§
    OŠtj‡; ã                  óF  — U d Z ddlmZ ddlZddlmZ ddlmZmZ ddl	m
Z
 ddlmZ ddlmZ dd	lmZ dd
lmZ ddlmZmZmZmZmZ ddlmZ ddlmZ ddlmZmZ ddl m!Z!m"Z"m#Z# ddl$m%Z%m&Z& ddl'm(Z(m)Z)m*Z*m+Z+ ddl,m-Z- ddl.m/Z/ ddl0m1Z1m2Z2m3Z3m4Z4m5Z5m6Z6m7Z7m8Z8m9Z9m:Z:m;Z; ddl<m=Z=m>Z>m?Z? ddl@mAZA ddlBmCZCmDZDmEZEmFZF ddlGmHZH ddlImJZJmKZK ddlLmMZMmNZNmOZOmPZP ddlQmRZRmSZSmTZTmUZU ddlVmWZWmXZX ddlYmZZZm[Z[ ddl\m]Z]m^Z^m_Z_m`Z`maZambZbmcZcmdZdmeZemfZfmgZg ddlhmiZi dd ljmkZkmlZl dd!lmmnZn d"d#lompZp dd$lqmrZrmsZsmtZtmuZumvZv dd%lwmxZxmyZy dd&lzm{Z{ dd'l|m}Z~ dd(l|mZ€  e(d)¦  «        Z�d*„ Z‚d+„ Zƒdd,l„m…Z…  e…d-¦  «        Z†d`d4„Z‡ G d5„ d6eˆ¦  «        Z‰d7„ ZŠd8„ Z‹d9„ ZŒd:„ Z�d;„ ZŽd<„ Z�d=„ Z�d>„ Z‘d?„ Z’d@„ Z“i Z”dAe•dB<   dC„ Z–dD„ Z—dE„ Z˜dadG„Z™dH„ ZšdbdJ„Z›dK„ ZœdbdL„Z�dM„ ZždbdN„ZŸdO„ Z dP„ Z¡dQ„ Z¢dR„ Z£dS„ Z¤da¥ee†dadT„¦   «         ¦   «         Z¦dadU„Z§dV„ Z¨dW„ Z©dX„ Zªe†dY„ ¦   «         Z«dZ„ Z¬d[„ Z­d\„ Z®d]„ Z¯e†dbd^„¦   «         Z°d_„ Z±dS )ca¿  
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                ó˜   ‡— t          | ¦  «        } t          | dd¦  «        r t          ˆfd„| j        D ¦   «         ¦  «        S  | j        ‰Ž S )NÚis_PiecewiseFc              3  ó2   •K  — | ]}t          |g‰¢R Ž V — Œd S ©N)Ú_has)Ú.0ÚiÚfs     €úW/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/sympy/integrals/meijerint.pyú	<genexpr>z_has.<locals>.<genexpr>T   s/   øè è € Ð1Ð1 1•4˜�;˜A�;�;�;Ð1Ð1Ð1Ð1Ð1Ð1ó    )r7   ÚgetattrÚallÚargsÚhas)Úresrg   s    `rh   rd   rd   O   sY   ø€ õ ˜Ñ
Ô
€CÝˆs�N EÑ*Ô*ð 2ÝÐ1Ð1Ð1Ð1¨¬Ð1Ñ1Ô1Ñ1Ô1Ð1Øˆ3Œ7�Aˆ;Ðrj   c                ó¸  ‡ ‡‡	‡
‡‡‡‡‡‡‡— d„ }t          t          |d¦  «        ¦  «        \  ŠŠŠ	Š}t          dd„ g¬¦  «        Š‰t          ‰z  z  Š‰t          j        ddfˆ fd„	Š
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<    G d„ dt          ¦  «        } ‰
t          ‰‰z
  ¦  «        ‰‰z
  ‰	dz
  z  z  ‰	gg g dg‰‰z  t          ‰	¦  «        ‰‰	dz
  z  z  t          ‰dk    ¦  «        ¦  «          ‰
t          ‰‰z
  ¦  «        ‰‰z
  ‰	dz
  z  z  g ‰	gdgg ‰‰z  t          ‰	¦  «        ‰‰	dz
  z  z  t          ‰dk    ¦  «        ¦  «          ‰
t          t          ‰‰z  d‰z  z  z
  ¦  «        ‰‰z
  ‰	dz
  z  z  ‰	gg g dg‰‰z  t          ‰	¦  «        ‰‰	dz
  z  z  t          ‰dk    ¦  «        ¦  «          ‰
t          ‰‰z  d‰z  z  t          z
  ¦  «        ‰‰z
  ‰	dz
  z  z  g ‰	gdgg ‰‰z  t          ‰	¦  «        ‰‰	dz
  z  z  t          ‰dk    ¦  «        ¦  «          ‰
‰‰z   ‰	 z  d‰	z
  gg dgg ‰‰z  ‰‰	 z  t          ‰	¦  «        z  t           |‰	¦  «        ¦  «        ¬¦  «          ‰
t          ‰‰z
  ¦  «        ‰	 z  d‰	z
  gd‰	z
  dz  gdgd‰	z
  dz  g‰‰z  dt          t          ‰	z  dz  ¦  «        z  t          d‰	z
  ¦  «        z  t          ‰¦  «        ‰	 z  z  t          ‰	¦  «        dk     ¦  «          ‰
‰‰	z  ‰‰	z  z
  ‰‰z
  z  d‰	gg d‰	gg ‰‰z  ‰‰	dz
  z  t          ‰	t          z  ¦  «        z  t          z  ¦  «         d„ Šˆˆ	ˆ
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t-          ‰¦  «        g g dgt          j        g‰dz  dz  t+          t          ¦  «        ¦  «          ‰
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  ¦  «        z  ‰d¦  «          |t1          ‰¦  «        ‰z  t          ‰dz
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tC          ‰	‰¦  «        g ‰	g‰	dz
  dgg ‰¦  «          ‰
tE          ‰¦  «        dgg t          j        gdg‰dz  dt+          t          ¦  «        z  ¦  «          ‰
tG          ‰¦  «        g dgdt          j        gg ‰dz  dt+          t          ¦  «        z  ¦  «          ‰
tI          ‰¦  «        t          j        gg dgt'          dd¦  «        g‰dz   ‰t+          t          ¦  «        z  ¦  «          ‰
tK          ‰¦  «        dgg t'          dd¦  «        gdt'          dd¦  «        gt          dz  ‰dz  z  dz  t          j        ¦  «          ‰
tM          ‰¦  «        dgg t'          dd¦  «        gdt'          dd¦  «        gt          dz  ‰dz  z  dz  t          j        ¦  «          ‰
tO          ‰	‰¦  «        g g ‰	dz  g‰	 dz  g‰dz  dz  ¦  «          ‰
tQ          ‰	‰¦  «        g ‰	dz    dz  g‰	dz  ‰	 dz  g‰	dz    dz  g‰dz  dz  ¦  «          ‰
