ó
    ‰*£h‡; ã                  ó*  • % S r SSKJr  SSKrSSKJr  SSKJrJr  SSK	J
r
  SSKJr  SSKJr  SS	KJr  SS
KJr  SSKJrJrJrJrJr  SSKJr  SSKJr  SSKJrJr  SSK J!r!J"r"J#r#  SSK$J%r%J&r&  SSK'J(r(J)r)J*r*J+r+  SSK,J-r-  SSK.J/r/  SSK0J1r1J2r2J3r3J4r4J5r5J6r6J7r7J8r8J9r9J:r:J;r;  SSK<J=r=J>r>J?r?  SSK@JArA  SSKBJCrCJDrDJErEJFrF  SSKGJHrH  SSKIJJrJJKrK  SSKLJMrMJNrNJOrOJPrP  SSKQJRrRJSrSJTrTJUrU  SSKVJWrWJXrX  SSKYJZrZJ[r[  SSK\J]r]J^r^J_r_J`r`JaraJbrbJcrcJdrdJereJfrfJgrg  SSKhJiri  SS KjJkrkJlrl  SS!KmJnrn  S"S#KoJprp  SS$KqJrrrJsrsJtrtJuruJvrv  SS%KwJxrxJyry  SS&KzJ{r{  SS'K|J}r~  SS(K|Jr€  \(" S)5      r�S* r‚S+ rƒSS,K„J…r…  \…" S-5      r†SXS. jr‡ " S/ S0\ˆ5      r‰S1 rŠS2 r‹S3 rŒS4 r�S5 rŽS6 r�S7 r�S8 r‘S9 r’S: r“0 r”S;\•S<'   S= r–S> r—S? r˜SYS@ jr™SA ršSZSB jr›SC rœSZSD jr�SE ržSZSF jrŸSG r SH r¡SI r¢SJ r£SK r¤Sq¥\\†SYSL j5       5       r¦SYSM jr§SN r¨SO r©SP rª\†SQ 5       r«SR r¬SS r­ST r®SU r¯\†SZSV j5       r°SW r±g)[a¿  
Integrate functions by rewriting them as Meijer G-functions.

There are three user-visible functions that can be used by other parts of the
sympy library to solve various integration problems:

- meijerint_indefinite
- meijerint_definite
- meijerint_inversion

They can be used to compute, respectively, indefinite integrals, definite
integrals over intervals of the real line, and inverse laplace-type integrals
(from c-I*oo to c+I*oo). See the respective docstrings for details.

The main references for this are:

[L] Luke, Y. L. (1969), The Special Functions and Their Approximations,
    Volume 1

[R] Kelly B. Roach.  Meijer G Function Representations.
    In: Proceedings of the 1997 International Symposium on Symbolic and
    Algebraic Computation, pages 205-211, New York, 1997. ACM.

[P] A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev (1990).
    Integrals and Series: More Special Functions, Vol. 3,.
    Gordon and Breach Science Publisher
é    )ÚannotationsN)ÚSYMPY_DEBUG)ÚSÚExpr)ÚAdd)ÚBasic)Úcacheit)ÚTuple)Úfactor_terms)ÚexpandÚ
expand_mulÚexpand_power_baseÚexpand_trigÚFunction)ÚMul)Úilcm)ÚRationalÚpi)ÚEqÚNeÚ_canonical_coeff)Údefault_sort_keyÚordered)ÚDummyÚsymbolsÚWildÚSymbol)Úsympify)Ú	factorial)ÚreÚimÚargÚAbsÚsignÚ
unpolarifyÚpolarifyÚ
polar_liftÚprincipal_branchÚunbranched_argumentÚperiodic_argument)ÚexpÚ	exp_polarÚlog)Úceiling)ÚcoshÚsinhÚ_rewrite_hyperbolics_as_expÚHyperbolicFunction©Úsqrt)Ú	PiecewiseÚpiecewise_fold)ÚcosÚsinÚsincÚTrigonometricFunction)ÚbesseljÚbesselyÚbesseliÚbesselk)Ú
DiracDeltaÚ	Heaviside)Ú
elliptic_kÚ
elliptic_e)ÚerfÚerfcÚerfiÚEiÚexpintÚSiÚCiÚShiÚChiÚfresnelsÚfresnelc)Úgamma)ÚhyperÚmeijerg)ÚSingularityFunctioné   )ÚIntegral)ÚAndÚOrÚBooleanAtomÚNotÚBooleanFunction)ÚcancelÚfactor)Úmultiset_partitions)Údebug)ÚdebugfÚzc                óš   ^• [        U 5      n [        U SS5      (       a  [        U4S jU R                   5       5      $ U R                  " T6 $ )NÚis_PiecewiseFc              3  ó<   >#   • U  H  n[        U/TQ76 v •  M     g 7f©N)Ú_has)Ú.0ÚiÚfs     €ÚV/home/mande/repo/quber/.venv/lib/python3.13/site-packages/sympy/integrals/meijerint.pyÚ	<genexpr>Ú_has.<locals>.<genexpr>T   s   øé € Ð1ª 1”4˜�;˜A—;ªùs   ƒ)r6   ÚgetattrÚallÚargsÚhas)Úresrf   s    `rg   rc   rc   O   sA   ø€ ô ˜Ó
€CÜˆs�N E×*Ñ*ÜÔ1¨¯ªÓ1Ó1Ð1Ø�7Š7�Aˆ;Ðó    c                óÄ  ^ ^^	^
^^^^^^^• S n[        [        US5      5      u  mmm	mn[        SS /S9mT[        T-  -  mT[        R
                  SS4U 4S jjm
SU 4S jjnS	 nT	U" T	5      SS4/T S
'    " S S[        5      nT
" [        TT-
  5      TT-
  T	S-
  -  -  T	// / S/TT-  [        T	5      TT	S-
  -  -  [        TS:„  5      5        T
" [        TT-
  5      TT-
  T	S-
  -  -  / T	/S// TT-  [        T	5      TT	S-
  -  -  [        TS:„  5      5        T
" [        [        TT-  ST-  -  -
  5      TT-
  T	S-
  -  -  T	// / S/TT-  [        T	5      TT	S-
  -  -  [        TS:„  5      5        T
" [        TT-  ST-  -  [        -
  5      TT-
  T	S-
  -  -  / T	/S// TT-  [        T	5      TT	S-
  -  -  [        TS:„  5      5        T
" TT-   T	* -  ST	-
  // S// TT-  TT	* -  [        T	5      -  [        U" T	5      5      S9  T
" [        TT-
  5      T	* -  ST	-
  /ST	-
  S-  /S/ST	-
  S-  /TT-  S[        [        T	-  S-  5      -  [        ST	-
  5      -  [        T5      T	* -  -  [        T	5      S:  5        T
" TT	-  TT	-  -
  TT-
  -  ST	// ST	// TT-  TT	S-
  -  [        T	[        -  5      -  [        -  5        S mUU	U
UU4S jnU" SS5        U" SS5        U" [        R                  S5        U" [        R                  S5        UU	U
UUU4S jnU" SS5        U" SS5        U" [        R                  S5        U" [        R                  S5        T
" [!        [#        S5      T-  5      / / S// 5        T
" [%        T5      / S/[        R                  /SS/TS-  S-  [        ['        SS5      -  5        T
" [)        T5      / [        R                  /S/[        R                  [        R                  /TS-  S-  [        ['        SS5      -  5        T
" [        T5      / / [        R                  /S/TS-  S-  [+        [        5      5        T
" [-        T5      / / S/[        R                  /TS-  S-  [+        [        5      5        T
" [/        T5      / / S/['        SS5      /TS-  S-  [+        [        5      S-  5        UU4S jmUU4S jmU" [1        T5      T-  [        ST-
  5      -  TS5        U" [1        T5      T-  [        TS-
  5      -  TS5        UU4S jnU" [1        T5      T-  US5        U" [1        TT	-   5      U" [1        T	5      5      [        R
                  [3        SS// S/S/TT	-  5      4/-   S5        U" [1        [        TT	-
  5      5      U" [1        [        T	5      5      5      [        [3        SS/[        R                  /S/S[        R                  /TT	-  5      4/-   S5        U" [5        T5      U" [        R6                  * [        -  5      [        R8                  [3        / S/SS// T[#        S5      -  5      4/-   S5        T
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" [=        T5      / S/SS/[        R                  /TS-  S-  [+        [        5      * S-  5        T
" [?        T5      [        R                  // S/['        SS5      ['        SS5      /[#        S5      TS-  -  S-  T[+        [        5      -  S-  5        T
" [A        T5      / [        R                  S/SS/[        R                  [        R                  /TS-  S-  [        [	        S5      -  * S-  5        T
" [C        T	T5      / T	/T	S-
  S// T5        T
" [E        T5      S// [        R                  /S/TS-  S[+        [        5      -  5        T
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" [K        T5      S// ['        SS5      /S['        SS5      /[        S-  TS-  -  S-  [        R                  5        T
" [M        T5      S// ['        SS5      /S['        SS5      /[        S-  TS-  -  S-  [        R                  5        T
" [O        T	T5      / / T	S-  /T	* S-  /TS-  S-  5        T
" [Q        T	T5      / T	S-   * S-  /T	S-  T	* S-  /T	S-   * S-  /TS-  S-  5        T
" [S        T	T5      / ST	-   S-  /T	S-  /T	* S-  ST	-   S-  /TS-  S-  [        5        T
" [U        T	T5      / / T	S-  T	* S-  // TS-  S-  [        R                  5        T
" [W        T5      [        R                  [        R                  // S/S/T* [        R                  5        T
" [Y        T5      [        R                  S[        R                  -  // S/S/T* ['        SS5      S-  5        g)z7Add formulae for the function -> meijerg lookup table. c                ó    • [        U [        /S9$ )N©Úexclude)r   r^   )Úns    rg   ÚwildÚ"_create_lookup_table.<locals>.wildZ   s   € Ü�A¤˜sÑ#Ð#ro   Úpqabcrt   c                ó2   • U R                   =(       a    U S:„  $ ©Nr   )Ú
is_Integer©Úxs    rg   Ú<lambda>Ú&_create_lookup_table.<locals>.<lambda>]   s   € ¨¯©×(>¸¸Q¹Ð(>ro   )Ú
propertiesTc	                ó†   >• T	R                  [        U [        5      / 5      R                  U U[	        XX4U5      4/Xx45        g rb   )Ú
setdefaultÚ_mytyper^   ÚappendrP   )
ÚformulaÚanÚapÚbmÚbqr"   ÚfacÚcondÚhintÚtables
            €rg   ÚaddÚ!_create_lookup_table.<locals>.add`   sD   ø€ Ø×Ñœ ¬!Ó,¨bÓ1×8Ñ8¸'Ø'*¬G°B¸BÀCÓ,HÐ&IÐ%JÈDð:Xõ 	Yro   c                óh   >• TR                  [        U [        5      / 5      R                  XX#45        g rb   )r�   r‚   r^   rƒ   )r„   ÚinstrŠ   r‹   rŒ   s       €rg   ÚaddiÚ"_create_lookup_table.<locals>.addid   s,   ø€ Ø×ÑÜ�GœQÓ ó	%ß%+¡V¨W¸DÐ,GÕ%Hro   c           	     ó^   • U [        S// / S/[        5      4U [        / S/S// [        5      4/$ ©NrR   r   )rP   r^   )Úas    rg   ÚconstantÚ&_create_lookup_table.<locals>.constanth   s>   € Ø”G˜Q˜C  R¨!¨¬aÓ0Ð1Ø”G˜B   a S¨"¬aÓ0Ð1ð3ð 	3ro   © c                  ó$   • \ rS rSr\S 5       rSrg)Ú2_create_lookup_table.<locals>.IsNonPositiveIntegerén   c                óB   • [        U5      nUR                  SL a  US:*  $ g )NTr   )r%   rz   )Úclsr"   s     rg   ÚevalÚ7_create_lookup_table.<locals>.IsNonPositiveInteger.evalp   s%   € ä˜S“/ˆCØ�~‰~ Ò%Ø˜a‘x�ð &ro   r˜   N)Ú__name__Ú
__module__Ú__qualname__Ú__firstlineno__Úclassmethodrž   Ú__static_attributes__r˜   ro   rg   ÚIsNonPositiveIntegerrš   n   s   † à	ñ	 ó 
ó	 ro   r¦   rR   r   )r‹   é   c                óN   • [         [        SS5      -  U* U-  S-  SSU -  -
  -  -  $ )Néÿÿÿÿr§   rR   )r   r   )Úrr$   Únus      rg   ÚA1Ú _create_lookup_table.<locals>.A1ˆ   s/   € Ü”8˜B “?Ñ" T E¨"¡H¨Q¡J°!°a¸±c±'Ñ#:Ñ:Ð:ro   c                óì   >• T" [        TS-  T-   5      UT-  -   T-  TS-  T-   U -  -  ST-   S-  SSU -  -
  TS-  -   // TUT-  -
  S-  /TUT-  -   S-  /TTS-  -  TTSU -  -
  -  T" XT5      -  5        g )Nr§   rR   r3   )rª   Úsgnr¬   r•   r�   ÚbÚts     €€€€€rg   ÚtmpaddÚ$_create_lookup_table.<locals>.tmpadd‹   sŸ   ø€ áŒT�!�Q‘$˜‘(‹^˜c !™eÑ# aÑ'¨¨A©°©°A©Ñ5Ø�!‰e�Q‰Y˜˜A˜a™C™ ! A¡#™Ð&¨Ø�#�a‘%‰i˜‰]ˆO˜q 3 q¡5™y¨!™m˜_¨a°°1±©fØ��A�a‘C‘‰L™˜A A›Ñ&õ	(ro   r©   c                óF  >• T" [        TT[        T-  -  -   5      U[        T5      -  [        TS-  -  -  -   T-  TT[        T-  -  -   U -  -  SU -
  UT-  S-  -   /SU -
  UT-  S-  -
  /S[        R                  // T[        T-  -  T-  TTS-  U -
  -  T" XT5      -  5        g )Nr§   rR   r   )r4   r^   r   ÚHalf)rª   r¯   r¬   r•   r�   r°   ÚpÚqs     €€€€€€rg   r²   r³   —   s±   ø€ ÙŒT�!�aœ˜1™‘f‘*Ó ¤D¨£G¡¬A°°!±©HÑ 4Ñ4°qÑ8¸!¸aÄÀ1Á¹f¹*Àq¹ÑHØ�‰U�S˜‘U˜1‘W‰_Ð  A¡¨¨A©¨a©¡Ð0°1´a·f±f°+¸rØŒa�‰d‰F�1‰H�a˜!˜A™# ™'‘l¡2 a¨a£=Ñ0õ	2ro   é   é   c                óˆ   >• U T   n[         R                  U-  [        U5      -  [        / S/US-   -  S/US-   -  / T5      4/$ r”   )r   ÚNegativeOner   rP   ©ÚsubsÚNrt   r±   s     €€rg   Ú	make_log1Ú'_create_lookup_table.<locals>.make_log1²   sV   ø€ Ø�‰GˆÜ—‘ Ñ!¤)¨A£,Ñ.Ü˜˜a˜S ! a¡%™[¨1¨#¨q°1©u©+°r¸1Ó=ð?ð @ð 	@ro   c           	     ó`   >• U T   n[        U5      [        S/US-   -  / / S/US-   -  T5      4/$ r”   )r   rP   r¼   s     €€rg   Ú	make_log2Ú'_create_lookup_table.<locals>.make_log2·   sH   ø€ Ø�‰GˆÜ˜1“Ü˜!˜˜a !™e™ b¨"¨q¨c°1°q±5©k¸1Ó=ð?ð @ð 	@ro   c                ó&   >• T" U 5      T" U 5      -   $ rb   r˜   )r½   r¿   rÂ   s    €€rg   Ú	make_log3Ú'_create_lookup_table.<locals>.make_log3Á   s   ø€ Ù˜‹¡¨4£Ñ0Ð0ro   z3/2é   N©T)-ÚlistÚmapr   r^   r   ÚOner   r@   rN   rT   rW   r#   r8   r   r    rµ   r+   r'   r0   r   r/   r4   r7   r9   r-   rP   rF   ÚImaginaryUnitr»   rH   rI   rJ   rK   rG   rC   rD   rE   rL   rM   r;   r<   r=   r>   rA   rB   )rŒ   ru   Úcr‘   r–   r¦   r²   rÅ   r¬   r•   r�   r°   r¿   rÂ   rt   r¶   r·   r±   s   `       @@@@@@@@@@rg   Ú_create_lookup_tablerÎ   X   så	  ÿú€ ò$äœ˜T 7Ó+Ó,�M€A€qˆ!ˆQ�ÜˆSÑ>Ð?Ñ@€AØ	Œ!ˆQ‰$‰€Aà)*´·±¸DÀt÷ Y÷Iò3ð ‘X˜a“[ $¨Ð-Ð.€Eˆ"�Iô œxô  ñ Œ	�!�a‘%Ó˜!˜a™% 1 q¡5Ñ)Ñ)¨A¨3°°B¸¸¸Q¸q¹SÜˆa‹��Q˜‘U‘ÑœS  Q¡›Zô)áŒ	�!�a‘%Ó˜!˜a™% 1 q¡5Ñ)Ñ)¨2°¨s°Q°C¸¸Q¸q¹SÜˆa‹��Q˜‘U‘ÑœS  Q¡›Zô)áŒ	”!�q˜‘s˜a ™c‘lÑ"Ó# Q¨¡U¨a°!©eÑ$4Ñ4°q°c¸2¸rÀAÀ3ÈÈ!ÉÜˆa‹��Q˜‘U‘ÑœS  Q¡›Zô)áŒ	�1�Q‘3˜!˜A™#‘,¤Ñ"Ó# Q¨¡U¨a°!©eÑ$4Ñ4°b¸1¸#À¸sÀBÈÈ!ÉÜˆa‹��Q˜‘U‘ÑœS  Q¡›Zô)áˆˆQ‰�1�"‰˜˜A™�w  Q C¨¨Q¨q©S°!°q°b±'¼%À»(Ñ2BÜÑ% aÓ(Ó)ò+áŒˆA�‰E‹
�a�RÑ˜1˜q™5˜' Q¨¡U¨A¡I ;°°°q¸1±u¸a±i°[À!ÀAÁ#Ø	Œ#Œb�‰d�1‰f‹+‰”e˜A ™E“lÑ"¤3 q£6¨Q¨B¡<Ñ/´°A³¸±ô<áˆˆA‰��1‘‰�q˜1‘uÑ  1˜v r¨A¨q¨6°2°q¸±sØ	ˆA�‰E‰
”3�qœ‘t“9ÑœRÑô!ò;÷(ñ (ñ ˆ1ˆa„LÙ
ˆ1ˆb„MÙ
Œ1�6‰6�1ÔÙ
Œ1�6‰6�2Ô÷2ò 2ñ ˆ1ˆa„LÙ
ˆ1ˆb„MÙ
Œ1�6‰6�1ÔÙ
Œ1�6‰6�2Ôñ ŒŒJ�r‹N˜1ÑÓ˜r 2¨ s¨BÔ/ñ ŒˆQ‹��a�Sœ1Ÿ6™6˜( Q¨ F¨A¨q©D°©F´B¼ÀÀA»Ñ4FÔGÙŒˆQ‹�”a—f‘f�X ˜s¤Q§V¡V¬Q¯V©VÐ$4°a¸±d¸1±f¼bÄ(È1ÈaÃ.Ñ>PÔQñ ŒˆA‹��BœŸ™˜ 1 # q¨!¡t¨A¡v¬t´B«xÔ8ÙŒˆA‹��B˜˜œaŸf™f˜X q¨!¡t¨A¡v¬t´B«xÔ8ñ ŒˆQ‹��R˜!˜œx¨¨A›Ð/°°A±°a±¼¼b»À!¹ÔDö@ö
@ñ 	ŒˆQ‹�‰”9˜Q ™UÓ#Ñ	# Y°Ô5ÙŒˆQ‹�‰”9˜Q ™UÓ#Ñ	# Y°Ô5ö1áŒˆQ‹�‰�I˜tÔ$ÙŒˆQ�‰U‹Ù	”#�a“&Ó	œaŸe™e¤W¨a°¨V°R¸!¸¸q¸cÀ1ÀQÁ3Ó%GÐHÐIÑ	IØ	ôñ 	ŒŒS��Q‘‹Z‹™(¤3¤s¨1£v£;Ó/ÜŒw˜˜1�v¤§¡˜x¨!¨¨q´!·&±&¨k¸1¸Q¹3Ó?Ð
@Ð	AñBà	ôñ 	ŒˆA‹Ù	”1—?‘?Ð"¤2Ñ%Ó	&¬1¯=©=¼'À"ÀqÀcÈAÈqÈ6ÐSUØ”J˜r“NÑ"ó;$ð +%ð *&ñ 
&à	ôñ Œˆ1‹�ˆs�BœŸ™˜ 1 a &¨!¨Q©$¨q©&´$´r³(¸1±*Ô=ÙŒˆ1‹ˆr�A�3˜˜A˜¤§¡ ¨!¨Q©$¨q©&´4¼³8°)¸A±+Ô>ñ ŒˆA‹”—‘�˜"˜q˜c¤H¨R°£O´X¸bÀ!³_Ð#EÄzÐRTÃ~ÐVWÐYZÑVZÑGZÐ[\ÑG\Ø	Œ$Œr‹(‰
�1‰ôáŒˆA‹�”Q—V‘V˜Q�K ! Q ¬!¯&©&´!·&±&Ð)9¸1¸a¹4À¹6Ü
ŒAˆe‹H‰ðDØñDô ñ Œˆq�!‹�b˜1˜#  A¡ q˜z¨2¨qÔ1ñ ŒˆA‹���Rœ!Ÿ&™&˜ A 3¨¨1©¨a´´R³©jÔ9áŒˆQ‹��a�S˜1œaŸf™f˜+ r¨1¨a©4°´4¼³8±Ô<ñ ŒˆQ‹”!—&‘&�˜2 ˜s¤X¨b°!£_Ð$5¸¸1¹°u¸aÄÄRÃ¹jÔIñ Œ�‹�a�S˜"œx¨¨1›~Ð.°´H¸QÀ³NÐ0CÄRÈÁUÈ1ÈaÉ4ÁZÐPRÁ]ÔTU×TZÑTZÔ[ÙŒ�‹�a�S˜"œx¨¨1›~Ð.°´H¸QÀ³NÐ0CÄRÈÁUÈ1ÈaÉ4ÁZÐPRÁ]ÔTU×TZÑTZÔ[ñ Œ��1‹�r˜2  !¡˜u¨ r¨!¡t f¨a°©d°1©fÔ5ñ Œ��1‹�r˜a !™e˜H Q™J˜<¨!¨A©#°¨r°!©t¨¸¸Q¹°xÀ±z°lÀAÀqÁDÈÁFÔKñ6 Œ��1‹�r˜Q ™U A™I˜;¨¨1©¨°°°1±°q¸1±u¸a±iÐ/@À!ÀQÁ$ÀqÁ&Ì"ÔMñ Œ��1‹�r˜2  !¡ a R¨¡T˜{¨B°°1±°Q±¼¿¹Ô?ñ Œ
�1‹œŸ™¤§¡Ð'¨¨a¨S°1°#¸°r¼1¿6¹6ÔBÙŒ
�1‹œŸ™ ¤!§&¡&¡Ð)¨2°¨s°Q°C¸!¸¼XÀbÈ!»_ÈQÑ=NÕOro   )ÚtimethisrP   c                ó¸   ^• SS jnTU R                   ;  a  gU R                  (       a  [        U 5      4$ [        [	        U4S jU R
                   5       US95      $ )z3Create a hashable entity describing the type of f. c                ó"   • U R                  5       $ rb   )Ú	class_keyr{   s    rg   ÚkeyÚ_mytype.<locals>.key/  s   € Ø�{‰{‹}Ðro   r˜   c              3  óN   >#   • U  H  n[        UT5        H  o"v •  M     M     g 7frb   )r‚   )rd   r•   r±   r|   s      €rg   rh   Ú_mytype.<locals>.<genexpr>6  s   øé € ÐB¢F˜q´G¸A¸q·M¨qœ±M™¢Fùs   ƒ"%©rÓ   )r|   ztype[Basic]Úreturnztuple[int, int, str])Úfree_symbolsÚis_FunctionÚtypeÚtupleÚsortedrl   )rf   r|   rÓ   s    ` rg   r‚   r‚   -  sG   ø€ ôð 	�—‘ÓØØ	
��Ü�A‹wˆxˆÜ”ÔB A§F¢FÓBÈÑLÓMÐMro   c                  ó   • \ rS rSrSrSrg)Ú_CoeffExpValueErrori9  z<
Exception raised by _get_coeff_exp, for internal use only.
r˜   N)r    r¡   r¢   r£   Ú__doc__r¥   r˜   ro   rg   rß   rß   9  s   † ñò 	ro   rß   c                óH  • SSK Jn  [        U" U 5      5      R                  U5      u  p4U(       d  U[        R
                  4$ Uu  nUR                  (       a(  UR                  U:w  a  [        S5      eX4R                  4$ XA:X  a  U[        R                  4$ [        SU -  5      e)aP  
When expr is known to be of the form c*x**b, with c and/or b possibly 1,
return c, b.

