '       d…   V^ 8”  d3   \        Wr4      \        Wr,           V4      ,          \        Wu4      ,          # V^ 8  dC   \        Wr4      \        Wu,           V) 4      ,          \        Wr,           V,           V) 4      ,          # R# R# R# u upi u upi )r   N)	r^   r   r\   r‰   r   rs   rQ   r†   r/   )Úclsr$   r9   r2   r!   r�   Ú_bÚ_as   &&&     r)   ÚevalÚ_rf.evalÐ  s~  € à�<�<ˆ<ßÜ—u‘u�ä�A“ˆAà�1ŒuÜ¬E°!¬HÓ5©H q ŸU˜U©HÑ5Ñ6Ð6Ø�Q”Øœ¬e°A¸°r¸AµvÔ.>Ó?Ñ.>¨ 1Ÿu˜uÑ.>Ñ?Ñ@Õ@Ð@ñ ð �x�xˆxØŸ™Ó(‘�à—<—<�<Ø˜1”uÜ" 1›z¬#¨a­f°a«.Õ8Ð8Ø˜QœÜ" 1›z¬#¨a­f°q­j¸1¸"Ó*=Õ=Ð=à�x�xˆxØŸ™Ó(‘�à—<—<�<Ø˜1”uÜ" 2›z¬#¨b­f°a«.Õ8¼¸R»ÕCÐCØ˜QœÜ" 2›z¬#¨b­f°q°b«/Õ9¼#¸b½fÀq½jÈ1È"Ó:MÕMÐMñ ñ  ñ ùò 6ùâ?s   ÁG4ÂG9
rK   N)Ú__name__Ú
__module__Ú__qualname__Ú__firstlineno__ÚclassmethodrÒ   Ú__static_attributes__Ú__classdictcell__)Ú__classdict__s   @r)   r/   r/   Ï  s   ø‡ € ØñNó öNr5   r/   N)Ú
sympy.corer   r   r   r   r   Úsympy.core.sortingr   r	   Úsympy.core.functionr
   Úsympy.core.symbolr   Úsympy.functionsr   r   r   Úsympy.polysr   r   Úsympy.utilities.iterablesr   r   r*   r   r/   rK   r5   r)   Ú<module>rã      s<   ðß 1Õ 1ß 8Ý +Ý #ß ,Ñ ,ß &ß 0òG+òTxôvNˆ(ö Nr5   