tS          ‰	‰¦  «        g d‰	z   dz  g‰	dz  g‰	 dz  d‰	z   dz  g‰dz  dz  t          ¦  «          ‰
tU          ‰	‰¦  «        g g ‰	dz  ‰	 dz  gg ‰dz  dz  t          j        ¦  «          ‰
tW          ‰¦  «        t          j        t          j        gg dgdg‰ t          j        ¦  «          ‰
tY          ‰¦  «        t          j        dt          j        z  gg dgdg‰ t'          dd¦  «        dz  ¦  «         dS )z8 Add formulae for the function -> meijerg lookup table. c                ó0   — t          | t          g¬¦  «        S )N©Úexclude)r   r_   )Úns    rh   Úwildz"_create_lookup_table.<locals>.wildZ   s   € Ý�A¥˜sÐ#Ñ#Ô#Ð#rj   Úpqabcrt   c                ó   — | j         o| dk    S ©Nr   )Ú
is_Integer©Úxs    rh   ú<lambda>z&_create_lookup_table.<locals>.<lambda>]   s   € ¨¬Ð(>¸¸Qº€ rj   )Ú
propertiesTc	                ó°   •— ‰	                      t          | t          ¦  «        g ¦  «                             | |t	          |||||¦  «        fg||f¦  «         d S rc   )Ú
setdefaultÚ_mytyper_   ÚappendrQ   )
ÚformulaÚanÚapÚbmÚbqr#   ÚfacÚcondÚhintÚtables
            €rh   Úaddz!_create_lookup_table.<locals>.add`   sl   ø€ Ø×Ò� ­!Ñ,Ô,¨bÑ1Ô1×8Ò8¸'Ø'*­G°B¸¸BÀÀCÑ,HÔ,HÐ&IÐ%JÈDÐRVð:Xñ 	Yô 	Yð 	Yð 	Yð 	Yrj   c                óˆ   •— ‰                      t          | t          ¦  «        g ¦  «                             | |||f¦  «         d S rc   )r   r€   r_   r�   )r‚   Úinstrˆ   r‰   rŠ   s       €rh   Úaddiz"_create_lookup_table.<locals>.addid   sD   ø€ Ø×ÒÝ�G�QÑÔ ñ	%ô 	%ß%+¢V¨W°d¸DÀ$Ð,GÑ%HÔ%HÐ%HÐ%HÐ%Hrj   c           	     ór   — | t          dgg g dgt          ¦  «        f| t          g dgdgg t          ¦  «        fgS ©NrS   r   )rQ   r_   )Úas    rh   Úconstantz&_create_lookup_table.<locals>.constanth   sD   € Ø•G˜Q˜C  R¨!¨­aÑ0Ô0Ð1Ø•G˜B   a S¨"­aÑ0Ô0Ð1ð3ð 	3rj   © c                  ó$   — e Zd Zed„ ¦   «         ZdS )ú2_create_lookup_table.<locals>.IsNonPositiveIntegerc                óB   — t          |¦  «        }|j        du r|dk    S d S )NTr   )r&   ry   )Úclsr#   s     rh   Úevalz7_create_lookup_table.<locals>.IsNonPositiveInteger.evalp   s+   € å˜S‘/”/ˆCØŒ~ Ð%Ð%Ø˜a’x�ð &Ð%rj   N)Ú__name__Ú
__module__Ú__qualname__Úclassmethodr˜   r“   rj   rh   ÚIsNonPositiveIntegerr•   n   s-   € € € € € à	ð	 ð 	 ñ 
Œð	 ð 	 ð 	 rj   r�   rS   r   )r‰   é   c                óX   — t           t          dd¦  «        z  | |z  dz  dd| z  z
  z  z  S )Néÿÿÿÿrž   rS   )r   r   )Úrr%   Únus      rh   ÚA1z _create_lookup_table.<locals>.A1ˆ   s1   € Ý•8˜B ‘?”?Ñ" T E¨"¡H¨Q¡J°!°a¸±c±'Ñ#:Ñ:Ð:rj   c                ó  •—  ‰t          ‰dz  ‰z   ¦  «        |‰z  z   ‰z  ‰dz  ‰z   | z  z  d‰z   dz  dd| z  z
  ‰dz  z   gg ‰|‰z  z
  dz  g‰|‰z  z   dz  g‰‰dz  z  ‰‰d| z  z
  z   ‰| |‰¦  «        z  ¦  «         d S )Nrž   rS   r4   )r¡   Úsgnr£   r‘   r‹   ÚbÚts     €€€€€rh   Útmpaddz$_create_lookup_table.<locals>.tmpadd‹   s½   ø€ àˆ�T�!�Q‘$˜‘(‰^Œ^˜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˜s A™œÑ&ñ	(ô 	(ð 	(ð 	(ð 	(rj   r    c                óf  •—  ‰t          ‰‰t          ‰z  z  z   ¦  «        |t          ‰¦  «        z  t          ‰dz  z  z  z   ‰z  ‰‰t          ‰z  z  z   | z  z  d| z
  |‰z  dz  z   gd| z
  |‰z  dz  z
  gdt          j        gg ‰t          ‰z  z  ‰z  ‰‰dz  | z
  z   ‰| |‰¦  «        z  ¦  «         d S )Nrž   rS   r   )r5   r_   r   ÚHalf)r¡   r¥   r£   r‘   r‹   r¦   ÚpÚqs     €€€€€€rh   r¨   z$_create_lookup_table.<locals>.tmpadd—   sÐ   ø€ Øˆ�T�!�a�˜1™‘f‘*ÑÔ ¥D¨¡G¤G¡­A°°!±©HÑ 4Ñ4°qÑ8¸!¸aÅÀ1Á¹f¹*Àq¹ÑHØ�‰U�S˜‘U˜1‘W‰_Ð  A¡¨¨A©¨a©¡Ð0°1µa´f°+¸rØ�a�‰d‰F�1‰H�a˜!˜A™# ™'‘l 2 2 a¨¨a¡=¤=Ñ0ñ	2ô 	2ð 	2ð 	2ð 	2rj   é   é   c                ó–   •— | ‰         }t           j        |z  t          |¦  «        z  t          g dg|dz   z  dg|dz   z  g ‰¦  «        fgS r�   )r   ÚNegativeOner    rQ   ©ÚsubsÚNrt   r§   s     €€rh   Ú	make_log1z'_create_lookup_table.<locals>.make_log1²   sY   ø€ Ø�ŒGˆÝ” Ñ!¥)¨A¡,¤,Ñ.Ý˜˜a˜S ! a¡%™[¨1¨#¨q°1©u©+°r¸1Ñ=Ô=ð?ð @ð 	@rj   c           	     óv   •— | ‰         }t          |¦  «        t          dg|dz   z  g g dg|dz   z  ‰¦  «        fgS r�   )r    rQ   r±   s     €€rh   Ú	make_log2z'_create_lookup_table.