Examples
========

>>> from sympy.abc import x, a, b
>>> from sympy.integrals.meijerint import _get_coeff_exp
>>> _get_coeff_exp(a*x**b, x)
(a, b)
>>> _get_coeff_exp(x, x)
(1, 1)
>>> _get_coeff_exp(2*x, x)
(2, 1)
>>> _get_coeff_exp(x**3, x)
(1, 3)
r   )Úpowsimpzexpr not of form a*x**bzexpr not of form a*x**b: %s)Úsympy.simplifyrâ   r   Úas_coeff_mulr   ÚZeroÚis_PowÚbaserß   r+   rË   )Úexprr|   râ   rÍ   Úms        rg   Ú_get_coeff_exprê   @  s†   € õ& 'Ü™w t›}Ó-×:Ñ:¸1Ó=�F€QÞØ”!—&‘&ˆyÐØ
�C€QØ‡x‡xØ�6‰6�Q‹;Ü%Ð&?Ó@Ð@Ø—%‘%ˆxˆØ	
‹Ø”!—%‘%ˆxˆä!Ð"?À$Ñ"FÓGÐGro   c                ó:   ^• U4S jm[        5       nT" XU5        U$ )aS  
Find the exponents of ``x`` (not including zero) in ``expr``.

Examples
========

>>> from sympy.integrals.meijerint import _exponents
>>> from sympy.abc import x, y
>>> from sympy import sin
>>> _exponents(x, x)
{1}
>>> _exponents(x**2, x)
{2}
>>> _exponents(x**2 + x, x)
{1, 2}
>>> _exponents(x**3*sin(x + x**y) + 1/x, x)
{-1, 1, 3, y}
c                óê   >• X:X  a  UR                  S/5        g U R                  (       a-  U R                  U:X  a  UR                  U R                  /5        g U R                   H  nT" X1U5        M     g ©NrR   )Úupdateræ   rç   r+   rl   )rè   r|   rn   ÚargumentÚ_exponents_s       €rg   rð   Ú_exponents.<locals>._exponents_u  sV   ø€ Ø‹9Ø�J‰J˜�sŒOØØ�;�;˜4Ÿ9™9¨›>Ø�J‰J˜Ÿ™�zÔ"ØØŸ	œ	ˆHÙ˜ SÖ)ò "ro   )Úset)rè   r|   rn   rð   s      @rg   Ú
_exponentsró   b  s    ø€ õ&*ô ‹%€CÙ�˜ÔØ€Jro   c                óŽ   • U R                  [        5       Vs1 s H   o!UR                  ;   d  M  UR                  iM"     sn$ s  snf )zAFind the types of functions in expr, to estimate the complexity. )Úatomsr   rÙ   Úfunc)rè   r|   Úes      rg   Ú
_functionsrø   ƒ  s4   € à ŸJ™J¤xÔ0ÓHÒ0�q¸¿¹Ñ4G‹FˆA�FŒFÑ0ÑHÐHùÒHs
   ˜A¯Ac                óŒ   ^^^^• S Vs/ s H  n[        UT/S9PM     snu  mmUUUU4S jm[        5       nT" X5        U$ s  snf )a8  
Find numbers a such that a linear substitution x -> x + a would
(hopefully) simplify expr.

Examples
========

>>> from sympy.integrals.meijerint import _find_splitting_points as fsp
>>> from sympy import sin
>>> from sympy.abc import x
>>> fsp(x, x)
{0}
>>> fsp((x-1)**3, x)
{1}
>>> fsp(sin(x+3)*x, x)
{-3, 0}
Úpqrr   c                ó  >• [        U [        5      (       d  g U R                  TT-  T-   5      nU(       a%  UT   S:w  a  UR                  UT   * UT   -  5        g U R                  (       a  g U R
                   H  nT" X15        M     g ry   )Ú
isinstancer   Úmatchr�   Úis_Atomrl   )rè   rn   ré   rï   Úcompute_innermostr¶   r·   r|   s       €€€€rg   rÿ   Ú1_find_splitting_points.<locals>.compute_innermostœ  ss   ø€ Ü˜$¤×%Ñ%ØØ�J‰J�q˜‘s˜Q‘wÓˆÞ��1‘˜“Ø�G‰G�Q�q‘T�E˜!˜A™$‘JÔØØ�<�<ØØŸ	œ	ˆHÙ˜hÖ,ò "ro   )r   rò   )rè   r|   rt   Ú	innermostrÿ   r¶   r·   s    `  @@@rg   Ú_find_splitting_pointsr  ˆ  sL   û€ ñ$ +/Ó/ª$ QŒD�˜Q˜CÔ ©$Ñ/�D€A€q÷
-ð 
-ô “€IÙ�dÔ&ØÐùò 0s   ‰Ac           	     ób  • [         R                  n[         R                  n[         R                  n[        U 5      n [        R                  " U 5      nU HÕ  nXa:X  a  X1-  nM  XR
                  ;  a  X&-  nM#  UR                  (       a�  XR                  R
                  ;  a„  UR                  R                  U5      u  pxX�4:w  a&  [        UR                  5      R                  U5      u  pxX�4:X  a5  X1UR                  -  -  nU[        [        XvR                  -  SS95      -  nMÑ  XF-  nM×     X#U4$ )aI  
Split expression ``f`` into fac, po, g, where fac is a constant factor,
po = x**s for some s independent of s, and g is "the rest".

Examples
========

>>> from sympy.integrals.meijerint import _split_mul
>>> from sympy import sin
>>> from sympy.abc import s, x
>>> _split_mul((3*x)**s*sin(x**2)*x, x)
(3**s, x*x**s, sin(x**2))
F)r½   )r   rË   r   r   Ú	make_argsrÙ   ræ   r+   rç   rä   r   r%   r&   )	rf   r|   r‰   ÚpoÚgrl   r•   rÍ   r±   s	            rg   Ú
_split_mulr  ¬  sô   € ô �%‰%€CÜ	
�‰€BÜ	�‰€AÜ˜!Ó€Aä�=Š=˜Ó€DÛˆØ‹6Ø‰GŠBØ—n‘nÓ$Ø‰HŠCà�x�x˜A§U¡U×%7Ñ%7Ó7Ø—v‘v×*Ñ*¨1Ó-‘�Ø˜“9Ü% a§f¡fÓ-×:Ñ:¸1Ó=‘D�AØ˜“9Ø˜QŸU™U™(‘N�BØœ:¤h¨q·%±%©x¸eÑ&DÓEÑE�CÙØ‰FŠAñ ð  �Aˆ:Ðro   c                ó"  • [         R                  " U 5      n/ nU Hp  nUR                  (       aK  UR                  R                  (       a0  UR                  nUR
                  nUS:  a  U* nSU-  nX%/U-  -  nM_  UR                  U5        Mr     U$ )zì
Return a list ``L`` such that ``Mul(*L) == f``.

If ``f`` is not a ``Mul`` or ``Pow``, ``L=[f]``.
If ``f=g**n`` for an integer ``n``, ``L=[g]*n``.
If ``f`` is a ``Mul``, ``L`` comes from applying ``_mul_args`` to all factors of ``f``.
r   rR   )r   r  ræ   r+   rz   rç   rƒ   )rf   rl   Úgsr  rt   rç   s         rg   Ú	_mul_argsr
  Ó  s|   € ô �=Š=˜Ó€DØ	€BÛˆØ�8�8˜Ÿ™×(×(Ø—‘ˆAØ—6‘6ˆDØ�1‹uØ�B�Ø˜‘v�Ø�&˜‘(‰NŠBà�I‰I�aŽLñ ð €Iro   c                óØ   • [        U 5      n[        U5      S:  a  g[        U5      S:X  a  [        U5      /$ [        US5       VVs/ s H  u  p#[	        U6 [	        U6 4PM     snn$ s  snnf )a_  
Find all the ways to split ``f`` into a product of two terms.
Return None on failure.

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

Although the order is canonical from multiset_partitions, this is
not necessarily the best order to process the terms. For example,
if the case of len(gs) == 2 is removed and multiset is allowed to
sort the terms, some tests fail.

Examples
========

>>> from sympy.integrals.meijerint import _mul_as_two_parts
>>> from sympy import sin, exp, ordered
>>> from sympy.abc import x
>>> list(ordered(_mul_as_two_parts(x*sin(x)*exp(x))))
[(x, exp(x)*sin(x)), (x*exp(x), sin(x)), (x*sin(x), exp(x))]
r§   N)r
  ÚlenrÜ   r[   r   )rf   r	  r|   Úys       rg   Ú_mul_as_two_partsr  ê  sb   € ô. 
�1‹€BÜ
ˆ2ƒw�ƒ{ØÜ
ˆ2ƒw�!ƒ|Ü�b“	ˆ{ÐÜ-@ÀÀQÔ-GÔHÒ-G¡6 AŒS�!ˆW”c˜1�gÓÑ-GÒHÐHùÓHs   ÁA&c                ó²  • S n[        [        U R                  5      [        U R                  5      -
  5      nUSU R                  -   US-  -   -  nUS[
        -  US-
  U R                  -  -  -  nU[        U" U R                  U5      U" U R                  U5      U" U R                  U5      U" U R                  U5      U R                  U-  XU-  -  -  5      4$ )zFReturn C, h such that h is a G function of argument z**n and
g = C*h. c                ó‚   • [         R                  " U [        U5      5       VVs/ s H  u  p#X#-   U-  PM     snn$ s  snnf )z4(a1, .., ak) -> (a1/n, (a1+1)/n, ..., (ak + n-1)/n) )Ú	itertoolsÚproductÚrange)Úparamsrt   r•   re   s       rg   ÚinflateÚ_inflate_g.<locals>.inflate  s5   € ä&/×&7Ò&7¸ÄÀaÃÔ&IÔJÒ&I™d˜a�‘˜”	Ñ&IÒJÐJùÓJs   ¤;rR   r§   )r   r  r†   rˆ   r«   r   ÚdeltarP   r…   Úaotherr‡   Úbotherrï   )r  rt   r  ÚvÚCs        rg   Ú
_inflate_gr  	  sÁ   € ò
Kô 	
Œ#ˆa�d‰d‹)”c˜!Ÿ$™$“iÑ
Ó €AØ	ˆA�—‘‰H�q˜‘s‰NÑ€AØˆ!ŒB‰$�1�q‘5˜!Ÿ'™'‘/Ñ	"Ñ"€AØŒg‘g˜aŸd™d AÓ&©°·±¸!Ó(<Ù˜aŸd™d AÓ&©°·±¸!Ó(<Ø—j‘j !‘m a¨A©#¡hÑ.ó0ð 0ð 0ro   c                óÀ   • S n[        U" U R                  5      U" U R                  5      U" U R                  5      U" U R                  5      SU R
                  -  5      $ )zHTurn the G function into one of inverse argument
(i.e. G(1/x) -> G'(x)) c                ó8   • U  Vs/ s H  nSU-
  PM
     sn$ s  snf rí   r˜   )Úlr•   s     rg   ÚtrÚ_flip_g.<locals>.tr  s   € Ù Ó!šq˜!��A”™qÑ!Ð!ùÒ!ó   …rR   )rP   r‡   r  r…   r  rï   )r  r   s     rg   Ú_flip_gr#    sB   € ò"ä‘2�a—d‘d“8™R §¡›\©2¨a¯d©d«8±R¸¿¹³\À1ÀQÇZÁZÁ<ÓPÐPro   c           	     óì  • US:  a  [        [        U 5      U* 5      $ [        UR                  5      n[        UR                  5      n[        X5      u  p@U R                  nUS[        -  SU-
  S-  -  U[        SS5      -  -  -  nXRU-  -  n[        U5       Vs/ s H
  ofS-   U-  PM     nnU[        U R                  U R                  U R                  [        U R                  5      U-   U5      4$ s  snf )a@  
Let d denote the integrand in the definition of the G function ``g``.
Consider the function H which is defined in the same way, but with
integrand d/Gamma(a*s) (contour conventions as usual).