<locals>.make_log2·   sM   ø€ Ø�ŒGˆÝ˜1‘”Ý˜!˜˜a !™e™ b¨"¨q¨c°1°q±5©k¸1Ñ=Ô=ð?ð @ð 	@rj   c                ó2   •—  ‰| ¦  «         ‰| ¦  «        z   S rc   r“   )r²   r´   r¶   s    €€rh   Ú	make_log3z'_create_lookup_table.<locals>.make_log3Á   s   ø€ Øˆy˜‰Œ  ¨4¡¤Ñ0Ð0rj   z3/2é   N©T)-ÚlistÚmapr   r_   r   ÚOner   rA   rO   rU   rX   r$   r9   r   r!   rª   r,   r(   r1   r   r0   r5   r8   r:   r.   rQ   rG   ÚImaginaryUnitr°   rI   rJ   rK   rL   rH   rD   rE   rF   rM   rN   r<   r=   r>   r?   rB   rC   )rŠ   ru   ÚcrŽ   r’   r�   r¨   r¸   r£   r‘   r‹   r¦   r´   r¶   rt   r«   r¬   r§   s   `       @@@@@@@@@@rh   Ú_create_lookup_tablerÀ   X   s$  øøøøøøøøøøø€ ð$ð $ð $å�˜T 7Ñ+Ô+Ñ,Ô,�M€A€qˆ!ˆQ�ÝˆSÐ>Ð>Ð?Ð@Ñ@Ô@€AØ	�!ˆQ‰$‰€Aà)*µ´¸DÀtð Yð Yð Yð Yð Yð YðIð Ið Ið Ið Ið Ið3ð 3ð 3ð �X�X˜a‘[”[ $¨Ð-Ð.€Eˆ"�Ið ð  ð  ð  ð  �xñ  ô  ð  ð €C�	�!�a‘%ÑÔ˜!˜a™% 1 q¡5Ñ)Ñ)¨A¨3°°B¸¸¸Q¸q¹SÝˆa‰Œ��Q˜‘U‘Ñ�S  Q¢™ZœZñ)ô )ð )à€C�	�!�a‘%ÑÔ˜!˜a™% 1 q¡5Ñ)Ñ)¨2°¨s°Q°C¸¸Q¸q¹SÝˆa‰Œ��Q˜‘U‘Ñ�S  Q¢™ZœZñ)ô )ð )à€C�	•!�q˜‘s˜a ™c‘lÑ"Ñ#Ô# Q¨¡U¨a°!©eÑ$4Ñ4°q°c¸2¸rÀAÀ3ÈÈ!ÉÝˆa‰Œ��Q˜‘U‘Ñ�S  Q¢™ZœZñ)ô )ð )à€C�	�1�Q‘3˜!˜A™#‘,¥Ñ"Ñ#Ô# Q¨¡U¨a°!©eÑ$4Ñ4°b¸1¸#À¸sÀBÈÈ!ÉÝˆa‰Œ��Q˜‘U‘Ñ�S  Q¢™ZœZñ)ô )ð )à€CˆˆQ‰�1�"‰˜˜A™�w  Q C¨¨Q¨q©S°!°q°b±'½%À¹(¼(Ñ2BÝÐ%Ð% aÑ(Ô(Ñ)Ô)ð+ñ +ô +ð +à€C�ˆA�‰E‰
Œ
�a�RÑ˜1˜q™5˜' Q¨¡U¨A¡I ;°°°q¸1±u¸a±i°[À!ÀAÁ#Ø	�#�b�‰d�1‰f‰+Œ+‰•e˜A ™E‘l”lÑ"¥3 q¡6¤6¨Q¨B¡<Ñ/µ°A±´¸²ñ<ô <ð <à€CˆˆA‰��1‘‰�q˜1‘uÑ  1˜v r¨A¨q¨6°2°q¸±sØ	ˆA�‰E‰
•3�q�‘t‘9”9Ñ�RÑñ!ô !ð !ð;ð ;ð ;ð(ð (ð (ð (ð (ð (ð (ð (ð (ð €Fˆ1ˆa�L„L€LØ
€Fˆ1ˆb�M„M€MØ
€F�1Œ6�1ÑÔÐØ
€F�1Œ6�2ÑÔÐð2ð 2ð 2ð 2ð 2ð 2ð 2ð 2ð 2ð 2ð €Fˆ1ˆa�L„L€LØ
€Fˆ1ˆb�M„M€MØ
€F�1Œ6�1ÑÔÐØ
€F�1Œ6�2ÑÔÐð €C��J�r‰NŒN˜1ÑÑÔ˜r 2¨ s¨BÑ/Ô/Ð/ð €C�ˆQ‰Œ��a�S�1œ6˜( Q¨ F¨A¨q©D°©FµB½ÀÀA¹¼Ñ4FÑGÔGÐGØ€C�ˆQ‰Œ�•a”f�X ˜s¥Q¤V­Q¬VÐ$4°a¸±d¸1±f½bÅ(È1ÈaÁ.Ä.Ñ>PÑQÔQÐQð €C�ˆA‰Œ��B�œ˜ 1 # q¨!¡t¨A¡v­tµB©x¬xÑ8Ô8Ð8Ø€C�ˆA‰Œ��B˜˜�aœf˜X q¨!¡t¨A¡v­tµB©x¬xÑ8Ô8Ð8ð €C�ˆQ‰Œ��R˜!˜�x¨¨A™œÐ/°°A±°a±½½b¹¼À!¹ÑDÔDÐDð@ð @ð @ð @ð @ð @ð
@ð @ð @ð @ð @ð @ð 	€D�ˆQ‰Œ�‰•9˜Q ™UÑ#Ô#Ñ	# Y°Ñ5Ô5Ð5Ø€D�ˆQ‰Œ�‰•9˜Q ™UÑ#Ô#Ñ	# Y°Ñ5Ô5Ð5ð1ð 1ð 1ð 1ð 1ð 1à€D�ˆQ‰Œ�‰�I˜tÑ$Ô$Ð$Ø€D�ˆQ�‰U‰ŒØ	ˆ•#�a‘&”&Ñ	Ô	�aœe¥W¨a°¨V°R¸!¸¸q¸cÀ1ÀQÁ3Ñ%GÔ%GÐHÐIÑ	IØ	ñô ð ð 	€D��S��Q‘‰ZŒZ‰Œ˜(˜(¥3¥s¨1¡v¤v¡;¤;Ñ/Ô/Ý�w˜˜1�v¥¤˜x¨!¨¨qµ!´&¨k¸1¸Q¹3Ñ?Ô?Ð
@Ð	AñBà	ñô ð ð 	€D�ˆA‰ŒØ	ˆ•1”?Ð"¥2Ñ%Ñ	&Ô	&­1¬=½'À"ÀqÀcÈAÈqÈ6ÐSUØ•J˜r‘N”NÑ"ñ;$ô ;$ð +%ð *&ñ 
&à	ñô ð ð €C�ˆ1‰Œ�ˆs�B�œ˜ 1 a &¨!¨Q©$¨q©&µ$µr±(´(¸1±*Ñ=Ô=Ð=Ø€C�ˆ1‰Œˆr�A�3˜˜A˜¥¤ ¨!¨Q©$¨q©&µ4½±8´8°)¸A±+Ñ>Ô>Ð>ð €C�ˆA‰Œ•”�˜"˜q˜c¥H¨R°¡O¤OµX¸bÀ!±_´_Ð#EÅzÐRTÁ~Ä~ÐVWÐYZÑVZÑGZÐ[\ÑG\Ø	�$�r‰(Œ(‰
�1‰ñô ð à€C�ˆA‰Œ�•Q”V˜Q�K ! Q ­!¬&µ!´&Ð)9¸1¸a¹4À¹6Ý
�Aˆe‰HŒH‰ðDØñDñ ô ð ð €C�ˆq�!‰Œ�b˜1˜#  A¡ q˜z¨2¨qÑ1Ô1Ð1ð €C�ˆA‰Œ���R�!œ&˜ A 3¨¨1©¨aµµR±´©jÑ9Ô9Ð9à€C�ˆQ‰Œ��a�S˜1�aœf˜+ r¨1¨a©4°µ4½±8´8±Ñ<Ô<Ð<ð €C�ˆQ‰Œ•!”&�˜2 ˜s¥X¨b°!¡_¤_Ð$5¸¸1¹°u¸aÅÅRÁÄ¹jÑIÔIÐIð €C��‰Œ�a�S˜"�x¨¨1™~œ~Ð.°µH¸QÀ±N´NÐ0CÅRÈÁUÈ1ÈaÉ4ÁZÐPRÁ]ÕTUÔTZÑ[Ô[Ð[Ø€C��‰Œ�a�S˜"�x¨¨1™~œ~Ð.°µH¸QÀ±N´NÐ0CÅRÈÁUÈ1ÈaÉ4ÁZÐPRÁ]ÕTUÔTZÑ[Ô[Ð[ð €C���1‰Œ�r˜2  !¡˜u¨ r¨!¡t f¨a°©d°1©fÑ5Ô5Ð5ð €C���1‰Œ�r˜a !™e˜H Q™J˜<¨!¨A©#°¨r°!©t¨¸¸Q¹°xÀ±z°lÀAÀqÁDÈÁFÑKÔKÐKð6 €C���1‰Œ�r˜Q ™U A™I˜;¨¨1©¨°°°1±°q¸1±u¸a±iÐ/@À!ÀQÁ$ÀqÁ&Í"ÑMÔMÐMð €C���1‰Œ�r˜2  !¡ a R¨¡T˜{¨B°°1±°Q±½¼Ñ?Ô?Ð?ð €C�
�1‰Œ�œ¥¤Ð'¨¨a¨S°1°#¸°r½1¼6ÑBÔBÐBØ€C�
�1‰Œ�œ ¥!¤&¡Ð)¨2°¨s°Q°C¸!¸½XÀbÈ!¹_¼_ÈQÑ=NÑOÔOÐOÐOÐOrj   )ÚtimethisrQ   rg   r   r{   r   Úreturnútuple[type[Basic], ...]c                ó®   ‡— d	d„}‰| j         vrdS | j        rt          | ¦  «        fS t          t	          ˆfd„| j        D ¦   «         |¬¦  «        ¦  «        S )
z4 Create a hashable entity describing the type of f. r{   útype[Basic]rÂ   útuple[int, int, str]c                ó*   — |                       ¦   «         S rc   )Ú	class_keyrz   s    rh   Úkeyz_mytype.<locals>.key/  s   € Ø�{Š{‰}Œ}Ðrj   r“   c              3  óB   •K  — | ]}t          |‰¦  «        D ]}|V — ŒŒd S rc   )r€   )re   r‘   r§   r{   s      €rh   ri   z_mytype.<locals>.<genexpr>6  s8   øè è € ÐBÐB˜qµG¸A¸q±M´MÐBÐB¨q˜ÐBÐBÐBÐBÐBÐBÐBrj   ©rÉ   )r{   rÅ   rÂ   rÆ   )Úfree_symbolsÚis_FunctionÚtypeÚtupleÚsortedrm   )rg   r{   rÉ   s    ` rh   r€   r€   -  st   ø€ ðð ð ð ð 	�”ÐÐØˆrØ	
Œð Ý�A‰wŒwˆxˆÝ•ÐBÐBÐBÐB A¤FÐBÑBÔBÈÐLÑLÔLÑMÔMÐMrj   c                  ó   — e Zd ZdZdS )Ú_CoeffExpValueErrorzD
    Exception raised by _get_coeff_exp, for internal use only.
    N)r™   rš   r›   Ú__doc__r“   rj   rh   rÒ   rÒ   9  s   € € € € € ðð ð 	€Drj   rÒ   c                ó2  — ddl m} t           || ¦  «        ¦  «                             |¦  «        \  }}|s|t          j        fS |\  }|j        r#|j        |k    rt          d¦  «        ‚||j	        fS ||k    r|t          j
        fS t          d| z  ¦  «        ‚)aŒ  
    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.simplifyrÕ   r   Úas_coeff_mulr   ÚZeroÚis_PowÚbaserÒ   r,   r½   )Úexprr{   rÕ   r¿   Úms        rh   Ú_get_coeff_exprÝ   @  s°   € ð& 'Ð&Ð&Ð&Ð&Ð&Ý˜w˜w t™}œ}Ñ-Ô-×:Ò:¸1Ñ=Ô=�F€QˆØð Ø•!”&ˆyÐØ
�C€QØ„xð HØŒ6�QŠ;ˆ;Ý%Ð&?Ñ@Ô@Ð@Ø�!”%ˆxˆØ	
ˆaŠˆØ•!”%ˆxˆå!Ð"?À$Ñ"FÑGÔGÐGrj   c                óH   ‡— ˆfd„Št          ¦   «         } ‰| ||¦  «         |S )a�  
    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                óÐ   •— | |k    r|                      dg¦  «         d S | j        r(| j        |k    r|                      | j        g¦  «         d S | j        D ]} ‰|||¦  «         Œd S ©NrS   )ÚupdaterÙ   rÚ   r,   rm   )rÛ   r{   ro   ÚargumentÚ_exponents_s       €rh   rã   z_exponents.<locals>._exponents_u  sƒ   ø€ Ø�1Š9ˆ9Ø�JŠJ˜�s‰OŒOˆOØˆFØŒ;ð 	˜4œ9¨š>˜>Ø�JŠJ˜œ�zÑ"Ô"Ð"ØˆFØœ	ð 	*ð 	*ˆHØˆK˜ ! SÑ)Ô)Ð)Ð)ð	*ð 	*rj   ©Úset)rÛ   r{   ro   rã   s      @rh   Ú
_exponentsræ   b  s@   ø€ ð&*ð *ð *ð *ð *õ ‰%Œ%€CØ€K��a˜ÑÔÐØ€Jrj   c                óP   ‡— ˆfd„|                       t          ¦  «        D ¦   «         S )zB Find the types of functions in expr, to estimate the complexity. c                ó0   •— h | ]}‰|j         v ¯|j        ’ŒS r“   )rÌ   Úfunc)re   Úer{   s     €rh   ú	<setcomp>z_functions.<locals>.<setcomp>…  s'   ø€ ÐHÐHÐH�q°A¸¼Ð4GÐ4GˆAŒFÐ4GÐ4GÐ4Grj   )Úatomsr   )rÛ   r{   s    `rh   Ú
_functionsrí   ƒ  s)   ø€ àHÐHÐHÐH˜DŸJšJ¥xÑ0Ô0ÐHÑHÔHÐHrj   c                ót   ‡‡‡‡— ˆfd„dD ¦   «         \  ŠŠˆˆˆˆfd„Št          ¦   «         } ‰| |¦  «         |S )ap  
    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}
    c                ó4   •— g | ]}t          |‰g¬ ¦  «        ‘ŒS )rr   )r   )re   rt   r{   s     €rh   ú
<listcomp>z*_find_splitting_points.<locals>.<listcomp>š  s(   ø€ Ð/Ð/Ð/ Q�D�˜Q˜CÐ Ñ Ô Ð/Ð/Ð/rj   Úpqc                ó  •— t          | t          ¦  «        sd S |                      ‰‰z  ‰z   ¦  «        }|r3|‰         dk    r'|                     |‰          |‰         z  ¦  «         d S | j        rd S | j        D ]} ‰||¦  «         Œd S rx   )Ú
isinstancer   Úmatchr‹   Úis_Atomrm   )rÛ   ro   rÜ   râ   Úcompute_innermostr«   r¬   r{   s       €€€€rh   rö   z1_find_splitting_points.<locals>.compute_innermostœ  s¬   ø€ Ý˜$¥Ñ%Ô%ð 	ØˆFØ�JŠJ�q˜‘s˜Q‘wÑÔˆØð 	��1”˜’�Ø�GŠG�Q�q”T�E˜!˜Aœ$‘JÑÔÐØˆFØŒ<ð 	ØˆFØœ	ð 	-ð 	-ˆHØÐ˜h¨Ñ,Ô,Ð,Ð,ð	-ð 	-rj   rä   )rÛ   r{   Ú	innermoströ   r«   r¬   s    ` @@@rh   Ú_find_splitting_pointsrø   ˆ  sq   øøøø€ ð$ 0Ð/Ð/Ð/¨$Ð/Ñ/Ô/�D€A€qð
-ð 
-ð 
-ð 
-ð 
-ð 
-ð 
-ð 
-õ ‘”€IØÐ�d˜IÑ&Ô&Ð&ØÐrj   c                ó&  — t           j        }t           j        }t           j        }t          | ¦  «        } t          j        | ¦  «        }|D ]Ã}||k    r||z  }Œ||j        vr||z  }Œ|j        rš||j        j        vrŒ|j         	                    |¦  «        \  }}||fk    r*t          |j        ¦  «         	                    |¦  «        \  }}||fk    r7|||j        z  z  }|t          t          ||j        z  d¬¦  «        ¦  «        z  }Œ¾||z  }ŒÄ|||fS )aq  
    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©r²   )r   r½   r   r   Ú	make_argsrÌ   rÙ   r,   rÚ   r×   r   r&   r'   )	rg   r{   r‡   ÚpoÚgrm   r‘   r¿   r§   s	            rh   Ú
_split_mulrþ   ¬  s'  € õ Œ%€CÝ	
Œ€BÝ	Œ€AÝ˜!ÑÔ€AåŒ=˜ÑÔ€DØð ð ˆØ�Š6ˆ6Ø�!‰GˆBˆBØ�a”nÐ$Ð$Ø�1‰HˆCˆCàŒxð ˜A Q¤UÔ%7Ð7Ð7Ø”v×*Ò*¨1Ñ-Ô-‘��1Ø˜˜’9�9Ý% a¤fÑ-Ô-×:Ò:¸1Ñ=Ô=‘D�A�qØ˜˜’9�9Ø˜!˜QœU™(‘N�BØ�:¥h¨q°!´%©x¸eÐ&DÑ&DÔ&DÑEÔEÑE�CØØ�‰FˆAˆAà��Aˆ:Ðrj   c                óØ   — t          j        | ¦  «        }g }|D ]P}|j        r2|j        j        r&|j        }|j        }|dk     r| }d|z  }||g|z  z  }Œ;|                     |¦  «         ŒQ|S )a   
    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   rS   )r   rû   rÙ   r,   ry   rÚ   r�   )rg   rm   Úgsrý   rt   rÚ   s         rh   Ú	_mul_argsr  Ó  s‹   € õ Œ=˜ÑÔ€DØ	€BØð 	ð 	ˆØŒ8ð 	˜œÔ(ð 	Ø”ˆAØ”6ˆDØ�1ŠuˆuØ�B�Ø˜‘v�Ø�4�&˜‘(‰NˆBˆBà�IŠI�a‰LŒLˆLˆLØ€Irj   c                óÄ   — t          | ¦  «        }t          |¦  «        dk     rdS t          |¦  «        dk    rt          |¦  «        gS d„ t          |d¦  «        D ¦   «         S )aŸ  
    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))]
    rž   Nc                ó8   — g | ]\  }}t          |Ž t          |Ž f‘ŒS r“   r   )re   r{   Úys      rh   rð   z%_mul_as_two_parts.<locals>.<listcomp>  s)   € ÐHÐHÐH¡6 A q�S�!ˆW•c˜1�gÐÐHÐHÐHrj   )r  ÚlenrÏ   r\   )rg   r   s     rh   Ú_mul_as_two_partsr  ê  s_   € õ. 