If ``a`` is rational, the function H can be written as C*G, for a constant C
and a G-function G.

This function returns C, G.
r   r§   rR   r©   )Ú_inflate_fox_hr#  r   r¶   r·   r  rï   r   r   r  rP   r…   r  r‡   rÉ   r  )r  r•   r¶   r·   ÚDr^   rt   Úbss           rg   r%  r%  "  sÙ   € ð 	ˆ1ƒuÜœg a›j¨1¨"Ó-Ð-Ü	ˆ!�#‰#‹€AÜ	ˆ!�#‰#‹€Aô �aÓ�D€AØ	�
‰
€AØˆ!ŒB‰$�1�q‘5˜!‘)Ñ	˜Q¤¨¨Q£Ñ/Ñ	/Ñ/€AØˆA‰�I€AÜ" 1œXÓ	&šX˜ˆq‰5�!Œ)™X€BÐ	&ØŒg�a—d‘d˜AŸH™H a§d¡d¬D°·±«N¸RÑ,?ÀÓCÐCÐCùò 
's   ÂC1zdict[tuple[str, str], Dummy]Ú_dummiesc                óT   • [        X40 UD6nXBR                  ;   a  [        U 40 UD6$ U$ )z¦
Return a dummy. This will return the same dummy if the same token+name is
requested more than once, and it is not already in expr.
This is for being cache-friendly.
)Ú_dummy_rÙ   r   )ÚnameÚtokenrè   ÚkwargsÚds        rg   Ú_dummyr/  ?  s4   € ô 	�Ñ&˜vÑ&€AØ×ÑÓÜ�TÑ$˜VÑ$Ð$Ø€Hro   c                óT   • X4[         ;  a  [        U 40 UD6[         X4'   [         X4   $ )zT
Return a dummy associated to name and token. Same effect as declaring
it globally.
)r(  r   )r+  r,  r-  s      rg   r*  r*  K  s1   € ð
 ˆ=œHÓ$Ü"'¨Ñ"7°Ñ"7Œ�$�ÑÜ�T�MÑ"Ð"ro   c                óh   ^• [        U4S jU R                  [        [        5       5       5      (       + $ )z�Check if f(x), when expressed using G functions on the positive reals,
will in fact agree with the G functions almost everywhere c              3  óB   >#   • U  H  nTUR                   ;   v •  M     g 7frb   )rÙ   ©rd   rè   r|   s     €rg   rh   Ú_is_analytic.<locals>.<genexpr>X  s   øé € ÐNÒ6M¨d�1˜×)Ñ)Ö)Ò6Mùó   ƒ)Úanyrõ   r@   r#   )rf   r|   s    `rg   Ú_is_analyticr7  U  s$   ø€ ô ÔN°a·g±g¼iÌÔ6MÓNÓNÔNÐNro   c                ó^  ^^• U(       a  U R                  S [        5      n Sn[        U [        5      (       d  U $ [	        S[
        S9u  mmn[        TT:  [        TT5      5      TT:*  4[        [        [        T5      5      [        :*  [        [        T5      S[        -  -
  5      [        :*  5      [        [        T5      [        -
  S5      4[        [        S[        T5      -  [        -   5      [        :*  [        S[        T5      -  [        -
  5      [        :*  5      [        [        T5      S5      4[        [        S[        T5      -  [        -   5      [        :  [        S[        T5      -  [        -
  5      [        :*  5      [        R                  4[        [        [        T5      [        S-  -
  5      [        S-  :*  [        [        T5      [        S-  -   5      [        S-  :*  5      [        [        T5      S5      4[        [        [        T5      [        S-  -
  5      [        S-  :*  [        [        T5      [        S-  -   5      [        S-  :  5      [        R                  4[        [        [        TS-  S-  S-   5      5      [        :  [        [        [        TS-  S-  S-   5      5      [        5      5      [        R                  4[        [        [        TS-  S-  S-   5      5      [        :  [        STS-  S-  S-   -  S5      5      [        R                  4[        [        [!        T5      5      [        :*  [        [!        [#        S[        -  [        R$                  -  5      T-  5      5      [        :*  5      [        [!        [#        [        R$                  * [        -  5      T-  5      S5      4[        [        [!        T5      5      [        S-  :*  [        [!        [#        [        * [        R$                  -  5      T-  5      5      [        S-  :*  5      [        [!        [#        [        R$                  * [        -  S-  5      T-  5      S5      4[        TT:*  [        TT:  U5      5      TT:*  4[        TS-  S5      TS-  S:„  -  TS-  S:„  4[        ST-  S5      ['        [        [        T5      5      5      [        T5      -  S:„  -  [        T5      S:„  4[        TS5      ['        [        [        T5      5      5      [        T5      -  S:„  -  [        T5      S:„  4[        [        T5      5      [        S-  :  ['        [        [        T5      5      5      [)        [        TS-  5      5      -  S:„  -  TS-  S:„  4/nU R*                  " U R,                   Vs/ s H  n[/        XA5      PM     sn6 n S	nU(       Ga«  Sn[1        U5       GH�  u  nu  pxUR*                  U R*                  :w  a  M%  [1        U R,                  5       GHP  u  pšX'R,                  S   R2                  ;   a!  U
R5                  UR,                  S   5      nSnO SnU
R5                  UR,                  S   5      nU(       d  Ml  UR,                  S
U UR,                  US-   S
 -    Vs/ s H  oÝR7                  U5      PM     nnU	/nU GH  n[1        U R,                  5       Hê  u  nnUU;   a  M  UU:X  a	  UU/-  n  M6  [        U[        5      (       aN  UR,                  S   U:X  a;  [        U[        5      (       a&  UR,                  S   UR,                  ;   a	  UU/-  n  M™  [        U[        5      (       d  M—  UR,                  S   U:X  d  M¬  [        U[        5      (       d  MÃ  UR,                  S   UR,                  ;   d  Mâ  UU/-  n  GM     GM
     [9        U5      [9        U5      S-   :w  a  GMÛ  [1        U R,                  5       VVs/ s H  u  nnUU;  d  M  UPM     snnUR7                  U5      /-   n[:        (       a  US;  a  [=        SU5        U R*                  " U6 n S	n  GM�     GM’     U(       a  GM«  UU4S jnU R                  S U5      n [:        (       a  [=        SU 5        U $ s  snf s  snf s  snnf )az  
Do naive simplifications on ``cond``.

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

Note that this routine is completely ad-hoc, simplification rules being
added as need arises rather than following any logical pattern.

Examples
========

>>> from sympy.integrals.meijerint import _condsimp as simp
>>> from sympy import Or, Eq
>>> from sympy.abc import x, y
>>> simp(Or(x < y, Eq(x, y)))
x <= y
c                ó   • U R                   $ rb   ©Úis_Relational©Ú_s    rg   r}   Ú_condsimp.<locals>.<lambda>o  s   €  a§o¢oro   Fzp q r)r�   r§   r   rR   éþÿÿÿTN)
r   r§   r¸   é   é   é   é   é   é   é   zused new rule:c                óà  >• U R                   S:w  d  U R                  S:w  a  U $ U R                  nUR                  [	        T5      T-  5      nU(       d&  UR                  [        [        T5      T-  5      5      nU(       dg  [        U[        5      (       aP  UR                  S   R                  (       d2  UR                  S   [        R                  L a  UR                  S   S:„  $ U $ UT   S:„  $ )Nz==r   rR   )Úrel_opÚrhsÚlhsrý   r"   r)   r'   rü   r*   rl   Úis_polarr   ÚInfinity)ÚrelÚLHSré   r¶   r·   s      €€rg   Úrel_touchupÚ_condsimp.<locals>.rel_touchupÂ  s¹   ø€ Ø�:‰:˜Ó §¡¨A£ØˆJð �g‰gˆØ�I‰I”c˜!“f˜a‘iÓ ˆÞØ—	‘	Ô-¬j¸«m¸QÑ.>Ó?Ó@ˆAÞÜ˜#Ô0×1Ñ1¸#¿(¹(À1¹+×:N×:NØŸ™ ™¤q§z¡zÒ1ØŸ™ ™ a™Ð(ØˆJØ�!‘�q‘Ðro   c                ó   • U R                   $ rb   r:  r<  s    rg   r}   r>  Ñ  s   €  !§/¢/ro   z_condsimp: )Úreplacer   rü   rX   r   r   rU   r   rT   r#   r"   r   r   Úfalser   Útruer)   r,   rÌ   r7   r4   rö   rl   Ú	_condsimpÚ	enumeraterÙ   rý   r½   r  r   Úprint)rŠ   Úfirstrª   Úrulesr=  ÚchangeÚiruleÚfroÚtort   Úarg1ré   Únumr|   Ú	otherargsÚ	otherlistÚarg2ÚkÚarg3Úarg_ÚnewargsrO  r¶   r·   s                         @@rg   rU  rU  [  s  ù€ ö& Ø�|‰|Ñ5Ô7GÓHˆØˆÜ�dœO×,Ñ,ØˆÜ�g¤4Ñ(�G€A€qˆ!ô
 
ˆA�‰E”2�a˜“8Ó	˜a 1™fÐ%ô 
ŒS”�Q“‹[œBÑ¤¤C¨£F¨Q¬r©T¡MÓ 2´bÑ 8Ó	9Ü	ŒC�‹F”R‰K˜Ó	ð	ä	ŒS�”3�q“6‘œB‘Ó¤2Ñ%¤s¨1¬S°«V©8´b©=Ó'9¼RÑ'?Ó	@Ü	ŒC�‹F�A‹ð	ä	ŒS�”3�q“6‘œB‘Ó¤"Ñ$¤c¨!¬C°«F©(´R©-Ó&8¼BÑ&>Ó	?Ü	
�‰ð	ä	ŒS”�Q“œ"˜Q™$‘Ó¤2 a¡4Ñ'¬¬S°«V´b¸±d©]Ó);¼rÀ!¹tÑ)CÓ	DÜ	ŒC�‹F�A‹ð	ä	ŒS”�Q“œ"˜Q™$‘Ó¤2 a¡4Ñ'¬¬S°«V´b¸±d©]Ó);¼bÀ¹dÑ)BÓ	CÜ	
�‰ð	ä	ŒS”�Q˜‘T˜!‘V˜a‘Z“Ó!¤BÑ&¬¬3¬s°1°a±4¸±6¸A±:«Ó+?ÄÓ(DÓ	EÜ	
�‰ð	ä	ŒC”�A�q‘D˜‘F˜Q‘J“Ó ¤2Ñ%¤r¨!¨Q°©T°!©V°a©Z©.¸!Ó'<Ó	=Ü	
�‰ð	ä	ŒSÔ$ QÓ'Ó(¬BÑ.ÜÔ"¤9¨R´©U´1·?±?Ñ-BÓ#CÀAÑ#EÓFÓGÌ2ÑMó
Oä	Ô¤	¬1¯?©?Ð*:¼2Ñ*=Ó >¸qÑ @ÓAÀ1Ó	Eð	Gô 
ŒSÔ$ QÓ'Ó(¬B¨q©DÑ0ÜÔ"¤9¬b¨S´·±Ñ-@Ó#AÀ!Ñ#CÓDÓEÌÈAÉÑMó
Oä	Ô¤	¬1¯?©?Ð*:¼2Ñ*=¸aÑ*?Ó @ÀÑ BÓCÀQÓ	Gð	Iô 
ˆA�‰F”C˜˜A™˜q“MÓ	" A¨¡FÐ+Ü	ˆAˆq‰D�!‹˜˜1™˜q™Ñ	! 1 a¡4¨!¡8Ð,Ü	ˆAˆa‰C�‹”sœ3œs 1›v›;Ó'¬¨A«Ñ.°Ñ2Ñ	3´S¸³V¸a±ZÐ@Ü	ˆAˆq‹”SœœS ›V›Ó%¤c¨!£fÑ,¨qÑ0Ñ	1´3°q³6¸A±:Ð>Ü
Œc�!‹f‹+œ˜1™Ñ
¤¤S¬¨Q«£[Ó!1´$´s¸1¸a¹4³y³/Ñ!AÀAÑ!EÑ	FÈÈ1ÉÈqÉÐQð9€Eð< �9Š9°D·I²IÓ>²I¨q”y Ö*±IÑ>Ð?€DØ€Fß
ØˆÜ )¨%× 0ÑˆE‘9�CØ�x‰x˜4Ÿ9™9Ó$ÙÜ$ T§Y¡Y×/‘�ØŸ™ ™×0Ñ0Ó0ØŸ
™
 3§8¡8¨A¡;Ó/�AØ‘Cà�CØŸ
™
 3§8¡8¨A¡;Ó/�AÞÙØ03·±¸¸#°ÀÇÁÈ#ÐPQÉ'È(ÐASÒ0SÓTÒ0S¨1ŸV™V AžYÑ0S�	ÐTØ˜C�	Ü%�DÜ#,¨T¯Y©YÖ#7™˜˜4Ø 	›>Ù$Ø 4›<Ø%¨!¨Ñ,˜IÚ!Ü% d¬C×0Ñ0°T·Y±Y¸q±\ÀQÓ5FÜ *¨4´× 5Ñ 5¸$¿)¹)ÀA¹,È$Ï)É)Ó:SØ%¨!¨Ñ,˜IÚ!Ü% d¬C×0Ó0°T·Y±Y¸q±\ÀQÕ5FÜ *¨4´× 5Ó 5¸$¿)¹)ÀA¹,È$Ï)É)Õ:SØ%¨!¨Ñ,˜IÛ!ô $8ñ &ô �y“>¤S¨£^°aÑ%7Ó7ÚÜ1:¸4¿9¹9Ô1Eô 2Ò1E¡I Q¨Ø yÑ0÷  Ñ1Eò 2Ø57·W±W¸Q³Z°LñA�ç’;ØÐ$FÓFÜÐ.°Ô6Ø—y’y 'Ð*�Ø�ÛôG 0ñ !1÷ ‰&öVð �<‰<Ñ1°;Ó?€Dß‚{Üˆm˜TÔ"Ø€Kùò ?ùò  Uùó&2s   ÖbÚb$ß;b)
àb)
c                ób   • [        U [        5      (       a  U $ [        U R                  5       5      $ )zRe-evaluate the conditions. )rü   ÚboolrU  Údoit)rŠ   s    rg   Ú
_eval_condrj  Ö  s%   € ä�$œ×ÑØˆÜ�T—Y‘Y“[Ó!Ð!ro   c                óX   • [        X5      nU(       d  UR                  [         S 5      nU$ )zÚBring expr nearer to its principal branch by removing superfluous
factors.
This function does *not* guarantee to yield the principal branch,
to avoid introducing opaque principal_branch() objects,
unless full_pb=True. c                ó   • U $ rb   r˜   )r|   r  s     rg   r}   Ú&_my_principal_branch.<locals>.<lambda>é  s   € ¹ro   )r(   rR  )rè   ÚperiodÚfull_pbrn   s       rg   Ú_my_principal_branchrp  á  s'   € ô ˜4Ó
(€CÞØ�k‰kÔ*©NÓ;ˆØ€Jro   c           	     óx  ^	^
• [        X5      u  nm
[        UR                  U5      u  nm	UR                  5       n[        XV5      nU [	        T	5      UT
S-   T	-  S-
  -  -  -  nU	U
4S jnU[        U" UR                  5      U" UR                  5      U" UR                  5      U" UR                  5      XS-  5      4$ )z„
Rewrite the integral fac*po*g dx, from zero to infinity, as
integral fac*G, where G has argument a*x. Note po=x**s.
Return fac, G.
rR   c                óJ   >• U  Vs/ s H  oST-   T-  -   S-
  PM     sn$ s  snf rí   r˜   )r  r•   r°   Úss     €€rg   r   Ú_rewrite_saxena_1.<locals>.trü  s*   ø€ Ù+,Ó-ª1 a�Q˜‘U˜A‘I‘ Ô!©1Ñ-Ð-ùÒ-s   † )
rê   rï   Ú
get_periodrp  r#   rP   r…   r  r‡   r  )r‰   r  r  r|   r=  r•   rn  r  r   r°   rs  s            @@rg   Ú_rewrite_saxena_1rv  í  s¨   ù€ ô ˜"Ó �D€A€qÜ˜!Ÿ*™* aÓ(�D€A€qØ�\‰\‹^€FÜ˜QÓ'€Að 	ŒS�‹V�A˜˜Q™ ™	 A™Ñ&Ñ&Ñ'€Aö.àŒg‘b˜Ÿ™“h¡ 1§8¡8£©b°·±«h¹¸1¿8¹8»Ø‘cóð ð ro   c                óà
  • U R                   n[        U R                  U5      u  pE[        [	        U R
                  5      [	        U R                  5      [	        U R                  5      [	        U R                  5      /5      u  pgp‰X‰:”  a^  S n
[        [        U
" U R
                  5      U
" U R                  5      U
" U R                  5      U
" U R                  5      X-  5      U5      $ U R
                   Vs/ s H  n[        U5      * S:  PM     snU R                   Vs/ s H  nSS[        U5      -
  :  PM     sn-   n[        U6 nXÐR                   Vs/ s H  n[        U5      * S:  PM     sn-  nXÐR                   Vs/ s H  nSS[        U5      -
  :  PM     sn-  n[        U6 n[        U R                  5      * U	S-   U-
  S-  -   X˜-
  :„  nS nS nU" S5        U" SX4XgX‰45        U" S[!        U R                  5      [!        U R                  5      45        U" S	[!        U R
                  5      [!        U R                  5      45        U" S
XïU45        / n/ nSU:*  X‰:  SU:*  /nSU:*  SU:*  [#        X˜S-   5      [%        [        [#        US5      [#        XhS-   5      5      5      /nSU:*  [#        X˜5      /n['        [)        US-  5      S-   5       H2  nU[+        [-        [/        U5      5      USU-  -
  [0        -  5      /-  nM4     US:„  [-        [/        U5      5      U[0        -  :  /n[+        US5      U/nU(       a  / nUUU4 H  nU[        UU-   U-   6 /-  nM     UU-  nU" SU5        U/nU(       a  / n[        [#        US5      US-   U:*  Xi:*  [-        [/        U5      5      U[0        -  :  /UQ76 /nUU-  nU" SU5        UU/nU(       a  / n[        X‰:  SU:*  US:„  [#        [-        [/        U5      5      U[0        -  5      /UQ76 /nU[        X‰S-
  :*  [#        US5      [#        [-        [/        U5      5      S5      /UQ76 /-  nUU-  nU" SU5        / nU[#        X‰5      [#        US5      [#        [/        U5      S5      [+        US5      /-  nU(       d  UU/-  n/ n[3        U R                  U R                  5       H  u  pËUX¼-
  /-  nM     U[        [5        U6 5      S:  /-  n[        U6 nUU/-  nU" SU/5        [        US:„  [-        [/        U5      5      U[0        -  :  5      /nU(       d  UU/-  n[        U6 nUU/-  nU" SU/5        [7        U6 $ s  snf s  snf s  snf s  snf )a:  
Return a condition under which the mellin transform of g exists.
Any power of x has already been absorbed into the G function,
so this is just $\int_0^\infty g\, dx$.