�1‰Œ€BÝ
ˆ2�w„w�‚{€{ØˆtÝ
ˆ2�w„w�!‚|€|Ý�b‘	”	ˆ{ÐØHÐHÕ-@ÀÀQÑ-GÔ-GÐHÑHÔHÐHrj   c                ó”  — d„ }t          t          | j        ¦  «        t          | j        ¦  «        z
  ¦  «        }|d| j        z   |dz  z   z  }|dt
          z  |dz
  | j        z  z  z  }|t           || j        |¦  «         || j	        |¦  «         || j
        |¦  «         || j        |¦  «        | j        |z  |||z  z  z  ¦  «        fS )zO Return C, h such that h is a G function of argument z**n and
        g = C*h. c                ó`   ‡— ˆfd„t          j        | t          ‰¦  «        ¦  «        D ¦   «         S )z5 (a1, .., ak) -> (a1/n, (a1+1)/n, ..., (ak + n-1)/n) c                ó&   •— g | ]\  }}||z   ‰z  ‘ŒS r“   r“   )re   r‘   rf   rt   s      €rh   rð   z/_inflate_g.<locals>.inflate.<locals>.<listcomp>  s%   ø€ ÐJÐJÐJ™d˜a ��Q‘˜‘	ÐJÐJÐJrj   )Ú	itertoolsÚproductÚrange)Úparamsrt   s    `rh   Úinflatez_inflate_g.<locals>.inflate  s0   ø€ àJÐJÐJÐJ¥iÔ&7¸ÅÀaÁÄÑ&IÔ&IÐJÑJÔJÐJrj   rS   rž   )r   r  r„   r†   r¢   r   ÚdeltarQ   rƒ   Úaotherr…   Úbotherrâ   )rý   rt   r  ÚvÚCs        rh   Ú
_inflate_gr  	  sÝ   € ð
Kð Kð Kõ 	
�#ˆaŒd‰)Œ)•c˜!œ$‘i”iÑ
Ñ Ô €AØ	ˆA�”‰H�q˜‘s‰NÑ€AØˆ!�B‰$�1�q‘5˜!œ'‘/Ñ	"Ñ"€AØ�g�g�g˜aœd AÑ&Ô&¨¨°´¸!Ñ(<Ô(<Ø�g˜aœd AÑ&Ô&¨¨°´¸!Ñ(<Ô(<Ø”j !‘m a¨!¨A©#¡hÑ.ñ0ô 0ð 0ð 0rj   c                ó®   — d„ }t           || j        ¦  «         || j        ¦  «         || j        ¦  «         || j        ¦  «        d| j        z  ¦  «        S )zQ Turn the G function into one of inverse argument
        (i.e. G(1/x) -> G'(x)) c                ó   — d„ | D ¦   «         S )Nc                ó   — g | ]}d |z
  ‘ŒS ©rS   r“   ©re   r‘   s     rh   rð   z'_flip_g.<locals>.tr.<locals>.<listcomp>  s   € Ð!Ð!Ð!˜!��A‘Ð!Ð!Ð!rj   r“   ©Úls    rh   Útrz_flip_g.<locals>.tr  s   € Ø!Ð!˜qÐ!Ñ!Ô!Ð!rj   rS   )rQ   r…   r  rƒ   r  râ   )rý   r  s     rh   Ú_flip_gr    sU   € ð"ð "ð "å�2�2�a”d‘8”8˜R˜R ¤™\œ\¨2¨2¨a¬d©8¬8°R°R¸¼±\´\À1ÀQÄZÁ<ÑPÔPÐPrj   c           	     óÜ  ‡— |dk     rt          t          | ¦  «        | ¦  «        S t          |j        ¦  «        Št          |j        ¦  «        }t          | |¦  «        \  }} | j        }|dt          z  d‰z
  dz  z  ‰t          dd¦  «        z  z  z  }|‰‰z  z  }ˆfd„t          ‰¦  «        D ¦   «         }|t          | j        | j        | j        t          | j        ¦  «        |z   |¦  «        fS )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   rž   rS   r    c                ó    •— g | ]
}|d z   ‰z  ‘ŒS r  r“   )re   rt   r«   s     €rh   rð   z"_inflate_fox_h.<locals>.<listcomp>8  s!   ø€ Ð	&Ð	&Ð	&˜ˆ1ˆq‰5�!‰)Ð	&Ð	&Ð	&rj   )Ú_inflate_fox_hr  r   r«   r¬   r  râ   r   r   r  rQ   rƒ   r  r…   r»   r  )rý   r‘   r¬   ÚDr_   Úbsr«   s         @rh   r   r   "  sã   ø€ ð 	ˆ1‚u€uÝ�g a™jœj¨1¨"Ñ-Ô-Ð-Ý	ˆ!Œ#‰Œ€AÝ	ˆ!Œ#‰Œ€Aõ �a˜ÑÔ�D€A€qØ	Œ
€AØˆ!�B‰$�1�q‘5˜!‘)Ñ	˜Q¥¨¨Q¡¤Ñ/Ñ	/Ñ/€AØˆˆA‰�I€AØ	&Ð	&Ð	&Ð	&�U 1™XœXÐ	&Ñ	&Ô	&€BØ�g�a”d˜AœH a¤d­D°´©N¬N¸RÑ,?ÀÑCÔCÐCÐCrj   zdict[tuple[str, str], Dummy]Ú_dummiesc                óN   — t          | |fi |¤Ž}||j        v rt          | fi |¤ŽS |S )z¶
    Return a dummy. This will return the same dummy if the same token+name is
    requested more than once, and it is not already in expr.
    This is for being cache-friendly.
    )Ú_dummy_rÌ   r   )ÚnameÚtokenrÛ   ÚkwargsÚds        rh   Ú_dummyr*  ?  sD   € õ 	��eÐ&Ð&˜vÐ&Ð&€AØˆDÔÐÐÝ�TÐ$Ð$˜VÐ$Ð$Ð$Ø€Hrj   c                ód   — | |ft           vrt          | fi |¤Žt           | |f<   t           | |f         S )z`
    Return a dummy associated to name and token. Same effect as declaring
    it globally.
    )r#  r   )r&  r'  r(  s      rh   r%  r%  K  s@   € ð
 �%ˆ=�HÐ$Ð$Ý"'¨Ð"7Ð"7°Ð"7Ð"7��$˜�ÑÝ�T˜5�MÔ"Ð"rj   c                óx   ‡— t          ˆfd„|                      t          t          ¦  «        D ¦   «         ¦  «         S )zŠ 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  ó*   •K  — | ]}‰|j         v V — Œd S rc   )rÌ   ©re   rÛ   r{   s     €rh   ri   z_is_analytic.<locals>.<genexpr>X  s+   øè è € ÐNÐN¨d�1˜Ô)Ð)ÐNÐNÐNÐNÐNÐNrj   )Úanyrì   rA   r$   )rg   r{   s    `rh   Ú_is_analyticr0  U  s9   ø€ õ ÐNÐNÐNÐN°a·g²g½iÍÑ6MÔ6MÐNÑNÔNÑNÔNÐNÐNrj   Tc                ó  ‡‡‡‡‡— ‰r|                       d„ t          ¦  «        } dŠt          | t          ¦  «        s| S t	          dt
          ¬¦  «        \  ŠŠ}t          ‰‰k     t          ‰‰¦  «        ¦  «        ‰‰k    ft          t          t          ‰¦  «        ¦  «        t          k    t          t          ‰¦  «        dt          z  z
  ¦  «        t          k    ¦  «        t          t          ‰¦  «        t          z
  d¦  «        ft          t          dt          ‰¦  «        z  t          z   ¦  «        t          k    t          dt          ‰¦  «        z  t          z
  ¦  «        t          k    ¦  «        t          t          ‰¦  «        d¦  «        ft          t          dt          ‰¦  «        z  t          z   ¦  «        t          k     t          dt          ‰¦  «        z  t          z
  ¦  «        t          k    ¦  «        t          j        ft          t          t          ‰¦  «        t          dz  z
  ¦  «        t          dz  k    t          t          ‰¦  «        t          dz  z   ¦  «        t          dz  k    ¦  «        t          t          ‰¦  «        d¦  «        ft          t          t          ‰¦  «        t          dz  z
  ¦  «        t          dz  k    t          t          ‰¦  «        t          dz  z   ¦  «        t          dz  k     ¦  «        t          j        ft          t          t          ‰dz  dz  dz   ¦  «        ¦  «        t          k     t          t          t          ‰dz  dz  dz   ¦  «        ¦  «        t          ¦  «        ¦  «        t          j        ft          t          t          ‰dz  dz  dz   ¦  «        ¦  «        t          k     t          d‰dz  dz  dz   z  d¦  «        ¦  «        t          j        ft          t          t!          ‰¦  «        ¦  «        t          k    t          t!          t#          dt          z  t          j        z  ¦  «        ‰z  ¦  «        ¦  «        t          k    ¦  «        t          t!          t#          t          j         t          z  ¦  «        ‰z  ¦  «        d¦  «        ft          t          t!          ‰¦  «        ¦  «        t          dz  k    t          t!          t#          t           t          j        z  ¦  «        ‰z  ¦  «        ¦  «        t          dz  k    ¦  «        t          t!          t#          t          j         t          z  dz  ¦  «        ‰z  ¦  «        d¦  «        ft          ‰‰k    t          ‰‰k     |¦  «        ¦  «        ‰‰k    ft          ‰dz  d¦  «        ‰dz  dk    z  ‰dz  dk    ft          d‰z  d¦  «        t'          t          t          ‰¦  «        ¦  «        ¦  «        t          ‰¦  «        z  dk    z  t          ‰¦  «        dk    ft          ‰d¦  «        t'          t          t          ‰¦  «        ¦  «        ¦  «        t          ‰¦  «        z  dk    z  t          ‰¦  «        dk    ft          t          ‰¦  «        ¦  «        t          dz  k     t'          t          t          ‰¦  «        ¦  «        ¦  «        t)          t          ‰dz  ¦  «        ¦  «        z  dk    z  ‰dz  dk    fg} | j        ˆfd	„| j        D ¦   «         Ž } d
}|�r;d}t/          |¦  «        D �]%\  }\  }}|j        | j        k    rŒt/          | j        ¦  «        D �]õ\  }}	||j        d         j        v r#|	                     |j        d         ¦  «        Šd}
n"d}
|	                     |j        d         ¦  «        Š‰sŒbˆfd„|j        d|
…         |j        |
dz   d…         z   D ¦   «         }|gŠ|D ]Ý}t/          | j        ¦  «        D ]Æ\  }}|‰v rŒ
||k    r‰|gz  Š n¯t          |t          ¦  «        rB|j        d         |k    r1t          |t          ¦  «        r|j        d         |j        v r‰|gz  Š nXt          |t          ¦  «        rB|j        d         |k    r1t          |t          ¦  «        r|j        d         |j        v r‰|gz  Š nŒÇŒÞt5          ‰¦  «        t5          |¦  «        dz   k    r�Œ˜ˆfd„t/          | j        ¦  «        D ¦   «         |                     ‰¦  «        gz   }t8          r|dvrt;          d|¦  «          | j        |Ž } d
} �Œ'|�°;ˆˆfd„}|                       d„ |¦  «        } t8          rt;          d| ¦  «         | S )a®  
    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   — | j         S rc   ©Úis_Relational©Ú_s    rh   r|   z_condsimp.<locals>.<lambda>o  s   €  a¤o€ rj   Fzp q r)r—   rž   r   rS   éþÿÿÿc                ó0   •— g | ]}t          |‰¦  «        ‘ŒS r“   )Ú	_condsimp)re   r6  Úfirsts     €rh   rð   z_condsimp.<locals>.<listcomp>•  s#   ø€ Ð>Ð>Ð>¨q•y  EÑ*Ô*Ð>Ð>Ð>rj   Tc                ó:   •— g | ]}|                      ‰¦  «        ‘ŒS r“   rú   )re   r{   rÜ   s     €rh   rð   z_condsimp.<locals>.<listcomp>¥  s#   ø€ ÐTÐTÐT¨1˜QŸVšV A™YœYÐTÐTÐTrj   Nc                ó"   •— g | ]\  }}|‰v¯	|‘ŒS r“   r“   )re   ÚkÚarg_Ú	otherlists      €rh   rð   z_condsimp.<locals>.<listcomp>¸  s1   ø€ ð 2ð 2ð 2¡I Q¨Ø yÐ0Ð0ð  Ø0Ð0Ð0rj   )
r   rž   r­   é   é   é   é   é   é   é   zused new rule:c                ó¶  •— | j         dk    s| j        dk    r| S | j        }|                     t	          ‰¦  «        ‰z  ¦  «        }|s2|                     t          t          ‰¦  «        ‰z  ¦  «        ¦  «        }|sSt          |t          ¦  «        r<|j	        d         j
        s*|j	        d         t          j        u r|j	        d         dk    S | S |‰         dk    S )Nz==r   rS   )Úrel_opÚrhsÚlhsrô   r#   r*   r(   ró   r+   rm   Úis_polarr   ÚInfinity)ÚrelÚLHSrÜ   r«   r¬   s      €€rh   Úrel_touchupz_condsimp.<locals>.rel_touchupÂ  sÒ   ø€ ØŒ:˜ÒÐ ¤¨A¢ ØˆJð ŒgˆØ�IŠI•c˜!‘f”f˜a‘iÑ Ô ˆØð 	AØ—	’	Õ-­j¸©m¬m¸QÑ.>Ñ?Ô?Ñ@Ô@ˆAØð 	Ý˜#Õ0Ñ1Ô1ð )¸#¼(À1¼+Ô:Nð )Øœ œ¥q¤zÐ1Ð1Øœ œ ašÐ(ØˆJØ�!”�q’Ðrj   c                ó   — | j         S rc   r3  r5  s    rh   r|   z_condsimp.<locals>.<lambda>Ñ  s   €  !¤/€ rj   z_condsimp: )Úreplacer   ró   rY   r   r   rV   r   rU   r$   r#   r   r   Úfalser   Útruer*   r-   r¾   r8   r5   ré   rm   Ú	enumeraterÌ   rô   r  r²   r   Úprint)rˆ   r:  r¡   ÚrulesÚchangeÚiruleÚfroÚtort   Úarg1ÚnumÚ	otherargsÚarg2r=  Úarg3ÚnewargsrO  rÜ   r?  r«   r¬   s    `               @@@@rh   r9  r9  [  sb  øøøøø€ ð& ð Ø�|Š|Ð5Ð5Õ7GÑHÔHˆØˆÝ�d�OÑ,Ô,ð ØˆÝ�g¥4Ð(Ñ(Ô(�G€A€qˆ!õ
 