See [L, section 5.6.1]. (Note that s=1.)

If ``helper`` is True, only check if the MT exists at infinity, i.e. if
$\int_1^\infty g\, dx$ exists.
c                ó8   • U  Vs/ s H  nSU-
  PM
     sn$ s  snf rí   r˜   ©r  r|   s     rg   r   Ú _check_antecedents_1.<locals>.tr  s   € Ù#$Ó%¢1˜a�A˜”E¡1Ñ%Ð%ùÒ%r"  rR   r§   c                 ó   • [        U 6   g rb   )Ú_debug)Úmsgs    rg   r\   Ú#_check_antecedents_1.<locals>.debug"  s	   € Ü�Šro   c                ó   • [        X5        g rb   ©Ú_debugf)Ústringr"   s     rg   r]   Ú$_check_antecedents_1.<locals>.debugf%  s
   € Ü�Õro   z$Checking antecedents for 1 function:z*  delta=%s, eta=%s, m=%s, n=%s, p=%s, q=%sz  ap = %s, %sz  bq = %s, %sz"  cond_3=%s, cond_3*=%s, cond_4=%sr   z	  case 1:z	  case 2:z	  case 3:z  extra case:z  second extra case:)r  rê   rï   r   r  r‡   r…   r†   rˆ   Ú_check_antecedents_1rP   r  r  r    rT   r«   rÉ   r   rW   r  r.   r   r#   r)   r   Úzipr   rU   ) r  r|   Úhelperr  Úetar=  ré   rt   r¶   r·   r   r°   r•   ÚtmpÚcond_3Úcond_3_starÚcond_4r\   r]   ÚcondsÚcase1Útmp1Útmp2Útmp3rc  Úextrar±   Úcase2Úcase3Ú
case_extrars  Úcase_extra_2s                                    rg   r„  r„    sG  € ð �G‰G€EÜ˜AŸJ™J¨Ó*�F€CÜ”C˜Ÿ™“Iœs 1§4¡4›y¬#¨a¯d©d«)´S¸¿¹³YÐ?Ó@�J€Aˆ!àƒuò	&ä#¤G©B¨q¯t©t«H±b¸¿¹³lÙ,.¨q¯t©t«H±b¸¿¹³lÀAÁEó%Kà$%ó'ð 	'ð  !ŸtštÓ
$št˜!ŒBˆq‹Eˆ6�AŒ:™tÑ
$¸q¿tºtÓ'Dºt¸!¨¨A´°1³©I¬¹tÑ'DÑ
D€CÜ�#ˆY€Fà§¢Ó)¢˜1ŒR�‹UˆF�QŒJ¡Ñ)Ñ)€CØ§8¢8Ó,¢8˜aˆA�”B�q“E‘	ŒM¡8Ñ,Ñ,€CÜ�s�)€Kä�!—$‘$‹xˆi˜1˜q™5 1™9 a™-Ñ'¨!©%Ñ/€Fòòñ 
Ð
0Ô1Ù
Ð7Ø˜˜aÐ#ô%á
ˆ?œT !§$¡$›Z¬¨a¯h©h«Ð8Ô9Ù
ˆ?œT !§$¡$›Z¬¨a¯h©h«Ð8Ô9Ù
Ð/°&ÀvÐ1NÔOà€Eð €EØ�‰F�A‘E˜1 ™6Ð"€DØ�‰F�A˜‘FœB˜q a¡%›L¬#¬c´"°Q¸³(¼B¸qÀaÁ%»LÓ.IÓ*JÐK€DØ�‰F”B�q“HÐ€DÜ”7˜5 ™7Ó# aÑ'Ö(ˆØ””CÔ+¨CÓ0Ó1°E¸A¸a¹C±KÄÑ3CÓDÐEÑEŠñ )à�1‰9”cÔ-¨cÓ2Ó3°e¼B±hÑ>Ð
?€CÜ��Q‹Z˜Ð €EÞØˆØ�D˜$ÓˆØ”#˜˜C™ %™Ð)Ð*Ñ*Šñ  à	ˆU�N€EÙ	ˆ+�uÔð ˆH€EÞØˆÜ”�A�q“˜1˜q™5 A™: q¡vÜÔ(¨Ó-Ó.°´r±Ñ9ðCØ<AòCð D€Eà	ˆU�N€EÙ	ˆ+�uÔð �VÐ€EÞØˆÜ�‘˜˜Q™ ¨¡	¬2¬cÔ2EÀcÓ2JÓ.KÈUÔSUÉXÓ+Vð Øòð €Eà	Œc�!˜1‘u‘*œb ¨›l¬B¬sÔ3FÀsÓ3KÓ/LÈaÓ,PÐYÐSXÒYÐZÑZ€EØ	ˆU�N€EÙ	ˆ+�uÔð €JØ”2�a“8œR  q›\¬2Ô.AÀ#Ó.FÈÓ+JÌBÈsÐTUËJÐWÑW€JÞØ�v�hÑˆ
Ø
€AÜ�A—D‘D˜!Ÿ$™$–‰ˆØ	ˆa‰eˆW‰Šñ  à”2”c˜1�g“; ‘?Ð#Ñ#€JÜ�jÐ!€JØ	ˆjˆ\Ñ€EÙ	ˆ/˜J˜<Ô(ä˜ ™	¤3Ô':¸3Ó'?Ó#@À5ÌÁ8Ñ#KÓLÐM€LÞØ˜˜Ñ ˆÜ˜Ð%€LØ	ˆlˆ^Ñ€EÙ	Ð
  < .Ô1ô
 ˆuˆ:Ðùòm %ùÒ'Dùò *ùÚ,s   Ã4UÄU!ÅU&Å?U+c                ó–  • SSK Jn  [        U R                  U5      u  p4SU-  nU R                   H  nU[        US-   5      -  nM     U R                   H  nU[        SU-
  S-
  5      -  nM     U R                   H  nU[        SU-
  S-
  5      -  nM     U R                   H  nU[        US-   5      -  nM     U" [        U5      5      $ )a[  
Evaluate $\int_0^\infty g\, dx$ using G functions,
assuming the necessary conditions are fulfilled.

Examples
========

>>> from sympy.abc import a, b, c, d, x, y
>>> from sympy import meijerg
>>> from sympy.integrals.meijerint import _int0oo_1
>>> _int0oo_1(meijerg([a], [b], [c], [d], x*y), x)
gamma(-a)*gamma(c + 1)/(y*gamma(-d)*gamma(b + 1))
r   )Ú	gammasimprR   )
rã   r—  rê   rï   r‡   rN   r…   r  r  r%   )r  r|   r—  r‡  r=  rn   r°   r•   s           rg   Ú	_int0oo_1r˜  r  s»   € õ )ä˜AŸJ™J¨Ó*�F€CØ
ˆC‰%€Cà�TŒTˆØŒu�Q˜‘U‹|ÑŠñ à�TŒTˆØŒu�Q˜‘U˜Q‘YÓÑŠñ à�XŒXˆØŒu�Q˜‘U˜Q‘YÓÑŠñ à�XŒXˆØŒu�Q˜‘U‹|ÑŠñ á”Z “_Ó%Ð%ro   c                ó  ^^^• UU4S jn[        UT5      u  px[        UR                  T5      u  py[        UR                  T5      u  pzU	S:  S:X  a  U	* n	[        U5      nU
S:  S:X  a  U
* n
[        U5      nU	R                  (       a  U
R                  (       d  gU	R                  U	R
                  pËU
R                  U
R
                  pí[        X¾-  XÜ-  5      nXûU-  -  nXýU-  -  n[        UU5      u  nn[        UU5      u  nnU" U5      nU" U5      nU UU-  -  n [        UR                  T5      u  nn[        UR                  T5      u  nnUS-   U-  S-
  mU [        U5      UT-  -  -  n U4S jn[        U" UR                  5      U" UR                  5      U" UR                  5      U" UR                  5      UT-  5      n[        UR                  UR                  UR                  UR                  UT-  5      nSSKJn  U" U SS9X#4$ )	a�  
Rewrite the integral ``fac*po*g1*g2`` from 0 to oo in terms of G
functions with argument ``c*x``.

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

Return C, f1, f2 such that integral C f1 f2 from 0 to infinity equals
integral fac ``po``, ``g1``, ``g2`` from 0 to infinity.

Examples
========

>>> from sympy.integrals.meijerint import _rewrite_saxena
>>> from sympy.abc import s, t, m
>>> from sympy import meijerg
>>> g1 = meijerg([], [], [0], [], s*t)
>>> g2 = meijerg([], [], [m/2], [-m/2], t**2/4)
>>> r = _rewrite_saxena(1, t**0, g1, g2, t)
>>> r[0]
s/(4*sqrt(pi))
>>> r[1]
meijerg(((), ()), ((-1/2, 0), ()), s**2*t/4)
>>> r[2]
meijerg(((), ()), ((m/2,), (-m/2,)), t/4)
c                óâ   >• [        U R                  T5      u  pU R                  5       n[        U R                  U R
                  U R                  U R                  [        XT5      TU-  -  5      $ rb   )	rê   rï   ru  rP   r…   r  r‡   r  rp  )r  r•   r°   Úperro  r|   s       €€rg   ÚpbÚ_rewrite_saxena.<locals>.pb«  sZ   ø€ Ü˜aŸj™j¨!Ó,‰ˆØ�l‰l‹nˆÜ�q—t‘t˜QŸX™X q§t¡t¨Q¯X©XÜ+¨A°GÓ<¸QÀ¹TÑAóCð 	Cro   r   TNrR   c                ó8   >• U  Vs/ s H  oT-   PM	     sn$ s  snf rb   r˜   )r  r•   r+   s     €rg   r   Ú_rewrite_saxena.<locals>.trÐ  s   ø€ Ù!"Ó#¢˜A�C”¡Ñ#Ð#ùÒ#ó   †©Ú	powdenest©Úpolar)rê   rï   r#  Úis_Rationalr¶   r·   r   r  r#   rP   r…   r  r‡   r  rã   r¢  )r‰   r  Úg1Úg2r|   ro  rœ  r=  rs  Úb1Úb2Úm1Ún1Úm2Ún2ÚtauÚr1Úr2ÚC1ÚC2Úa1r°   Úa2r   r¢  r+   s       ``                   @rg   Ú_rewrite_saxenarµ  �  sÐ  ú€ ö6Cô ˜"˜aÓ �D€AÜ˜2Ÿ;™;¨Ó*�E€AÜ˜2Ÿ;™;¨Ó*�E€AØ
ˆQ‰�4ÓØˆSˆÜ�R‹[ˆØ
ˆQ‰�4ÓØˆSˆÜ�R‹[ˆØ�>�> §§ØØ�T‰T�2—4‘4ˆØ�T‰T�2—4‘4ˆÜ
ˆr‰u�b‘eÓ
€CØ	�"‰u‰€BØ	�"‰u‰€Bä˜˜BÓ�F€BˆÜ˜˜BÓ�F€BˆÙ	ˆB‹€BÙ	ˆB‹€Bàˆ2ˆb‰5�L€CÜ˜2Ÿ;™;¨Ó*�E€BˆÜ˜2Ÿ;™;¨Ó*�E€Bˆð ˆq‰5�!‰)�a‰-€CØ
Œs�1‹v˜˜C™ÑÑ
 €Cõ$ä	‘�B—E‘E“™B˜rŸy™y›M©2¨b¯e©e«9±b¸¿¹³mÀRÈÁTÓ	J€BÜ	�—‘˜Ÿ	™	 2§5¡5¨"¯)©)°R¸±TÓ	:€Bå(Ù�S Ñ% rÐ-Ð-ro   c                óœ.  ^ ^^.^/^0^1^2^3^4^5^6• [        T R                  U5      u  m3n[        TR                  U5      u  m/n[        [        T R                  5      [        T R
                  5      [        T R                  5      [        T R                  5      /5      u  pEm5m6[        [        TR                  5      [        TR
                  5      [        TR                  5      [        TR                  5      /5      u  pgm0m2XE-   T5T6-   S-  -
  nXg-   T0T2-   S-  -
  n	T R                  T5T6-
  S-  -   S-   n
TR                  T0T2-
  S-  -   S-   nT2T0-
  T6T5-
  -
  nST6T5-
  -
  U-
  U
-
  n[        T2U-
  U-
  -  [        [        T/5      5      -   T2T0-
  -  m1[        T6U-
  U-
  -  [        [        T35      5      -   T6T5-
  -  m4[        S5        [        ST3XET5T6XŠ45        [        ST/XgT0T2X›45        [        SXÍT1T445        U U4S jnU" 5       n[        T R                   VVs/ s H+  nTR                    H  n[        SU-   U-   5      S:„  PM     M-     snn6 n[        T R
                   VVs/ s H+  nTR
                    H  n[        SU-   U-   5      S:  PM     M-     snn6 n[        T R
                   Vs/ s H3  nT0T2-
  [        SU-   S-
  5      -  [        U5      -
  [!        S	S5      :„  PM5     sn6 n[        T R                   Vs/ s H0  nT0T2-
  [        SU-   5      -  [        U5      -
  [!        S	S5      :„  PM2     sn6 n[        TR
                   Vs/ s H3  nT5T6-
  [        SU-   S-
  5      -  [        U
5      -
  [!        S	S5      :„  PM5     sn6 n[        TR                   Vs/ s H0  nT5T6-
  [        SU-   5      -  [        U
5      -
  [!        S	S5      :„  PM2     sn6 n[        U5      S[        U
S-
  T2T0-
  -  T6T5-
  T2T0-
  -  -   US-
  T6T5-
  -  -   5      -  -   S:„  n[        U5      S[        U
S-
  T2T0-
  -  T6T5-
  T2T0-
  -  -   US-
  T6T5-
  -  -   5      -  -
  S:„  n[        [        T35      5      U[        -  :  n[#        [        [        T35      5      U[        -  5      n[        [        T/5      5      U	[        -  :  n[#        [        [        T/5      5      U	[        -  5      n[%        X‰-   * [        -  [        R&                  -  5      n[)        UT/-  T3-  5      n[)        UT3-  T/-  5      n USU -  :X  aZ  [        [#        US5      X‰-   S:*  [+        [-        US5      [        Xº-   T6-   T5-
  5      S:  [        Xº-   T2-   T0-
  5      S:  5      5      n!OçS
 n"[        [#        US5      US-
  U	-   S:*  [+        [        [-        US5      U"" U5      5      [        [        Xº-   T6-   T5-
  5      S:  [#        US5      5      5      5      n![        [#        US5      U	S-
  U-   S:*  [+        [        [-        U S5      U"" U 5      5      [        [        Xº-   T2-   T0-
  5      S:  [#        U S5      5      5      5      n#[+        U!U#5      n!  T2T0-
  [        T/5      ST2T0-
  -  -  -  [/        T15      -  T6T5-
  [        T35      ST6T5-
  -  -  -  [/        T45      -  -   n$[1        U$S:„  5      S:w  a  U$S:„  n%GOÀU/U0U1U2U3U4U5U64S jn&[3        U&" SS5      U&" SS5      -  [        [#        [        T35      S5      [#        [        T/5      S5      5      4U&" [5        [        T/5      5      S5      U&" [5        [        T/5      5      S5      -  [        [#        [        T35      S5      [-        [        T/5      S5      5      4U&" S[5        [        T35      5      5      U&" S[5        [        T35      5      5      -  [        [-        [        T35      S5      [#        [        T/5      S5      5      4U&" [5        [        T/5      5      [5        [        T35      5      5      S45      n'U$S:„  [        [#        U$S5      [-        U'S5      [        U5      S:„  5      [        [#        U$S5      [#        U'S5      [        U5      S:„  5      /n([+        U(6 n% US4US4US4US4US4US4US4US4US4US4US4US4US4U!S4U%S44 H  u  n)n[        SUU)45        M     / m.U.4S jn*T.[        Xg-  U-  U-  S:g  UR8                  SL U	R8                  SL UUUUU5      /-  m.U*" S5        T.[        [#        T5T65      [#        US5      U	R8                  SL T3R8                  SL [        U
5      S:  UUUU5	      /-  m.U*" S5        T.[        [#        T0T25      [#        U	S5      UR8                  SL T/R8                  SL [        U5      S:  UUUU5	      /-  m.U*" S5        T.[        [#        T0T25      [#        T5T65      [#        US5      [#        U	S5      T3R8                  SL T/R8                  SL [        U5      S:  [        U
5      S:  [-        T3T/5      UUU5      /-  m.U*" S5        T.[        [#        T0T25      [#        T5T65      [#        US5      [#        U	S5      T3R8                  SL T/R8                  SL [        Xº-   5      S:  [-        T/T35      UUU5      /-  m.U*" S5        T.[        T0T2:„  UR8                  SL UR8                  SL U	S:¬  UUUUUU5
      /-  m.U*" S5        T.[        T0T2:  UR8                  SL UR8                  SL U	S:¬  UUUUUU5
      /-  m.U*" S5        T.[        T5T6:„  UR8                  SL U	R8                  SL US:¬  UUUUUU5
      /-  m.U*" S5        T.[        T5T6:  UR8                  SL U	R8                  SL US:¬  UUUUUU5
      /-  m.U*" S5        T.[        T0T2:„  [#        T5T65      [#        US5      U	S:¬  T3R8                  SL [        U
5      S:  UUUUU5      /-  m.U*" S5        T.[        T0T2:  [#        T5T65      [#        US5      U	S:¬  T3R8                  SL [        U
5      S:  UUUUU5      /-  m.U*" S5        T.[        [#        T0T25      T5T6:„  US:¬  [#        U	S5      T/R8                  SL [        U5      S:  UUUUU5      /-  m.U*" S5        T.[        [#        T0T25      T5T6:  US:¬  [#        U	S5      T/R8                  SL [        U5      S:  UUUUU5      /-  m.U*" S5        T.[        T0T2:  T5T6:„  US:¬  U	S:¬  UUUUUUU5      /-  m.U*" S5        T.[        T0T2:„  T5T6:  US:¬  U	S:¬  UUUUUUU5      /-  m.U*" S5        T.[        T0T2:„  T5T6:„  US:¬  U	S:¬  UUUUUUUUU!5      /-  m.U*" S5        T.[        T0T2:  T5T6:  US:¬  U	S:¬  UUUUUUUUU!5      /-  m.U*" S5        T.[        [#        US5      UR8                  SL UR8                  SL UR8                  SL UUU5      /-  m.U*" S 5        T.[        [#        US5      UR8                  SL UR8                  SL UR:                  SL UUU5      /-  m.U*" S!5        T.[        [#        US5      UR8                  SL U	R8                  SL UR:                  SL UUU5      /-  m.U*" S"5        T.[        [#        US5      UR8                  SL U	R8                  SL UR8                  SL UUU5      /-  m.U*" S#5        T.[        [#        XE-  S5      UR8                  SL U	R8                  SL UUUUU5      /-  m.U*" S$5        T.[        [#        Xg-  S5      UR8                  SL U	R8                  SL UUUUU5      /-  m.U*" S%5        [=        T USS&9n+[=        TUSS&9n,T.[        U,[#        US5      T5U:  UR8                  SL UUUU5      /-  m.U*" S'5        T.[        U,[#        US5      T6U:  UR8                  SL UUUU5      /-  m.U*" S(5        T.[        U+[#        US5      T0U:  U	R8                  SL UUUU5      /-  m.U*" S)5        T.[        U+[#        US5      T2U:  U	R8                  SL UUUU5      /-  m.U*" S*5        [+        T.6 n-[1        U-5      S:w  a  U-$ T.[        Xg-   T0:„  [#        US5      [#        US5      UR8                  SL UR8                  SL U	R:                  SL [        [        T/5      5      Xg-   T0-
  S-   [        -  :  UUUU!U%5      /-  m.U*" S+5        T.[        Xg-   T2:„  [#        US5      [#        US5      UR8                  SL UR8                  SL U	R:                  SL [        [        T/5      5      Xg-   T2-
  S-   [        -  :  UUUU!U%5      /-  m.U*" S,5        T.[        [#        T0T2S-
  5      [#        US5      [#        US5      UR8                  SL UR8                  SL U	S:¬  U	[        -  [        [        T/5      5      :  UUUU!U%5      /-  m.U*" S-5        T.[        [#        T0T2S-   5      [#        US5      [#        US5      UR8                  SL UR8                  SL U	S:¬  U	[        -  [        [        T/5      5      :  UUUU!U%5      /-  m.U*" S.5        T.[        T0T2S-
  :  [#        US5      [#        US5      UR8                  SL UR8                  SL U	S:¬  U	[        -  [        [        T/5      5      :  [        [        T/5      5      Xg-   T0-
  S-   [        -  :  UUUU!U%5      /-  m.U*" S/5        T.[        T0T2S-   :„  [#        US5      [#        US5      UR8                  SL UR8                  SL U	S:¬  U	[        -  [        [        T/5      5      :  [        [        T/5      5      Xg-   T2-
  S-   [        -  :  UUUU!U%5      /-  m.U*" S05        T.[        [#        US5      [#        US5      XE-   S:„  UR8                  SL U	R8                  SL UR:                  SL [        [        T35      5      XE-   T5-
  S-   [        -  :  UUUU!U%5      /-  m.U*" S15        T.[        [#        US5      [#        US5      XE-   T6:„  UR8                  SL U	R8                  SL UR:                  SL [        [        T35      5      XE-   T6-
  S-   [        -  :  UUUU!U%5      /-  m.U*" S25        T.[        [#        US5      [#        US5      [#        T5T6S-
  5      UR8                  SL U	R8                  SL US:¬  U[        -  [        [        T35      5      :  [        [        T35      5      US-   [        -  :  UUUU!U%5      /-  m.U*" S35        T.[        [#        US5      [#        US5      [#        T5T6S-   5      UR8                  SL U	R8                  SL US:¬  U[        -  [        [        T35      5      :  [        [        T35      5      US-   [        -  :  UUUU!U%5      /-  m.U*" S45        T.[        [#        US5      [#        US5      T5T6S-
  :  UR8                  SL U	R8                  SL US:¬  U[        -  [        [        T35      5      :  [        [        T35      5      XE-   T5-
  S-   [        -  :  UUUU!U%5      /-  m.U*" S55        T.[        [#        US5      [#        US5      T5T6S-   :„  UR8                  SL U	R8                  SL US:¬  U[        -  [        [        T35      5      :  [        [        T35      5      XE-   T6-
  S-   [        -  :  UUUU!U%5      /-  m.U*" S65        [+        T.6 $ s  snnf s  snnf s  snf s  snf s  snf s  snf ! [6         a    Sn% GN½f = f)7z=Return a condition under which the integral theorem applies. r§   rR   zChecking antecedents:z1  sigma=%s, s=%s, t=%s, u=%s, v=%s, b*=%s, rho=%sz1  omega=%s, m=%s, n=%s, p=%s, q=%s, c*=%s, mu=%s,z"  phi=%s, eta=%s, psi=%s, theta=%sc                 óÚ   >• TT4 Hb  n [         R                  " U R                  U R                  5       H0  u  pX-
  nUR                  (       d  M  UR
                  (       d  M/      g   Md     g)NFT)r  r  r…   r‡   Ú
is_integerÚis_positive)r  re   ÚjÚdiffr¦  r§  s       €€rg   Ú_c1Ú_check_antecedents.<locals>._c1ú  sU   ø€ Ø�b“ˆAÜ!×)Ò)¨!¯$©$°·±Ö5‘�Ø‘u�Ø—?—?‘? t×'7×'7Ñ'7Ú ó 6ñ ð
 ro   r   éýÿÿÿc                óV   • U S:g  =(       a    [        [        SU -
  5      5      [        :  $ )aw  Returns True if abs(arg(1-z)) < pi, avoiding arg(0).