ˆA�ŠE•2�a˜‘8”8Ñ	Ô	˜a 1šfÐ%õ 
�S•�Q‘”‰[Œ[�BÒ¥¥C¨¡F¤F¨Q­r©T¡MÑ 2Ô 2µbÒ 8Ñ	9Ô	9Ý	�C�‰FŒF•R‰K˜Ñ	Ô	ð	å	�S�•3�q‘6”6‘�B‘ÑÔ¥2Ò%¥s¨1­S°©V¬V©8µb©=Ñ'9Ô'9½RÒ'?Ñ	@Ô	@Ý	�C�‰FŒF�A‰Œð	å	�S�•3�q‘6”6‘�B‘ÑÔ¥"Ò$¥c¨!­C°©F¬F©(µR©-Ñ&8Ô&8½BÒ&>Ñ	?Ô	?Ý	
Œð	å	�S•�Q‘”�"˜Q™$‘ÑÔ¥2 a¡4Ò'­­S°©V¬Vµb¸±d©]Ñ);Ô);½rÀ!¹tÒ)CÑ	DÔ	DÝ	�C�‰FŒF�A‰Œð	å	�S•�Q‘”�"˜Q™$‘ÑÔ¥2 a¡4Ò'­­S°©V¬Vµb¸±d©]Ñ);Ô);½bÀ¹dÒ)BÑ	CÔ	CÝ	
Œð	å	�S•�Q˜‘T˜!‘V˜a‘Z‘”Ñ!Ô!¥BÒ&­­3­s°1°a±4¸±6¸A±:©¬Ñ+?Ô+?ÅÑ(DÔ(DÑ	EÔ	EÝ	
Œð	å	�C•�A�q‘D˜‘F˜Q‘J‘”Ñ Ô ¥2Ò%¥r¨!¨Q°©T°!©V°a©Z©.¸!Ñ'<Ô'<Ñ	=Ô	=Ý	
Œð	å	�SÕ$ QÑ'Ô'Ñ(Ô(­BÒ.ÝÕ"¥9¨Rµ©Uµ1´?Ñ-BÑ#CÔ#CÀAÑ#EÑFÔFÑGÔGÍ2ÒMñ
Oô 
Oå	Õ¥	­1¬?Ð*:½2Ñ*=Ñ >Ô >¸qÑ @ÑAÔAÀ1Ñ	EÔ	Eð	Gõ 
�SÕ$ QÑ'Ô'Ñ(Ô(­B¨q©DÒ0ÝÕ"¥9­b¨Sµ´Ñ-@Ñ#AÔ#AÀ!Ñ#CÑDÔDÑEÔEÍÈAÉÒMñ
Oô 
Oå	Õ¥	­1¬?Ð*:½2Ñ*=¸aÑ*?Ñ @Ô @ÀÑ BÑCÔCÀQÑ	GÔ	Gð	Iõ 
ˆA�ŠF•C˜˜Aš˜q‘M”MÑ	"Ô	" A¨¢FÐ+Ý	ˆAˆq‰D�!‰Œ˜˜1™˜qšÑ	! 1 a¡4¨!¢8Ð,Ý	ˆAˆa‰C�‰Œ•s�3�s 1™vœv™;œ;Ñ'Ô'­¨A©¬Ñ.°Ò2Ñ	3µS¸±V´V¸a²ZÐ@Ý	ˆAˆq‰Œ•S��S ™VœV™œÑ%Ô%¥c¨!¡f¤fÑ,¨qÒ0Ñ	1µ3°q±6´6¸A²:Ð>Ý
�c�!‰fŒf‰+Œ+�˜1™Ò
¥¥S­¨Q©¬¡[¤[Ñ!1Ô!1µ$µs¸1¸a¹4±y´y±/´/Ñ!AÀAÒ!EÑ	FÈÈ1ÉÈqÊÐQð9€Eð< ˆ4Œ9Ð>Ð>Ð>Ð>°D´IÐ>Ñ>Ô>Ð?€DØ€FØ
ñ (ØˆÝ )¨%Ñ 0Ô 0ð &	ñ &	ÑˆE‘9�C˜ØŒx˜4œ9Ò$Ð$ØÝ$ T¤YÑ/Ô/ð #ñ #‘��4Ø˜œ œÔ0Ð0Ð0ØŸ
š
 3¤8¨A¤;Ñ/Ô/�AØ�C�Cà�CØŸ
š
 3¤8¨A¤;Ñ/Ô/�AØð ØØTÐTÐTÐT°´¸¸#¸´ÀÄÈ#ÐPQÉ'È(È(ÔASÑ0SÐTÑTÔT�	Ø˜C�	Ø%ð "ð "�DÝ#,¨T¬YÑ#7Ô#7ð "ð "™˜˜4Ø 	˜>˜>Ø$Ø 4š<˜<Ø%¨!¨Ñ,˜IØ!˜EÝ% d­CÑ0Ô0ð "°T´Y¸q´\ÀQÒ5FÐ5FÝ *¨4µÑ 5Ô 5ð 6GØ:>¼)ÀA¼,È$Ì)Ð:SÐ:SØ%¨!¨Ñ,˜IØ!˜EÝ% d­CÑ0Ô0ð "°T´Y¸q´\ÀQÒ5FÐ5FÝ *¨4µÑ 5Ô 5ð 6GØ:>¼)ÀA¼,È$Ì)Ð:SÐ:SØ%¨!¨Ñ,˜IØ!˜EøøÝ�y‘>”>¥S¨¡^¤^°aÑ%7Ò7Ð7Ùð2ð 2ð 2ð 2µ¸4¼9Ñ1EÔ1Eð 2ñ 2ô 2Ø57·W²W¸Q±Z´Z°LñA�åð 7ØÐ$FÐFÐFÝÐ.°Ñ6Ô6Ð6Ø �t”y 'Ð*�Ø�ØùðQ ñ (ðVð ð ð ð ð ð �<Š<Ð1Ð1°;Ñ?Ô?€DÝð #Ýˆm˜TÑ"Ô"Ð"Ø€Krj   c                ór   — t          | t          ¦  «        r| S t          |                      ¦   «         ¦  «        S )z Re-evaluate the conditions. )ró   Úboolr9  Údoit)rˆ   s    rh   Ú
_eval_condrd  Ö  s/   € å�$�ÑÔð ØˆÝ�T—Y’Y‘[”[Ñ!Ô!Ð!rj   Fc                ób   — t          | |¦  «        }|s|                     t           d„ ¦  «        }|S )zû Bring expr nearer to its principal branch by removing superfluous
        factors.
        This function does *not* guarantee to yield the principal branch,
        to avoid introducing opaque principal_branch() objects,
        unless full_pb=True. c                ó   — | S rc   r“   )r{   r  s     rh   r|   z&_my_principal_branch.<locals>.<lambda>é  s   € ¸€ rj   )r)   rQ  )rÛ   ÚperiodÚfull_pbro   s       rh   Ú_my_principal_branchri  á  s6   € õ ˜4 Ñ
(Ô
(€CØð <Ø�kŠkÕ*¨N¨NÑ;Ô;ˆØ€Jrj   c           	     ó’  ‡	‡
— t          ||¦  «        \  }Š
t          |j        |¦  «        \  }Š	|                     ¦   «         }t          ||¦  «        }| t	          ‰	¦  «        |‰
dz   ‰	z  dz
  z  z  z  }ˆ	ˆ
fd„}|t           ||j        ¦  «         ||j        ¦  «         ||j        ¦  «         ||j	        ¦  «        ||z  ¦  «        fS )z”
    Rewrite the integral fac*po*g dx, from zero to infinity, as
    integral fac*G, where G has argument a*x. Note po=x**s.
    Return fac, G.
    rS   c                ó"   •— ˆˆfd„| D ¦   «         S )Nc                ó,   •— g | ]}|d ‰z   ‰z  z   d z
  ‘ŒS r  r“   )re   r‘   r¦   Úss     €€rh   rð   z1_rewrite_saxena_1.<locals>.tr.<locals>.<listcomp>ý  s*   ø€ Ð-Ð-Ð- a��Q˜‘U˜A‘I‘ Ñ!Ð-Ð-Ð-rj   r“   ©r  r¦   rm  s    €€rh   r  z_rewrite_saxena_1.<locals>.trü  s    ø€ Ø-Ð-Ð-Ð-Ð-¨1Ð-Ñ-Ô-Ð-rj   )
rÝ   râ   Ú
get_periodri  r$   rQ   rƒ   r  r…   r  )r‡   rü   rý   r{   r6  r‘   rg  r  r  r¦   rm  s            @@rh   Ú_rewrite_saxena_1rp  í  sà   øø€ õ ˜"˜aÑ Ô �D€A€qÝ˜!œ* aÑ(Ô(�D€A€qØ�\Š\‰^Œ^€FÝ˜Q Ñ'Ô'€Að 	�S�‰VŒV�A˜˜Q™ ™	 A™Ñ&Ñ&Ñ'€Að.ð .ð .ð .ð .ð .à�g�b�b˜œ‘h”h   1¤8¡¤¨b¨b°´©h¬h¸¸¸1¼8¹¼Ø˜‘cñô ð ð rj   c                ó0  — | j         }t          | j        |¦  «        \  }}t          t	          | j        ¦  «        t	          | j        ¦  «        t	          | j        ¦  «        t	          | j        ¦  «        g¦  «        \  }}}}	||	k    r_d„ }
t          t           |
| j        ¦  «         |
| j        ¦  «         |
| j        ¦  «         |
| j        ¦  «        ||z  ¦  «        |¦  «        S d„ | j        D ¦   «         d„ | j        D ¦   «         z   }t          |Ž }|d„ | j        D ¦   «         z  }|d„ | j        D ¦   «         z  }t          |Ž }t          | j        ¦  «         |	dz   |z
  dz  z   |	|z
  k    }d„ }d	„ } |d
¦  «          |d||||||	f¦  «          |dt!          | j        ¦  «        t!          | j        ¦  «        f¦  «          |dt!          | j        ¦  «        t!          | j        ¦  «        f¦  «          |d|||f¦  «         g }g }d|k    ||	k     d|k    g}d|k    d|k    t#          |	|dz   ¦  «        t%          t          t#          |d¦  «        t#          ||dz   ¦  «        ¦  «        ¦  «        g}d|k    t#          |	|¦  «        g}t'          t)          |dz  ¦  «        dz   ¦  «        D ]>}|t+          t-          t/          |¦  «        ¦  «        |d|z  z