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

If ``z`` is 1 then arg is NaN. This raises a
TypeError on `NaN < pi`. Previously this gave `False` so
this behavior has been hardcoded here but someone should
check if this NaN is more serious! This NaN is triggered by
test_meijerint() in test_meijerint.py:
`meijerint_definite(exp(x), x, 0, I)`
rR   )r#   r"   r   )r^   s    rg   Ú_condÚ!_check_antecedents.<locals>._cond$  s$   € ð ˜‘6×2œc¤# a¨!¡e£*›o´Ñ2Ð2ro   Fc                óª   >• U TT-
  -  [        T5      STT-
  -  -  -  [        T5      -  UT	T-
  -  [        T5      ST	T-
  -  -  -  [        T5      -  -   $ rí   )r#   r8   )
Úc1Úc2Úomegar¶   Úpsir·   ÚsigmaÚthetaÚur  s
     €€€€€€€€rg   Ú	lambda_s0Ú%_check_antecedents.<locals>.lambda_s0S  sc   ø€ Ø˜1˜q™5‘z¤# e£*¨q°!°a±%©yÑ"9Ñ9¼#¸c»(ÑBØ˜!˜a™%‘j¤ U£¨a°°Q±©iÑ!8Ñ8¼¸U»ÑCñDð Dro   r©   Tr¹   r¸   r@  rA  rB  é   é	   é
   rC  rD  rE  rF  é   z	  c%s: %sc                ó(   >• [        SU TS   45        g )Nz  case %s: %sr©   r€  )ÚcountrŒ  s    €rg   ÚprÚ_check_antecedents.<locals>.prl  s   ø€ Ü� %¨¨r©Ð!3Õ4ro   rÇ   é   é   é   é   é   é   é   )r†  ÚE1ÚE2ÚE3ÚE4é   é   é   é   é   é   é   é   é    é!   é"   é#   )rê   rï   r   r  r‡   r…   r†   rˆ   r«   r   r#   r)   r|  r�  rT   r    r   r   r+   rÌ   r%   rU   r   r7   rj  r5   r$   Ú	TypeErrorr¹  Úis_negativer„  )7r¦  r§  r|   r=  rs  r±   ré   rt   ÚbstarÚcstarÚrhoÚmuÚphir‡  r¼  rÃ  re   rº  rÄ  Úc3Úc4Úc5Úc6Úc7Úc8Úc9Úc10Úc11Úc12Úc13Úz0ÚzosÚzsoÚc14rÀ  Úc14_altÚlambda_cÚc15rÊ  Úlambda_srˆ  rŠ   rÒ  Ú
mt1_existsÚ
mt2_existsrª   rŒ  rÅ  r¶   rÆ  r·   rÇ  rÈ  rÉ  r  s7   ``                                            @@@@@@@@@rg   Ú_check_antecedentsr  Ù  sz  ÿú€ ô ˜bŸk™k¨1Ó-�H€Eˆ1Ü˜bŸk™k¨1Ó-�H€Eˆ1Ü”C˜Ÿ™“J¤ B§E¡E£
¬C°·±«J¼¸B¿E¹E»
ÐCÓD�J€Aˆ!ˆQÜ”C˜Ÿ™“J¤ B§E¡E£
¬C°·±«J¼¸B¿E¹E»
ÐCÓD�J€Aˆ!ˆQØ‰E�Q˜‘U˜A‘IÑ€EØ‰E�Q˜‘U˜A‘IÑ€EØ
�%‰%�1�q‘5˜!‘)Ñ
˜aÑ
€CØ	�‰�!�a‘%˜‘Ñ	˜QÑ	€BØ
ˆa‰%�1�q‘5‰/€CØ
ˆq�1‰u‰+˜Ñ
˜SÑ
 €CÜˆq�1‰u�q‰y‰>œCÔ 3°EÓ :Ó;Ñ;¸aÀ!¹eÑ
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ñBØ�!‰eñAñ ó ñ ñ Øñ€Bä
ˆc‹(�Q”r˜3 ™7 Q¨¡UÑ+¨q°1©u°q¸1±u©oÑ=ÀØ
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7€Cô 
ˆu‰}ÐœbÑ ¤§¡Ñ0Ó	1€BÜ
�R˜‘X˜e‘^Ó
$€CÜ
�R˜‘X˜e‘^Ó
$€CØ
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¨E°Q©JØ�b˜"˜b " c¨3ó0ð 1ñ 1€Eá€r„FØ	Œc�!�a‘%˜˜Q™ ¨¡
¨E°Q©JØ�b˜"˜b " c¨3ó0ð 1ñ 1€Eá€r„FØ	Œc�!�a‘%˜˜Q™ ¨¡
¨E°Q©JØ�b˜"˜b " b¨#¨s°Có9ð :ñ :€Eá€r„FØ	Œc�!�a‘%˜˜Q™ ¨¡
¨E°Q©JØ�b˜"˜b " b¨#¨s°Có9ð :ñ :€Eá€r„FØ	Œc”"�Q˜“(˜AŸM™M¨TÐ1°5×3DÑ3DÈÐ3LÈcÏoÉoÐaeÐNeÐgiÐkmÐorÓsÐtÑt€EÙ€r„FØ	Œc”"�Q˜“(˜AŸM™M¨TÐ1°5×3DÑ3DÈÐ3LÈcÏoÉoÐaeÐNeÐgiÐkmÐorÓsÐtÑt€EÙ€r„FØ	Œc”"�Q˜“(˜AŸM™M¨TÐ1°5×3DÑ3DÈÐ3LÈcÏoÉoÐaeÐNeÐgiÐkmÐorÓsÐtÑt€EÙ€r„FØ	Œc”"�Q˜“(˜AŸM™M¨TÐ1°5×3DÑ3DÈÐ3LÈcÏoÉoÐaeÐNeÐgiÐkmÐorÓsÐtÑt€EÙ€r„FØ	Œc”"�Q‘S˜!“*˜e×/Ñ/°4Ð7¸×9JÑ9JÈdÐ9RØ�b˜"˜c 3ó(ð )ñ )€Eá€r„FØ	Œc”"�Q‘S˜!“*˜e×/Ñ/°4Ð7¸×9JÑ9JÈdÐ9RØ�b˜"˜c 3ó(ð )ñ )€Eá€r„Fô & b¨!°DÑ9€JÜ% b¨!°DÑ9€JØ	Œc�*œb  A›h¨¨A©¨u×/@Ñ/@ÀDÐ/HÈ#ÈrÐSUÐWYÓZÐ[Ñ[€EÙ€t„HØ	Œc�*œb  A›h¨¨A©¨u×/@Ñ/@ÀDÐ/HÈ#ÈrÐSUÐWYÓZÐ[Ñ[€EÙ€t„HØ	Œc�*œb  A›h¨¨A©¨u×/@Ñ/@ÀDÐ/HÈ#ÈrÐSUÐWYÓZÐ[Ñ[€EÙ€t„HØ	Œc�*œb  A›h¨¨A©¨u×/@Ñ/@ÀDÐ/HÈ#ÈrÐSUÐWYÓZÐ[Ñ[€EÙ€t„Hô 	ˆEˆ
€AÜ�!ƒ}˜ÓØˆà	Œc�!‘%˜!‘)œR  1›X¤r¨#¨q£z°1·=±=ÀDÐ3HÈ%×J[ÑJ[Ð_cÐJcÐej×evÑevÐz~Ðe~ÜÔ)¨%Ó0Ó1°Q±U¸Q±YÀ±]ÄBÑ4FÑFØ�b˜#˜s Có)ð *ñ *€Eñ €r„FØ	Œc�!‘%˜!‘)œR  1›X¤r¨#¨q£z°1·=±=ÀDÐ3HÈ%×J[ÑJ[Ð_cÐJcÐej×evÑevÐz~Ðe~ÜÔ)¨%Ó0Ó1°Q±U¸Q±YÀ±]ÄBÑ4FÑFØ�b˜#˜s Có)ð *ñ *€Eñ €r„FØ	Œc”"�Q˜˜A™“,¤ 1 a£¬"¨S°!«*°a·m±mÀtÐ6KÈU×M^ÑM^ÐbfÐMfØ˜1‘*˜e¤B™h¬Ô-@ÀÓ-GÓ)HÑHØ�b˜#˜s Có)ð *ñ *€Eñ €r„FØ	Œc”"�Q˜˜A™“,¤ 1 a£¬"¨S°!«*°a·m±mÀtÐ6KÈU×M^ÑM^ÐbfÐMfØ˜1‘*˜e¤B™h¬Ô-@ÀÓ-GÓ)HÑHØ�b˜#˜s Có)ð *ñ *€Eñ €r„FØ	Œc�!�a˜!‘e‘)œR  1›X¤r¨#¨q£z°1·=±=ÀDÐ3HÈ%×J[ÑJ[Ð_cÐJcØ˜1‘*˜e¤B™h¬Ô-@ÀÓ-GÓ)HÑHÜÔ)¨%Ó0Ó1°Q±U¸Q±YÀ±]ÄBÑ4FÑFØ�b˜#˜s Có)ð *ñ *€Eñ €r„FØ	ŒcØ	ˆA�‰E‰	”2�a˜“8œR  Q›Z¨¯©¸$Ð)>À×@QÑ@QÐUYÐ@YÐ[`ÐdeÑ[eØœ‘(œSÔ!4°UÓ!;Ó<Ñ<ÜÔ)¨%Ó0Ó1°Q±U¸Q±YÀ±]ÄBÑ4FÑFØ�b˜#˜s Có	)ð *ñ *€Eñ
 €r„FØ	Œc”"�Q˜“(œB˜s A›J¨©°©	°1·=±=ÀDÐ3HÈ%×J[ÑJ[Ð_cÐJcÐej×evÑevÐz~Ðe~ÜÔ)¨%Ó0Ó1°Q±U¸Q±YÀ±]ÄBÑ4FÑFØ�b˜#˜s Có)ð *ñ *€Eñ €r„FØ	Œc”"�Q˜“(œB˜s A›J¨©°©	°1·=±=ÀDÐ3HÈ%×J[ÑJ[Ð_cÐJcÐej×evÑevÐz~Ðe~ÜÔ)¨%Ó0Ó1°Q±U¸Q±YÀ±]ÄBÑ4FÑFØ�b˜#˜s Có)ð *ñ *€Eñ €r„FØ	Œc”"�Q˜“(œB˜s A›J¬¨1¨a°!©e«°a·m±mÀtÐ6KÈU×M^ÑM^ÐbfÐMfØ˜1‘*˜e¤B™h¬Ô-@ÀÓ-GÓ)HÑHÜÔ)¨%Ó0Ó1°U¸Q±YÄ±NÑBØ�b˜#˜s Có)ð *ñ *€Eñ €r„FØ	Œc”"�Q˜“(œB˜s A›J¬¨1¨a°!©e«°a·m±mÀtÐ6KÈU×M^ÑM^ÐbfÐMfØ˜1‘*˜e¤B™h¬Ô-@ÀÓ-GÓ)HÑHÜÔ)¨%Ó0Ó1°U¸Q±YÄ±NÑBØ�b˜#˜s Có)ð *ñ *€Eñ €r„FØ	ŒcÜ
ˆ1ˆa‹”"�S˜!“*˜a ! a¡%™i¨¯©¸$Ð)>À×@QÑ@QÐUYÐ@YÐ[`ÐdeÑ[eØŒb‰”3Ô*¨5Ó1Ó2Ñ2ÜÔ Ó&Ó'¨1©5°1©9°q©=¼"Ñ*<Ñ<Ø
ˆB��S˜#ó	ð  ñ  €Eñ
 €r„FØ	ŒcÜ
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ˆB��S˜#ó	ð  ñ  €Eñ
 €r„Fäˆuˆ:ÐùóC @ùÛCùÚOùÚKùÚPùÚLøôv ó Ø‹ðúsJ   È2A\
É2A\!
Ê&:A\'Ë77A\,Í:A\1Î7A\6Ù'A$A\; ÛF?A\; Á\;A]Á]
A]c                óÌ  • [        U R                  U5      u  p4[        UR                  U5      u  pTS nU" U R                  5      [        UR                  5      -   n[        UR
                  5      U" U R                  5      -   nU" U R                  5      [        UR                  5      -   n	[        UR                  5      U" U R
                  5      -   n
[        XxXšXS-  5      U-  $ )a°  
Express integral from zero to infinity g1*g2 using a G function,
assuming the necessary conditions are fulfilled.

Examples
========

>>> from sympy.integrals.meijerint import _int0oo
>>> from sympy.abc import s, t, m
>>> from sympy import meijerg, S
>>> g1 = meijerg([], [], [-S(1)/2, 0], [], s**2*t/4)
>>> g2 = meijerg([], [], [m/2], [-m/2], t/4)
>>> _int0oo(g1, g2, t)
4*meijerg(((0, 1/2), ()), ((m/2,), (-m/2,)), s**(-2))/s**2
c                ó2   • U  Vs/ s H  o* PM     sn$ s  snf rb   r˜   ry  s     rg   ÚnegÚ_int0oo.<locals>.neg  s   € Ù‹šA�q“™A‰ÐùŠs   …)rê   rï   r‡   rÉ   r…   r  r  rP   )r¦  r§  r|   r‡  r=  rÅ  r
  r³  r´  r¨  r©  s              rg   Ú_int0oor  	  s«   € ô" ˜BŸK™K¨Ó+�F€CÜ˜bŸk™k¨1Ó-�H€Eòá	ˆR�U‰U‹”d˜2Ÿ5™5“kÑ	!€BÜ	ˆb�i‰i‹™3˜rŸy™y›>Ñ	)€BÙ	ˆR�U‰U‹”d˜2Ÿ5™5“kÑ	!€BÜ	ˆb�i‰i‹™3˜rŸy™y›>Ñ	)€BÜ�2˜2 5¡9Ó-¨cÑ1Ð1ro   c           	     ó>  ^^	• [        X5      u  nm	[        UR                  U5      u  nmUU	4S jnSSKJn  U" XT	T-  -  -  SS9[	        U" UR
                  5      U" UR                  5      U" UR                  5      U" UR                  5      UR                  5      4$ )zAbsorb ``po`` == x**s into g. c                ó>   >• U  Vs/ s H
  oTT-  -   PM     sn$ s  snf rb   r˜   )r  r±   r°   rs  s     €€rg   r   Ú_rewrite_inversion.<locals>.tr+  s!   ø€ Ù!"Ó#¢˜A�A�a‘C”¡Ñ#Ð#ùÒ#s   †r   r¡  Tr£  )	rê   rï   rã   r¢  rP   r…   r  r‡   r  )
r‰   r  r  r|   r=  r•   r   r¢  r°   rs  s
           @@rg   Ú_rewrite_inversionr  &  s�   ù€ ä˜"Ó �D€A€qÜ˜!Ÿ*™* aÓ(�D€A€qö$å(Ù�c˜a ™c™(‘l¨$Ñ/Ü‘B�q—t‘t“H™b §¡›l©B¨q¯t©t«H±b¸¿¹³lÀAÇJÁJÓOðQð Qro   c                óü	  ^ ^^^^^• [        S5        T R                  n[        UT5      u  p4US:  a   [        S5        [        [	        T 5      T5      $ U4S jmU4S jm[        [        T R                  5      [        T R                  5      [        T R                  5      [        T R                  5      /5      u  pVpxXV-   U-
  n	X…-
  U-
  n
Xš-
  S-  nX‡-
  mTS:X  a  [
        R                  nOTS:”  a  SnO[
        R                  nST-
  S-  [        T R                  6 -   [        T R                  6 -
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5        g[         R"                  " T R                  T R                  5       H,  u  pïXï-
  R$                  (       d  M  Xï:”  d  M!  [        S5          g   Xx:¼  a:  [        S5        ['        T R                   Vs/ s H  nT" US-
  SSU5      PM     sn6 $ U U4S jnUUU4S jnUUU4S jnUUU4S jn/ nU['        SU:*  SU:*  U[(        -  U-
  [(        S-  :¬  US:„  U" U[+        [
        R,                  [(        -  U
S-   -  5      -  5      5      /-  nU['        US-   U:*  US-   U:*  US:„  U[(        S-  :  US:H  XW-
  S-   [(        -  U-
  [(        S-  :¬  U" U[+        [
        R,                  [(        -  X…-
  -  5      -  5      U" U[+        [
        R,                  * [(        -  X…-
  -  5      -  5      5      /-  nU['        XX:H  US:H  US:„  TU-   [(        -  U-
  [(        S-  :¬  U" U5      5      /-  nU['        [/        ['        XxS-
  :*  SU	:*  U	TS-  :*  5      ['        US-   XV-   :*  XV-   Xx-   S-  :*  5      5      US:„  U[(        S-  :  U	S-   [(        -  U-
  [(        S-  :¬  U" U[+        [
        R,                  [(        -  U
-  5      -  5      U" U[+        [
        R,                  * [(        -  U
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  [(        S-  :¬  U" U[+        [
        R,                  [(        -  U
-  5      -  5      U" U[+        [
        R,                  * [(        -  U
-  5      -  5      5      /-  nUUS:H  /-  n[/        U6 $ s  snf )z6Check antecedents for the laplace inversion integral. z#Checking antecedents for inversion:r   z  Flipping G.c           
     óâ  >• [        UT5      u  pVX-  n XU-  -  nX&-  n/ nU[        [        R                  [	        U5      -  [
        -  S-  5      -  nU[        [        R                  * [	        U5      -  [
        -  S-  5      -  n	U(       a  Un
OU	n
U[        [        [        US5      [	        U5      S:*  5      [	        U 5      S:*  5      /-  nU[        [        US5      [        [        U5      S5      [	        U5      S:„  [	        U
5      S:  5      /-  nU[        [        US5      [        [        U5      S5      [	        U5      S:„  [	        U
5      S:*  [	        U 5      S:*  5      /-  n[        U6 $ )Nr§   r   r©   )rê   r+   r   rÌ   r    r   rT   rU   r   r   r!   )r•   r°   rÍ   r^   ÚplusÚcoeffÚexponentrŒ  ÚwpÚwmÚwr|   s              €rg   Ústatement_halfÚ4_check_antecedents_inversion.<locals>.statement_half<  sG  ø€ Ü(¨¨AÓ.‰ˆØ	‰ˆØ	�A‰X‰ˆØ	‰ˆØˆØŒs”1—?‘?¤2 a£5Ñ(¬Ñ+¨AÑ-Ó.Ñ.ˆØŒs”A—O‘OÐ#¤B q£EÑ)¬"Ñ,¨QÑ.Ó/Ñ/ˆÞØ‰AàˆAØ”#”bœ˜A˜q›¤2 a£5¨A¡:Ó.´°1³¸±Ó<Ð=Ñ=ˆØ”#”b˜˜A“h¤¤2 a£5¨!£¬b°«e°a©i¼¸A»À¹ÓCÐDÑDˆØ”#”b˜˜A“h¤¤2 a£5¨!£¬b°«e°a©i¼¸A»À!¹Ü˜“e˜r‘kó#ð $ñ 	$ˆä�5ˆzÐro   c           
     ó<   >• [        T" XX#S5      T" XX#S5      5      $ )zIProvide a convergence statement for z**a * exp(b*z**c),
c/f sphinx docs. TF)rT   )r•   r°   rÍ   r^   r  s       €rg   Ú	statementÚ/_check_antecedents_inversion.<locals>.statementN  s)   ø€ ô ‘> !¨¨dÓ3Ù! !¨¨eÓ4ó6ð 	6ro   r§   rR   z9  m=%s, n=%s, p=%s, q=%s, tau=%s, nu=%s, rho=%s, sigma=%sz   epsilon=%s, theta=%s, delta=%sz-  Computation not valid for these parameters.Fz  Not a valid G function.z$  Using asymptotic Slater expansion.c                ól   >• [        TR                   Vs/ s H  nT" US-
  SSU 5      PM     sn6 $ s  snf r”   )rT   r…   )r^   r•   r  r  s     €€rg   ÚEÚ'_check_antecedents_inversion.<locals>.E~  s3   ø€ Ü¸¿ºÓ=º°1‘Y˜q 1™u a¨¨AÖ.¹Ñ=Ð>Ð>ùÒ=s   •1c                ó"   >• T" TT* ST-  U 5      $ rí   r˜   )r^   rÇ  r  rÈ  s    €€€rg   ÚHÚ'_check_antecedents_inversion.<locals>.H�  s   ø€ Ù˜  ¨¨%©°Ó3Ð3ro   c                ó$   >• T" TT* ST-  U S5      $ )NrR   Tr˜   ©r^   rÇ  r  rÈ  s    €€€rg   ÚHpÚ(_check_antecedents_inversion.<locals>.Hp„  s   ø€ Ù˜e e V¨Q¨u©W°a¸Ó>Ð>ro   c                ó$   >• T" TT* ST-  U S5      $ )NrR   Fr˜   r%  s    €€€rg   ÚHmÚ(_check_antecedents_inversion.<locals>.Hm‡  s   ø€ Ù˜e e V¨Q¨u©W°a¸Ó?Ð?ro   )r|  rï   rê   Ú_check_antecedents_inversionr#  r   r  r‡   r…   r†   rˆ   rµ   ÚNaNr   r  r�  r  r  r¸  rT   r   r+   rÌ   rU   )r  r|   r^   r=  r÷   ré   rt   r¶   r·   r®  r«   rï  Úepsilonr  r•   r°   r  r"  r&  r)  rŒ  rÇ  r  r  rÈ  s   ``                   @@@@rg   r+  r+  2  sÂ  ý€ ä
Ð0Ô1Ø	�
‰
€AÜ˜!˜QÓ�D€AØˆ1ƒuÜˆÔä+¬G°A«J¸Ó:Ð:õõ$6ô ”C˜Ÿ™“Iœs 1§4¡4›y¬#¨a¯d©d«)´S¸¿¹³YÐ?Ó@�J€Aˆ!Ø
‰%�!‰)€CØ	
‰�‰€BØ‰8�Q‰,€CØ‰E€EØ�ƒzÜ—&‘&‰Ø	�‹Ø‰ä—%‘%ˆØ�%‰i˜‰]œS !§$¡$˜ZÑ'¬#¨q¯t©t¨*Ñ4°eÑ;€EØ�G‰G€EÜÐGØ�1˜ # uÐ-ô/äÐ.°¸%ÀÐ0GÔHð �G‰G�q˜‘s‹N˜q A›v¨!«&ÜÐ>Ô?Øô
 ×!Ò! !§$¡$¨¯©Ö-‰ˆØ‰E××Ñ !¥%ÜÐ.Ô/Ùñ .ð 	ƒvÜÐ5Ô6Ü¸¿ºÓ=º°1‘Y˜q 1™u a¨¨AÖ.¹Ñ=Ð>Ð>ö?÷4÷?÷@ð €Eà	Œc�!�q‘&˜!˜q™& #¤b¡&¨5¡.´B°q±DÑ"8¸%À!¹)Ù�A”cœ!Ÿ/™/¬"Ñ,¨b°1©fÑ5Ó6Ñ6Ó7ó9ð :ñ :€Eð 
Œc�!�a‘%˜1‘*˜a !™e q™j¨%°!©)°U¼RÀ¹T±\À1ÈÁ6Ø‘5˜1‘9œb‘. 5Ñ(¬B¨q©DÑ0Ù�Q”sœ1Ÿ?™?¬2Ñ-¨q©uÑ5Ó6Ñ6Ó7Ù�Q”sœAŸO™OÐ+¬BÑ.°±Ñ6Ó7Ñ7Ó8ó:ð ;ñ ;€Eð
 
Œc�!‘&˜!˜q™& %¨!¡)Ø˜7‘?¤BÑ&¨Ñ.´"°Q±$Ñ6¹¸!»ó>ð ?ñ ?€Eð 
Œc”"”S˜ !™e™ Q¨#¡X¨s°e¸A±g©~Ó>Ü˜˜Q™ !¡%™¨©°1±5¸!±)Ñ);Ó<ó>à˜!‘)˜U¤R¨¡T™\¨C°!©G´R©<¸%Ñ+?Ä2ÀaÁ4Ñ+GÙ�Q”sœ1Ÿ?™?¬2Ñ-¨bÑ0Ó1Ñ1Ó2Ù�Q”sœAŸO™OÐ+¬BÑ.¨rÑ1Ó2Ñ2Ó3ó	5ð 6ñ 6€Eð 
Œc�!�q‘&˜# ™' 5¨1¡9¨e¼"±f©n¼rÀ!¹tÑ.CØ‘=¤"Ñ$ uÑ,´°1±Ñ4Ù�Q”sœ1Ÿ?™?¬2Ñ-¨bÑ0Ó1Ñ1Ó2Ù�Q”sœAŸO™OÐ+¬BÑ.¨rÑ1Ó2Ñ2Ó3ó5ð 6ñ 6€Eð
 
ˆa�1‰fˆXÑ€Eô ˆuˆ:Ðùò[ >s   È
S9c                óÒ   • [        U R                  U5      u  p4[        [        U R                  U R
                  U R                  U R                  X2U-  -  5      U* 5      u  pPXR-  U -  $ )zG
Compute the laplace inversion integral, assuming the formula applies.
)rê   rï   r%  rP   r…   r  r‡   r  )r  r|   r±   r°   r•   r  s         rg   Ú_int_inversionr/  ¬  sT   € ô ˜!Ÿ*™* aÓ(�D€AÜœ' !§$¡$¨¯©°!·$±$¸¿¹À!ÀqÁDÁ&ÓIÈAÈ2ÓN�D€AØ‰3ˆq‰5€Lro   c                ó¢
  ^ ^!• SSK JnJm!JnJm   [
        (       d  0 q[        [
        5        [        U [        5      (       a¿  [        U R                  U5      R                  U5      u  pV[        U5      S:”  a  gUS   nUR                  (       a-  UR                  U:w  d  UR                  R                   (       d  gOXa:w  a  gSS[        U R"                  U R$                  U R&                  U R(                  XV-  5      4/S4$ U nU R+                  U[,        5      n [/        U [,        5      nU[
        ;   Ga  [
        U   n	U	 GH
  u  p«pÍU R1                  U
SS9nU(       d  M   0 nUR3                  5        H  u  nn[5        [7        USS9SS9UU'   M     Un[        U[8        5      (       d  UR+                  U5      nUS	:X  a  M�  [        U[8        [:        45      (       d  [5        UR+                  U5      5      n[=        U5      S	:X  a  MÇ  [        U[>        5      (       d  U" U5      n/ nU GH  u  nn[A        [5        UR+                  U5      R+                  [,        U5      SS9U5      n UR+                  U5      R+                  [,        U5      n[E        UU4-   6 RG                  [H        RJ                  [H        RL                  [H        RN                  5      (       a  M²  [        UR"                  UR$                  UR&                  UR(                  [5        UR                  SS95      nURQ                  UU4-   5        GM     U(       d  GM  UU4s  $    U(       d  g[S        S
5        U U!4S jnUn [U        SSU 5      nS n U" XUUS	SS9u  nnnU" UUUU5      nUcj  [W        SS5      nUU RX                  ;  aN  [[        X5      (       a>   U" U R+                  UUU-  5      UUUSS	S9u  nnnU" UUUU5      R+                  US5      nUbB  URG                  [H        RJ                  [H        R\                  [H        RL                  5      (       a  [S        S5        g[^        R`                  " U5      n/ nU H£  n U R                  U5      u  nn[        U5      S:”  a  [c        S5      eUS   n[A        UR                  U5      u  nnUUS[        UR"                  UR$                  UR&                  UR(                  [5        [7        USS9SS9UU-  -  5      4/-  nM¥     [S        SU5        US4$ ! [B         a     GM
  f = f! U a    Sn GN¬f = f! U a    Sn GNNf = f)a,  
Try to rewrite f as a sum of single G functions of the form
C*x**s*G(a*x**b), where b is a rational number and C is independent of x.
We guarantee that result.argument.as_coeff_mul(x) returns (a, (x**b,))
or (a, ()).
Returns a list of tuples (C, s, G) and a condition cond.
Returns None on failure.
rR   )Úmellin_transformÚinverse_mellin_transformÚIntegralTransformErrorÚMellinTransformStripErrorNr   T)Úold)Úlift)Úexponents_onlyFz)Trying recursive Mellin transform method.c           
     ó€   >•  T" XX#SSS9$ ! T a+    SSK Jn  T" U" [        [        U 5      5      5      XUSSS9s $ f = f)z»Calling simplify() all the time is slow and not helpful, since
most of the time it only factors things in a way that has to be
un-done anyway. But sometimes it can remove apparent poles. T)Ú
as_meijergÚneedevalr   )Úsimplify)rã   r;  rY   r   )ÚFrs  r|   Ústripr;  r4  r2  s        €€rg   Úmy_imtÚ_rewrite_single.<locals>.my_imt
  sY   ø€ ð
	0Ù+¨A°!Ø7;ÀdñLð Løà(ó 	0Ý/Ù+Ùœ¤ q£	Ó*Ó+¨Q°5Ø¨$ñ0ò 0ð	0ús   ƒ Œ.=¼=rs  zrewrite-singlec           	     ó  • [        XSS9nUbQ  SSKJn  Uu  pE[        U" USS95      n[	        XE4[        X[        R                  [        R                  45      S45      $ [        X[        R                  [        R                  45      $ )NT)Úonly_doubler   ©ÚhyperexpandÚnonrepsmall)Úrewrite)	Ú_meijerint_definite_4rã   rC  Ú_my_unpolarifyr5   rS   r   rå   rL  )rf   r|   rª   rC  rn   rŠ   s         rg   Úmy_integratorÚ&_rewrite_single.<locals>.my_integrator  sw   € Ü! !°DÑ9ˆØ‰=Ý2Ø‰IˆCÜ ¡¨S¸-Ñ!HÓIˆCÜ˜c˜[Ü& q¬a¯f©f´a·j±jÐ*AÓBÀDÐIóKð Kä˜œqŸv™v¤q§z¡zÐ2Ó3Ð3ro   )Ú
integratorr;  r:  r•   )rJ  r:  r;  z"Recursive Mellin transform failed.zUnexpected form...z"Recursive Mellin transform worked:)2Ú
transformsr1  r2  r3  r4  Ú_lookup_tablerÎ   rü   rP   rZ   rï   rä   r  ræ   rç   r+   r¥  r…   r  r‡   r  r½   r^   r‚   rý   Úitemsr%   r&   rh  rV   rj  rÉ   rê   Ú
ValueErrorr
   rm   r   rL  ÚComplexInfinityÚNegativeInfinityrƒ   r|  r/  r*  rÙ   r7  r,  r   r  ÚNotImplementedError)"rf   r|   Ú	recursiver1  r3  r  ré   Úf_r±   r  r„   ÚtermsrŠ   r‹   r½   Úsubs_r\  r]  rn   r‰   r  r¯  r>  rs  rH  r<  r=  r=  r•   rl   rÍ   r°   r4  r2  s"                                   @@rg   Ú_rewrite_singlerV  ¼  sØ  ù€ ÷;ó ;÷ Š=ØˆÜœ]Ô+ä�!”W×ÑÜ˜!Ÿ*™* aÓ(×5Ñ5°aÓ8‰ˆÜˆq‹6�A‹:ØØˆa‰DˆØ�8�8Ø�v‰v˜‹{ !§%¡%×"3×"3Øð #4à‹VØØ�A”w˜qŸt™t Q§X¡X¨q¯t©t°Q·X±X¸u¹wÓGÐHÐIÈ4ÐOÐOà	
€BØ	�‰ˆq”!‹€AÜ�”1‹€AØŒMÔÜ˜!ÑˆÜ*+Ñ&ˆG˜DØ—7‘7˜7¨�7Ð-ˆDßˆtØ�Ø#Ÿz™zž|‘G�C˜Ü!+¬H°R¸dÑ,CØ;?ñ"A�E˜#“Jñ  ,ð �Ü! $¬×-Ñ-ØŸ9™9 T›?�DØ˜5“=ÙÜ! $¬¬{Ð(;×<Ñ<Ü% d§i¡i°£oÓ6�DÜ˜dÓ# uÓ,ÙÜ! %¬×.Ñ.Ù! $›K�EØ�Ü#‘F�C˜Ü'¬
°3·8±8¸D³>×3FÑ3FÄqÈ!Ó3LØBFñ)HØIJóL�Bð!ØŸF™F 4›L×-Ñ-¬a°Ó3˜ô ˜r Q D™yÐ*×.Ñ.¬q¯z©z¼1×;LÑ;LÌa×N`ÑN`×aÑaÙ Ü §¡ a§h¡h°·±°a·h±hÜ *¨1¯:©:ÀdÑ KóM�Aà—J‘J˜r Q D™y×)ñ $÷ ‘3Ø ˜9Ò$ñG +,öL ØÜ
Ð6Ô7ö0ð 	€AÜˆsÐ$ aÓ(€Aò4ðÙ& q¨Q¸=Ø05ÀñF‰ˆˆ5�!á�1�a˜˜EÓ"ˆð 	�yô �CÐ)Ó*ˆØ�A—N‘NÓ"¤|°A×'9Ñ'9ðÙ.¨q¯v©v°a¸¸1¹«~¸qÀ!Ø:GØ8<ÀuñN‘��5˜!ñ ˜1˜a  EÓ*×/Ñ/°°1Ó5�ð 	�y�A—E‘Eœ!Ÿ*™*¤a§e¡e¬Q×->Ñ->×?Ñ?ÜÐ3Ô4ØÜ�=Š=˜Ó€DØ
€CÛˆØ�~‰~˜aÓ ‰ˆˆ1Üˆq‹6�A‹:Ü%Ð&:Ó;Ð;Øˆa‰DˆÜ˜aŸj™j¨!Ó,‰ˆˆ1Ø��A”w˜qŸt™t Q§X¡X¨q¯t©t°Q·X±XÜ)¬(Ø#$¨4ñ+1ØAEñ Gà ! 1¡ñ %ó&ð 'ð (ñ 	(Šñ ô Ð/°Ô3Ø�ˆ9Ðøô_ &ó !Û ð!ûð` "ó Ø‹ðûð *ó Ø“ðús6   É$%T!Í9T3 Ï=U Ô!
T0Ô/T0Ô3T?Ô>T?ÕUÕUc                ó\   • [        X5      u  p4n[        XQU5      nU(       a  X4US   US   4$ g)zç
Try to rewrite ``f`` using a (sum of) single G functions with argument a*x**b.
Return fac, po, g such that f = fac*po*g, fac is independent of ``x``.
and po = x**s.
Here g is a result from _rewrite_single.
Return None on failure.
r   rR   N)r  rV  )rf   r|   rR  r‰   r  r  s         rg   Ú	_rewrite1rX  J  s;   € ô ˜AÓ!�J€CˆQÜ˜˜iÓ(€AÞØ˜˜!™˜a ™dÐ"Ð"ð 	ro   c           	     óÄ  ^• [        U T5      u  p#n[        U4S j[        U5       5       5      (       a  g[        U5      nU(       d  g[	        [        UU4S jU4S jU4S j/5      5      n[        R                  " SU5       HZ  u  nu  px[        UTU5      n	[        UTU5      n
U	(       d  M+  U
(       d  M4  [        U	S   U
S   5      nUS:w  d  MN  X#U	S	   U
S	   U4s  $    g)
zô
Try to rewrite ``f`` as a product of two G functions of arguments a*x**b.
Return fac, po, g1, g2 such that f = fac*po*g1*g2, where fac is
independent of x and po is x**s.
Here g1 and g2 are results of _rewrite_single.
Returns None on failure.
c              3  óB   >#   • U  H  n[        UTS 5      SL v •  M     g7f)FN)rV  r3  s     €rg   rh   Ú_rewrite2.<locals>.<genexpr>a  s   øé € Ð
Lº|°tŒ?˜4  EÓ*¨dÕ2º|ùr5  Nc           	     ót   >• [        [        [        U S   T5      5      [        [        U S   T5      5      5      $ ©Nr   rR   )Úmaxr  ró   ©r¶   r|   s    €rg   r}   Ú_rewrite2.<locals>.<lambda>g  ó,   ø€ ”#”cœ* Q q¡T¨1Ó-Ó.´´J¸qÀ¹tÀQÓ4GÓ0HÔIro   c           	     ót   >• [        [        [        U S   T5      5      [        [        U S   T5      5      5      $ r]  )r^  r  rø   r_  s    €rg   r}   r`  h  ra  ro   c           	     ót   >• [        [        [        U S   T5      5      [        [        U S   T5      5      5      $ r]  )r^  r  r  r_  s    €rg   r}   r`  i  s1   ø€ ”#”cÔ0°°1±°qÓ9Ó:ÜÔ0°°1±°qÓ9Ó:ô<ro   ©FTrR   Fr   )
r  r6  r
  r  rÉ   r   r  r  rV  rT   )rf   r|   r‰   r  r  r  rR  Úfac1Úfac2r¦  r§  rŠ   s    `          rg   Ú	_rewrite2rg  X  sØ   ø€ ô ˜A˜qÓ!�J€CˆQÜ
Ô
L¼yÈ¼|Ó
L×LÑLØÜ˜!Ó€AÞØÜŒW�QÜIÜIô	<ð=ó >ó 	?€Aô $-×#4Ò#4°]ÀAÖ#FÑˆ	‘<�DÜ˜T 1 iÓ0ˆÜ˜T 1 iÓ0ˆßˆ2—"�"Ü�r˜!‘u˜b ™eÓ$ˆDØ�u�}Ø  1¡ r¨!¡u¨dÐ2Ò2ò $Gro   c                ó¬  • [        U 5      n / n[        [        X5      [        R                  1-  [
        S9 Hm  n[        U R                  XU-   5      U5      nU(       d  M*  UR                  XU-
  5      n[        U[        [        5      (       a  UR                  U5        Mk  Us  $    U R                  [        5      (       ax  [        S5        [        [!        U 5      U5      nU(       aQ  [#        U[$        5      (       d+  SSKJn  U" [+        U5      UR-                  [.        5      5      $ UR1                  U5        U(       a  [3        [5        U5      5      $ g)zÿ
Compute an indefinite integral of ``f`` by rewriting it as a G function.

Examples
========

>>> from sympy.integrals.meijerint import meijerint_indefinite
>>> from sympy import sin
>>> from sympy.abc import x
>>> meijerint_indefinite(sin(x), x)
-cos(x)
r×   ú*Try rewriting hyperbolics in terms of exp.r   ©ÚcollectN)r   rÝ   r  r   rå   r   Ú_meijerint_indefinite_1r½   rc   rO   rP   rƒ   rm   r2   r|  Úmeijerint_indefiniter1   rü   rÉ   Úsympy.simplify.radsimprk  r   rõ   r+   ÚextendÚnextr   )rf   r|   Úresultsr•   rn   Úrvrk  s          rg   rm  rm  u  s  € ô 	�‹
€AØ€GÜÔ*¨1Ó0´A·F±F°8Ñ;ÔAQÔRˆÜ% a§f¡f¨Q°A±Ó&6¸Ó:ˆÞÙØ�h‰h�q˜a™%Ó ˆÜ�”UœG×$Ñ$Ø�N‰N˜3ÖàŠJñ Sð 	‡u�uÔ× Ñ ÜÐ;Ô<Ü!Ü'¨Ó*¨Aó/ˆæÜ˜b¤$×'Ñ'Ý:Ùœ|¨BÓ/°·±¼#³Ó?Ð?Ø�N‰N˜2ÔÞÜ”G˜GÓ$Ó%Ð%ð ro   c           	     ó�  ^^• [        SU ST5        SSKJnJn  [	        U T5      nUc  gUu  pVpx[        SU5        [
        R                  n	U GHå  u  p«n[        UR                  T5      u  pÞ[        UT5      u  nnUU-  nXZ-  TSU-   -  -  U-  nUS-   U-  m[        SS	[
        R                  5      nU4S
 jn[        S U" UR                  5       5       5      (       ai  [        [        UR                  5      [        UR                   5      ST-
  /-   [        UR                  5      T* /-   [        UR"                  5      U5      * nOg[        [        UR                  5      ST-
  /-   [        UR                   5      [        UR                  5      [        UR"                  5      T* /-   U5      nUR$                  (       aF  U R'                  TS5      R)                  [
        R*                  [
        R,                  5      (       d  SnOSnU" UR'                  UUTU-  -  5      US9nX“" UU-  SS9-  n	GMè     U4S jn[/        U	SS9n	U	R0                  (       a=  / nU	R2                   H  u  nn[5        U" U5      5      nUUU4/-  nM!     [7        USS06n	O[5        U" U	5      5      n	[7        U	[5        U5      4[9        U T5      S45      $ )z/Helper that does not attempt any substitution. z,Trying to compute the indefinite integral ofÚwrtr   )rC  r¢  Nz could rewrite:rR   r±   zmeijerint-indefinitec                ó8   >• U  Vs/ s H  oT-   PM	     sn$ s  snf rb   r˜   )r¶   r•   rï  s     €rg   r   Ú#_meijerint_indefinite_1.<locals>.trº  s   ø€ Ù%&Ó'¢Q ˜”G¡QÑ'Ð'ùÒ'r   c              3  óV   #   • U  H  oR                   =(       a    US :*  S:H  v •  M!     g7f)r   TN)r¸  )rd   r°   s     rg   rh   Ú*_meijerint_indefinite_1.<locals>.<genexpr>¼  s#   é € ÐCº(°Q�|‰|×0  a¡¨DÑ 0Ô0º(ùs   ‚'))ÚplaceTr£  c                óz   >• [        [        U 5      SS9n [        R                  " U R	                  T5      S   5      $ )a9  This multiplies out superfluous powers of x we created, and chops off
constants:

    >> _clean(x*(exp(x)/x - 1/x) + 3)
    exp(x)

cancel is used before mul_expand since it is possible for an
expression to have an additive constant that does not become isolated
with simple expansion. Such a situation was identified in issue 6369:

Examples
========

>>> from sympy import sqrt, cancel
>>> from sympy.abc import x
>>> a = sqrt(2*x + 1)
>>> bad = (3*x*a**5 + 2*x - a**5 + 1)/a**2
>>> bad.expand().as_independent(x)[0]
0
>>> cancel(bad).expand().as_independent(x)[0]
1
F)ÚdeeprR   )r   rY   r   Ú
_from_argsÚas_coeff_add)rn   r|   s    €rg   Ú_cleanÚ'_meijerint_indefinite_1.<locals>._cleanÎ  s4   ø€ ô. œ ›¨5Ñ1ˆÜ�~Š~˜c×.Ñ.¨qÓ1°!Ñ4Ó5Ð5ro   )Úevaluater€  F)r|  rã   rC  r¢  rX  r   rå   rê   rï   r/  rË   r6  r‡   rP   rÉ   r…   r  r  Úis_extended_nonnegativer½   rm   r,  rO  r6   r`   rl   rG  r5   rS   )rf   r|   rC  r¢  r	  r‰   r  ÚglrŠ   rn   r  rs  r  r•   r°   r=  rÍ   Úfac_r±   r   rª   ry  r~  rf  r÷   rï  s    `                       @rg   rl  rl  š  sƒ  ù€ ä
Ð9¸1¸eÀQÔGß5ä	�1�a‹€BØ	�zààÑ€CˆRÜ
Ð˜bÔ!Ü
�&‰&€CÜ‰ˆˆaÜ˜aŸj™j¨!Ó,‰ˆÜ˜b !Ó$‰ˆˆ1Ø	ˆQ‰ˆð ‰w˜˜Q ™U™Ñ# aÑ'ˆØ�1‰u�a‰iˆô �3Ð.´·±Ó6ˆõ	(äÑC¹"¸Q¿T¹T¼(ÓC×CÑCÜÜ�Q—T‘T“
œD §¡›N¨a°©e¨WÑ4´d¸1¿4¹4³jÀSÀDÀ6Ñ6IÌ4ÐPQ×PXÑPXË>Ð[\ó^ð ^‰Aô Ü�Q—T‘T“
˜a ™e˜WÑ$¤d¨1¯8©8£n´d¸1¿4¹4³jÄ$ÀqÇxÁxÃ.ÐUXÐTXÐSYÑBYÐ[\ó^ˆAð ×$×$¨Q¯V©V°A°q«\×-=Ñ-=¼a¿e¹eÄQ×EVÑEV×-WÑ-WØ‰EàˆEÙ˜Ÿ™˜q ! A q¡D¡&Ó)°Ñ7ˆð 	ˆy˜˜a™ tÑ,Ñ,‹ñK õN6ô4 ˜ tÑ
,€CØ
××ØˆØ—H”H‰DˆAˆqÜ™v a›yÓ)ˆAØ˜˜A˜�xÑŠGñ ô ˜Ð1¨5Ñ1‰ä™V C›[Ó)ˆÜ�cœ>¨$Ó/Ð0´8¸A¸q³>À4Ð2HÓIÐIro   c                óæ	  • [        SXX#45        [        U 5      n U R                  [        5      (       a  [	        S5        gU R                  [
        5      (       a  [	        S5        gXX#4u  pEpg[        S5      nU R                  X5      n UnX#:X  a  [        R                  S4$ / n	U[        R                  L a2  U[        R                  La  [        U R                  X* 5      X* U* 5      $ U[        R                  L Ga	  [	        S5        [        X5      n
[	        SU
5        [        U
[        SS	9[        R                  /-    HÀ  n[	        S
U5        UR                   (       d  [	        S5        M-  [#        U R                  XU-   5      U5      nUc  [	        S5        M[  [#        U R                  XU-
  5      U5      nUc  [	        S5        M‰  Uu  pÎUu  pß[%        ['        Xï5      5      nUS:X  a  [	        S5        M¸  XÍ-   nUU4s  $    GO/U[        R                  L a&  [        XU[        R                  5      nUS   * US   4$ X#4[        R                  [        R                  4:X  aA  [#        X5      nU(       a-  [)        US   [*        5      (       a  U	R-                  U5        GO”U$ GO�U[        R                  L a’  [        X5       Hƒ  nUU-
  S:¬  S:X  d  M  [        SU5        [#        U R                  XU-   5      [/        UU-   U-
  5      -  U5      nU(       d  MV  [)        US   [*        5      (       a  U	R-                  U5        M�  Us  $    U R                  XU-   5      n X2-
  nSnU[        R                  Lah  [1        [        R2                  [5        U5      -  5      n[7        U5      nU R                  UUU-  5      n U [/        X1-
  5      U-  -  n [        R                  n[	        SX#5        [	        SU 5        [#        X5      nU(       a,  [)        US   [*        5      (       a  U	R-                  U5        OU$ UR                  [8        5      (       aˆ  [	        S5        [        [;        U5      XVU5      nU(       a`  [=        U[>        5      (       d:  SSK J!n  U" [E        US   5      US   RG                  [0        5      5      4USS -   nU$ U	RI                  U5        U	(       a  [K        [M        U	5      5      $ g)a   
Integrate ``f`` over the interval [``a``, ``b``], by rewriting it as a product
of two G functions, or as a single G function.

Return res, cond, where cond are convergence conditions.

Examples
========

>>> from sympy.integrals.meijerint import meijerint_definite
>>> from sympy import exp, oo
>>> from sympy.abc import x
>>> meijerint_definite(exp(-x**2), x, -oo, oo)
(sqrt(pi), True)

This function is implemented as a succession of functions
meijerint_definite, _meijerint_definite_2, _meijerint_definite_3,
_meijerint_definite_4. Each function in the list calls the next one
(presumably) several times. This means that calling meijerint_definite
can be very costly.
z$Integrating %s wrt %s from %s to %s.z+Integrand has DiracDelta terms - giving up.Nz5Integrand has Singularity Function terms - giving up.r|   Tz  Integrating -oo to +oo.z  Sensible splitting points:)rÓ   Úreversez  Trying to split atz  Non-real splitting point.z'  But could not compute first integral.z(  But could not compute second integral.Fz)  But combined condition is always false.r   rR   zTrying x -> x + %szChanged limits tozChanged function tori  rj  )'r�  r   rm   r?   r|  rQ   r   r½   r   rå   rP  rL  Úmeijerint_definiter  rÝ   r   Úis_extended_realÚ_meijerint_definite_2rU  rT   rc   rP   rƒ   r@   r+   rÌ   r"   r#   r2   r1   rü   rÉ   rn  rk  r   rõ   ro  rp  r   )rf   r|   r•   r°   rS  Úx_Úa_Úb_r.  rq  r  rÍ   Úres1Úres2Úcond1Úcond2rŠ   rn   Úsplitrñ  rr  rk  s                         rg   r†  r†  ô  s:  € ô< Ð2°Q¸1°LÔAÜ�‹
€AØ‡u�uŒZ×ÑÜÐ<Ô=Øà‡u�uÔ ×!Ñ!ÜÐFÔGØà˜1�Z�N€BˆBô 	ˆc‹
€AØ	�‰ˆq‹€AØ	€AàƒvÜ—‘˜ˆ~Ðà€GØŒA×ÑÒ 1¬A¯J©JÒ#6Ü! !§&¡&¨¨B£-°°B¸¸Ó;Ð;à	
Œa× Ñ Ó	 äÐ*Ô+Ü*¨1Ó0ˆ	ÜÐ-¨yÔ9Ü˜	Ô'7ÀÑFÌ!Ï&É&ÈÔQˆAÜÐ)¨1Ô-Ø×%×%ÜÐ4Ô5ÙÜ(¨¯©°°q±5Ó)9¸1Ó=ˆDØ‰|ÜÐ@ÔAÙÜ(¨¯©°°q±5Ó)9¸1Ó=ˆDØ‰|ÜÐAÔBÙØ‰KˆDØ‰KˆDÜœS Ó.Ó/ˆDØ�u‹}ÜÐBÔCÙØ‘+ˆCØ˜�9Òó) Rð, 
Œa�j‰jŠÜ   q¬!¯*©*Ó5ˆØ�A‘ˆw˜˜A™ˆÐà
ˆ”A—F‘FœAŸJ™JÐ'Ó	'ä# AÓ)ˆÞÜ�C˜‘FœG×$Ñ$Ø—‘˜sÖ#à�
ñ	 ð ”—
‘
Š?Ü/°Ö5�Ø˜‘I ‘N tÕ+ÜÐ0°%Ô8Ü/°·±°q¸e¹)Ó0DÜ1:¸1¸u¹9Àq¹=Ó1Iñ1JØKLóN�Cç�sÜ  A¡¬×0Ñ0Ø#ŸN™N¨3Ö/à#&šJñ 6ð �F‰F�1˜!‘eÓˆØ‰EˆØˆØ”A—J‘JÒÜ”a—o‘o¤c¨!£fÑ,Ó-ˆCÜ�A“ˆAØ—‘�q˜#˜a™%Ó ˆAØ”˜1™5Ó! #Ñ%Ñ%ˆAÜ—
‘
ˆAäÐ" AÔ)ÜÐ$ aÔ(Ü# AÓ)ˆÞÜ�C˜‘FœG×$Ñ$Ø—‘˜sÕ#à�
Ø	‡v�vÔ ×!Ñ!ÜÐ;Ô<ÜÜ'¨Ó+¨R°Ró9ˆæÜ˜b¤$×'Ñ'Ý:Ùœl¨2¨a©5Ó1°2°a±5·;±;¼sÓ3CÓDÐFÈÈAÈBÈÑO�Ø�	Ø�N‰N˜2ÔÞÜ”G˜GÓ$Ó%Ð%ð ro   c                óú  • U S4/nUS   S   nU1n[        U5      nXT;  a  X%S4/-  nUR                  U5        [        U5      nXT;  a  X%S4/-  nUR                  U5        UR                  [        [
        5      (       a1  [        [        U5      5      nXT;  a  X%S4/-  nUR                  U5        UR                  [        [        5      (       a+  SSK	J
n  U" U5      nXt;  a  X'S4/-  nUR                  U5        U$ )	z5Try to guess sensible rewritings for integrand f(x). zoriginal integrandr©   r   r   r   zexpand_trig, expand_mul)Úsincos_to_sumztrig power reduction)r   r�   r   rm   r:   r2   r   r7   r8   Úsympy.simplify.fur’  )rf   r|   rn   ÚorigÚsawÚexpandedr’  Úreduceds           rg   Ú_guess_expansionr˜    s  € àÐ#Ð$Ð
%€Càˆr‰7�1‰:€DØˆ&€CÜ˜$Ó€HØÓØ˜<Ð(Ð)Ñ)ˆØ�‰�Ôä�d‹|€HØÓØ˜8Ð$Ð%Ñ%ˆØ�‰�Ôà‡x�xÔ%Ô'9×:Ñ:Üœk¨$Ó/Ó0ˆØÓØÐ8Ð9Ð:Ñ:ˆCØ�G‰G�HÔà‡x�x””S×ÑÝ3Ù Ó%ˆØÓØÐ4Ð5Ð6Ñ6ˆCØ�G‰G�GÔà€Jro   c                óÞ   • [        SSU SS9nU R                  X5      n UnU S:X  a  [        R                  S4$ [	        X5       H'  u  p4[        SU5        [        X15      nU(       d  M%  Us  $    g)a`  
Try to integrate f dx from zero to infinity.

The body of this function computes various 'simplifications'
f1, f2, ... of f (e.g. by calling expand_mul(), trigexpand()
- see _guess_expansion) and calls _meijerint_definite_3 with each of
these in succession.
If _meijerint_definite_3 succeeds with any of the simplified functions,
returns this result.
r|   zmeijerint-definite2T)Úpositiver   ÚTryingN)r/  r½   r   rå   r˜  r|  Ú_meijerint_definite_3)rf   r|   Údummyr  Úexplanationrn   s         rg   rˆ  rˆ  Ÿ  sl   € ô  �3Ð-¨q¸4Ñ@€EØ	�‰ˆqÓ€AØ€AàˆAƒvÜ�v‰v�tˆ|Ðä*¨1Ö0‰ˆÜˆx˜Ô%Ü# AÓ)ˆßˆ3ØŠJò	 1ro   c                ój  • [        X5      nU(       a  US   S:w  a  U$ U R                  (       a€  [        S5        U R                   Vs/ s H  n[        X15      PM     nn[	        S U 5       5      (       a8  / n[
        R                  nU H  u  pgX&-  nXW/-  nM     [        U6 nUS:w  a  X'4$ gggs  snf )z¢
Try to integrate f dx from zero to infinity.

This function calls _meijerint_definite_4 to try to compute the
integral. If this fails, it tries using linearity.
rR   Fz#Expanding and evaluating all terms.c              3  ó(   #   • U  H  oS Lv •  M
     g 7frb   r˜   )rd   rª   s     rg   rh   Ú(_meijerint_definite_3.<locals>.<genexpr>Ê  s   é € Ð+¢d ˜�}¢dùs   ‚N)rF  Úis_Addr|  rl   rk   r   rå   rT   )rf   r|   rn   r  ÚressrŒ  rª   rÍ   s           rg   rœ  rœ  ½  s´   € ô   Ó
%€CÞ
ˆs�1‰v˜‹Øˆ
Ø‡x‡xÜÐ4Ô5Ø56·V²VÓ<²V°Ô% aÖ+±VˆÐ<ÜÑ+¡dÓ+×+Ñ+ØˆEÜ—&‘&ˆCÛ‘�Ø‘�Ø˜‘’ñ ô �U�ˆAØ�E‹zØ�v�ð ð ,ð ùâ<s   ÁB0c                ó*   • [        [        U 5      5      $ rb   )rj  r%   )rf   s    rg   rG  rG  Õ  s   € Ü”j “mÓ$Ð$ro   c                óè  • SSK Jn  [        SU 5        U(       dÂ  [        XSS9nUbµ  Uu  pVpx[        SXVU5        [        R
                  n	U HQ  u  p«n U
S:X  a  M  [        XZ-  XaU-  -  X5      u  p Xš[        X5      -  -  n	[        U[        X5      5      nUS:X  d  MQ    O   [        U5      nUS:X  a  [        S5        O[        S	U	5        [        U" U	5      5      U4$ [        X5      nUGb  S
 GH   nUu  pVpÞn[        SXVXÞ5        [        R
                  n	U H‰  u  nnnU Hz  u  nnn[        X_-  U-  XaUU-   -  -  UUX5      nUc  [        S5              gUu  n
nn[        SU
UU5        [        U[        UUU5      5      nUS:X  a    OXš[        UUU5      -  -  n	M|     M‰    O   [        U5      nUS:X  a  [        SU5        MÕ  [        SU	45        U(       a  X˜4s  $ [        U" U	5      5      U4s  $    gg)aa  
Try to integrate f dx from zero to infinity.

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

This function tries to apply the integration theorems found in literature,
i.e. it tries to rewrite f as either one or a product of two G-functions.

The parameter ``only_double`` is used internally in the recursive algorithm
to disable trying to rewrite f as a single G-function.
r   rB  ÚIntegratingF)rR  Nú#Could rewrite as single G function:úBut cond is always False.z&Result before branch substitutions is:rd  z!Could rewrite as two G functions:zNon-rational exponents.zSaxena subst for yielded:z&But cond is always False (full_pb=%s).z)Result before branch substitutions is: %s)rã   rC  r|  rX  r   rå   rv  r˜  rT   r„  rG  rg  rµ  r  r  r�  )rf   r|   rA  rC  r	  r‰   r  r  rŠ   rn   r  rs  ro  r¦  r§  r±  Ús1Úf1r²  Ús2Úf2rª   Úf1_Úf2_s                           rg   rF  rF  Ù  s  € õ +ä
ˆ=˜!ÔæÜ�q uÑ-ˆØ‰>Ø!ÑˆC�QÜÐ8¸#À1ÔEÜ—&‘&ˆCÛ‘��aØ˜“6ÙÜ(¨©°°a±4±¸Ó>‘�Øœ 1›Ñ(Ñ(�Ü˜4Ô!5°aÓ!;Ó<�Ø˜5•=Ùñ ô " $Ó'ˆDØ�u‹}ÜÐ2Õ3äÐ?ÀÔEÜ%¡k°#Ó&6Ó7¸Ð=Ð=ô 
�1‹€BØ	‚~Ü$ˆGØ$&Ñ!ˆC�R˜TÜÐ6¸ÀÔHÜ—&‘&ˆCÛ ‘
��B˜Û"$‘J�B˜˜BÜ'¨©¨r©	°2¸"¸r¹'±l±?Ø(*¨B°ó<�Aà‘yÜÐ8Ô9ÛØ"#‘K�A�s˜CÜÐ6¸¸3ÀÔDÜ˜tÔ%7¸¸SÀ!Ó%DÓE�DØ˜u“}ÙØœW S¨#¨qÓ1Ñ1Ñ1’Cñ #%ñ Ùñ !ô  " $Ó'ˆDØ�u‹}ÜÐ@À'ÖJäÐCÀcÀWÔMÞØ˜9Ò$Ü%¡k°#Ó&6Ó7¸Ð=Ò=ò7 %ð ro   c           	     óR  • U nUn[        SSS9nU R                  XB5      n [        SU 5        [        X5      (       d  [        S5        g[        R
                  nU R                  (       a  [        U R                  5      nO[        U [        5      (       a  U /nOSnU(       Ga|  / n/ nU(       Ga`  UR                  5       n	[        U	[        5      (       ar  [        U	5      n
U
R                  (       a  XjR                  -  nMY   [        U	R                  S   U5      u  p¼US:X  a  UR                  W5        OÑUR                  U	5        O¿U	R                   (       a�  [        U	5      n
U
R                  (       a  XjR                  -  nMÜ  XR"                  R$                  ;  aF   [        U	R                  U5      u  p¼US:X  a'  UR                  W['        U	R"                  5      -  5        UR                  U	5        OUR                  U	5        U(       a  GM`  [)        U6 n[+        U6 n XR$                  ;  ak  [        S	X5        [-        [/        U5      S5      nUS
:X  a  [        S5        gU [1        X%-   5      -  n[        SXí5        [3        UR                  X$5      U45      $ [5        X5      nUGb[  Uu  nnnn[        SUUU5        [        R
                  nU HO  u  nnn [7        UU-  UUU-  -  X5      u  nn UU[9        XU5      -  -  n[;        U[=        X5      5      nUS
:X  d  MO    O   [?        U5      nUS
:X  a  [        S5        g[        SU5        SSK J!n  [?        U" U5      5      nURE                  [F        5      (       d  U[G        U5      -  nUR                  X"U-   5      n[        U[H        5      (       d  UR                  X"U-   5      nSSK%J&n  [3        UR                  X$5      U4U" UR                  X$5      XS5      S45      $ g! [         a    Sn GN÷f = f! [         a    Sn GNqf = f)aº  
Compute the inverse laplace transform
$\int_{c+i\infty}^{c-i\infty} f(x) e^{tx}\, dx$,
for real c larger than the real part of all singularities of ``f``.

Note that ``t`` is always assumed real and positive.

Return None if the integral does not exist or could not be evaluated.

Examples
========

>>> from sympy.abc import x, t
>>> from sympy.integrals.meijerint import meijerint_inversion
>>> meijerint_inversion(1/x, x, t)
Heaviside(t)
r±   Tr£  zLaplace-invertingzBut expression is not analytic.Nr   rR   z.Expression consists of constant and exp shift:Fz3but shift is nonreal, cannot be a Laplace transformz1Result is a delta function, possibly conditional:r§  r¨  z"Result before branch substitution:rB  )ÚInverseLaplaceTransform)'r   r½   r|  r7  r   rå   Úis_MulrÉ   rl   rü   r+   Úpopr   rê   rß   rƒ   ræ   rç   rÙ   r-   r   r   r   r!   r?   r5   rX  r  r/  rT   r+  rG  rã   rC  rm   r@   rh  rK  r°  )rf   r|   r±   rS  Út_Úshiftrl   rf  Úexponentialsr"   rb  r•   r°   rŠ   rn   r	  r‰   r  r  r  rs  rC  r°  s                          rg   Úmeijerint_inversionr¶  !  s�  € ð$ 
€BØ	
€BÜˆc˜Ñ€AØ	�‰ˆr‹€AÜ
Ð Ô"Ü˜×ÑÜÐ0Ô1Øô �F‰F€Eà‡x‡xÜ�A—F‘F‹|‰Ü	�A”s×	Ñ	Øˆs‰àˆçØˆØˆßØ—(‘(“*ˆCÜ˜#œs×#Ñ#Ü˜c“{�Ø—;—;ØŸI™IÑ%�DÙðÜ)¨#¯(©(°1©+°qÓ9‘D�Að ˜“6Ø ×'Ñ'¨Õ*à—N‘N 3Õ'Ø——Ü˜c“{�Ø—;—;ØŸI™IÑ%�DÙØŸH™H×1Ñ1Ó1ðÜ-¨c¯g©g°qÓ9™˜ð ˜A“vØ$×+Ñ+¨A¬c°#·(±(«m©OÔ<Ø—‘˜sÕ#à—‘˜sÔ#÷; ‰dô< �\Ð"ˆÜ�ˆMˆà—‘ÓÜÐ?ÀÔJÜ”"�U“)˜QÓˆØ�5‹=ÜÐHÔIØØ”
˜1™9Ó%Ñ%ˆÜÐBÀCÔNä˜#Ÿ(™( 1›/¨4Ð0Ó1Ð1ä	�1‹€BØ	‚~ØÑˆˆR��DÜÐ4°c¸2¸qÔAÜ�f‰fˆÛ‰GˆAˆq�!Ü% c¨!¡e¨R°°1±©W°aÓ;‰DˆAˆqØ�1”^ A¨!Ó,Ñ,Ñ,ˆCÜ�tÔ9¸!Ó?Ó@ˆDØ�u�}Ùñ ô ˜dÓ#ˆØ�5‹=ÜÐ.Õ/äÐ7¸Ô=Ý2Ü ¡¨SÓ!1Ó2ˆCØ—7‘7œ9×%Ñ%Ø”y “|Ñ#�Ø—(‘(˜1 %™iÓ(ˆCÜ˜d¤D×)Ñ)Ø—y‘y ¨¡IÓ.�Ý;Ü˜cŸh™h q›o¨tÐ4Ù5°b·g±g¸a³nÀaÈTÓRÐTXÐYó[ð [ð/ øôI +ó Ø“Aðûô /ó Ø›ðús$   Ä P ÆP ÐPÐPÐP&Ð%P&)rf   r   r|   r   rØ   ztuple[type[Basic], ...]rÈ   )F)²rà   Ú
__future__r   r  Úsympyr   Ú
sympy.corer   r   Úsympy.core.addr   Úsympy.core.basicr   Úsympy.core.cacher	   Úsympy.core.containersr
   Úsympy.core.exprtoolsr   Úsympy.core.functionr   r   r   r   r   Úsympy.core.mulr   Úsympy.core.intfuncr   Úsympy.core.numbersr   r   Úsympy.core.relationalr   r   r   Úsympy.core.sortingr   r   Úsympy.core.symbolr   r   r   r   Úsympy.core.sympifyr   Ú(sympy.functions.combinatorial.factorialsr   Ú$sympy.functions.elementary.complexesr    r!   r"   r#   r$   r%   r&   r'   r(   r)   r*   Ú&sympy.functions.elementary.exponentialr+   r,   r-   Ú#sympy.functions.elementary.integersr.   Ú%sympy.functions.elementary.hyperbolicr/   r0   r1   r2   Ú(sympy.functions.elementary.miscellaneousr4   Ú$sympy.functions.elementary.piecewiser5   r6   Ú(sympy.functions.elementary.trigonometricr7   r8   r9   r:   Úsympy.functions.special.besselr;   r<   r=   r>   Ú'sympy.functions.special.delta_functionsr?   r@   Ú*sympy.functions.special.elliptic_integralsrA   rB   Ú'sympy.functions.special.error_functionsrC   rD   rE   rF   rG   rH   rI   rJ   rK   rL   rM   Ú'sympy.functions.special.gamma_functionsrN   Úsympy.functions.special.hyperrO   rP   Ú-sympy.functions.special.singularity_functionsrQ   Ú	integralsrS   Úsympy.logic.boolalgrT   rU   rV   rW   rX   Úsympy.polysrY   rZ   Úsympy.utilities.iterablesr[   Úsympy.utilities.miscr\   r|  r]   r�  r^   rc   rÎ   Úsympy.utilities.timeutilsrÏ   Útimeitr‚   rN  rß   rê   ró   rø   r  r  r
  r  r  r#  r%  r(  Ú__annotations__r/  r*  r7  rU  rj  rp  rv  r„  r˜  rµ  r  r  r  r+  r/  rL  rV  rX  rg  rm  rl  r†  r˜  rˆ  rœ  rG  rF  r¶  r˜   ro   rg   Ú<module>rÞ     s'  ðòõ8 #Û å ß Ý Ý "Ý $Ý 'Ý -÷8õ 8å Ý #ß +ß :Ñ :ß 8ß :Ó :Ý &Ý >÷÷ ÷ ñ ÷ GÑ FÝ 7÷9ó 9å 9ß J÷ó ç MÓ Mß Iß M÷6÷ 6÷ 6ñ 6å 9ß 8Ý MÝ ß JÕ Jß &Ý 9Ý 0Ý 2ñ 
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!òH$òNò.Iò>0ò QòDð4 +-€Ð
&Ó ,ò	ò#òOôyòv"ô	òô*mò`&ô<F.òRjò`	2ò:	Qòwòtð €ð 	ØóIó ó 	ðIôX#ò3ò:"&òJWJðt ñG&ó ðG&òTò@ò<ò0%ð óD>ó ðD>óNn[ro   