  t0          z  ¦  «        gz  }Œ?|dk    t-          t/          |¦  «        ¦  «        |t0          z  k     g}t+          |d¦  «        |g}|rg }|||fD ]}|t          ||z   |z   Ž gz  }Œ||z  } |d|¦  «         |g}|rg }t          t#          |d¦  «        |dz   |k    ||	k    t-          t/          |¦  «        ¦  «        |t0          z  k     g|¢R Ž g}||z  } |d|¦  «         ||g}|rg }t          ||	k     d|k    |dk    t#          t-          t/          |¦  «        ¦  «        |t0          z  ¦  «        g|¢R Ž g}|t          ||	dz
  k    t#          |d¦  «        t#          t-          t/          |¦  «        ¦  «        d¦  «        g|¢R Ž gz  }||z  } |d|¦  «         g }|t#          ||	¦  «        t#          |d¦  «        t#          t/          |¦  «        d¦  «        t+          |d¦  «        gz  }|s||gz  }g }t3          | j        | j        ¦  «        D ]\  }}|||z
  gz  }Œ|t          t5          |Ž ¦  «        dk     gz  }t          |Ž }||gz  } |d|g¦  «         t          |dk    t-          t/          |¦  «        ¦  «        |t0          z  k     ¦  «        g}|s||gz  }t          |Ž }||gz  } |d|g¦  «         t7          |Ž S )aV  
    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                ó   — d„ | D ¦   «         S )Nc                ó   — g | ]}d |z
  ‘ŒS r  r“   ©re   r{   s     rh   rð   z4_check_antecedents_1.<locals>.tr.<locals>.<listcomp>  s   € Ð%Ð%Ð%˜a�A˜‘EÐ%Ð%Ð%rj   r“   r  s    rh   r  z _check_antecedents_1.<locals>.tr  s   € Ø%Ð% 1Ð%Ñ%Ô%Ð%rj   c                ó6   — g | ]}t          |¦  «         d k     ‘ŒS r  ©r!   ©re   r¦   s     rh   rð   z(_check_antecedents_1.<locals>.<listcomp>  s$   € Ð
$Ð
$Ð
$˜!�Bˆq‰EŒEˆ6�AŠ:Ð
$Ð
$Ð
$rj   c                ó:   — g | ]}d d t          |¦  «        z
  k     ‘ŒS r  rv  r  s     rh   rð   z(_check_antecedents_1.<locals>.<listcomp>  s&   € Ð'DÐ'DÐ'D¸!¨¨Aµ°1±´©IªÐ'DÐ'DÐ'Drj   c                ó6   — g | ]}t          |¦  «         d k     ‘ŒS r  rv  rw  s     rh   rð   z(_check_antecedents_1.<locals>.<listcomp>  s$   € Ð)Ð)Ð)˜1�R�‰UŒUˆF�QŠJÐ)Ð)Ð)rj   c                ó:   — g | ]}d d t          |¦  «        z
  k     ‘ŒS r  rv  r  s     rh   rð   z(_check_antecedents_1.<locals>.<listcomp>  s&   € Ð,Ð,Ð,˜aˆA�•B�q‘E”E‘	ŠMÐ,Ð,Ð,rj   rS   rž   c                 ó   — t          | Ž  d S rc   )Ú_debug)Úmsgs    rh   r]   z#_check_antecedents_1.<locals>.debug"  s   € Ý�ˆˆˆˆrj   c                ó&   — t          | |¦  «         d S rc   ©Ú_debugf)Ústringr#   s     rh   r^   z$_check_antecedents_1.<locals>.debugf%  s   € Ý�˜ÑÔÐÐÐrj   z$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  rÝ   râ   r   r  r…   rƒ   r„   r†   Ú_check_antecedents_1rQ   r  r  rU   r!   r¢   r»   r   rX   r  r/   r   r$   r*   r   Úzipr   rV   ) rý   r{   Úhelperr  Úetar6  rÜ   rt   r«   r¬   r  ÚtmpÚcond_3Úcond_3_starÚcond_4r]   r^   ÚcondsÚcase1Útmp1Útmp2Útmp3r=  Úextrar§   Úcase2Úcase3Ú
case_extrarm  r‘   r¦   Úcase_extra_2s                                    rh   r‚  r‚    sX  € ð ŒG€EÝ˜AœJ¨Ñ*Ô*�F€CˆÝ•C˜œ‘I”I�s 1¤4™yœy­#¨a¬d©)¬)µS¸¼±Y´YÐ?Ñ@Ô@�J€A€qˆ!ˆQàˆ1‚u€uð	&ð 	&ð 	&å#¥G¨B¨B¨q¬t©H¬H°b°b¸¼±l´lØ,.¨B¨q¬t©H¬H°b°b¸¼±l´lÀAÀcÁEñ%Kô %Kà$%ñ'ô 'ð 	'ð %Ð
$˜qœtÐ
$Ñ
$Ô
$Ð'DÐ'D¸q¼tÐ'DÑ'DÔ'DÑ
D€CÝ�#ˆY€FàÐ)Ð) ¤Ð)Ñ)Ô)Ñ)€CØÐ,Ð, 1¤8Ð,Ñ,Ô,Ñ,€CÝ�s�)€Kå�!”$‰xŒxˆi˜1˜q™5 1™9 a™-Ñ'¨!¨a©%Ò/€Fðð ð ðð ð ð 
€EÐ
0Ñ1Ô1Ð1Ø
€FÐ7Ø�3˜˜1˜a Ð#ñ%ô %ð %à
€Fˆ?�T !¤$™ZœZ­¨a¬h©¬Ð8Ñ9Ô9Ð9Ø
€Fˆ?�T !¤$™ZœZ­¨a¬h©¬Ð8Ñ9Ô9Ð9Ø
€FÐ/°&¸+ÀvÐ1NÑOÔOÐOà€Eð €EØ�ŠF�A˜’E˜1 š6Ð"€DØ�ŠF�A˜’F�B˜q ! a¡%™LœL­#­cµ"°Q¸±(´(½B¸qÀ!ÀaÁ%¹L¼LÑ.IÔ.IÑ*JÔ*JÐK€DØ�ŠF•B�q˜!‘H”HÐ€DÝ•7˜5 ™7Ñ#Ô# aÑ'Ñ(Ô(ð Fð FˆØ••CÕ+¨CÑ0Ô0Ñ1Ô1°E¸A¸a¹C±KÅÑ3CÑDÔDÐEÑEˆˆØ�1Š9•cÕ-¨cÑ2Ô2Ñ3Ô3°e½B±hÒ>Ð
?€CÝ��Q‰ZŒZ˜Ð €EØð ØˆØ�D˜$Ðð +ð +ˆØ•#˜˜C™ %™Ð)Ð*Ñ*ˆˆØ	ˆU�N€EØ	€Eˆ+�uÑÔÐð ˆH€EØð ØˆÝ•�A�q‘”˜1˜q™5 Aš: q¨A¢vÝÕ(¨Ñ-Ô-Ñ.Ô.°µr±Ò9ðCØ<AðCð Cð Cð D€Eà	ˆU�N€EØ	€Eˆ+�uÑÔÐð �VÐ€EØð ØˆÝ��Q’˜˜Qš ¨¢	­2­cÕ2EÀcÑ2JÔ2JÑ.KÔ.KÈUÕSUÉXÑ+VÔ+Vð Øðð ð ð €Eà	�c�!�q˜1‘u’*�b ¨™lœl­B­sÕ3FÀsÑ3KÔ3KÑ/LÔ/LÈaÑ,PÔ,PÐYÐSXÐYÐYÐYÐZÑZ€EØ	ˆU�N€EØ	€Eˆ+�uÑÔÐð €JØ•2�a˜‘8”8�R  q™\œ\­2Õ.AÀ#Ñ.FÔ.FÈÑ+JÔ+JÍBÈsÐTUÉJÌJÐWÑW€JØð Ø�v�hÑˆ
Ø
€AÝ�A”D˜!œ$‘”ð ð ‰ˆˆ1Ø	ˆa�!‰eˆW‰ˆˆØ•2•c˜1�g‘;”; ’?Ð#Ñ#€JÝ�jÐ!€JØ	ˆjˆ\Ñ€EØ	€Eˆ/˜J˜<Ñ(Ô(Ð(å˜ š	¥3Õ':¸3Ñ'?Ô'?Ñ#@Ô#@À5ÍÁ8Ò#KÑLÔLÐM€LØð !Ø˜˜Ñ ˆÝ˜Ð%€LØ	ˆlˆ^Ñ€EØ	€EÐ
  < .Ñ1Ô1Ð1õ
 ˆuˆ:Ðrj   c                ó|  — ddl m} t          | j        |¦  «        \  }}d|z  }| j        D ]}|t          |dz   ¦  «        z  }Œ| j        D ]}|t          d|z
  dz
  ¦  «        z  }Œ| j        D ]}|t          d|z
  dz
  ¦  «        z  }Œ| j        D ]}|t          |dz   ¦  «        z  }Œ |t          |¦  «        ¦  «        S )aƒ  
    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	gammasimprS   )
rÖ   r•  rÝ   râ   r…   rO   rƒ   r  r  r&   )rý   r{   r•  r…  r6  ro   r¦   r‘   s           rh   Ú	_int0oo_1r–  r  sò   € ð )Ð(Ð(Ð(Ð(Ð(å˜AœJ¨Ñ*Ô*�F€CˆØ
ˆC‰%€CàŒTð ð ˆØ�u�Q˜‘U‰|Œ|ÑˆˆØŒTð  ð  ˆØ�u�Q˜‘U˜Q‘YÑÔÑˆˆØŒXð  ð  ˆØ�u�Q˜‘U˜Q‘YÑÔÑˆˆØŒXð ð ˆØ�u�Q˜‘U‰|Œ|ÑˆˆØˆ9•Z ‘_”_Ñ%Ô%Ð%rj   c                ó  ‡‡‡— ˆˆfd„}t          |‰¦  «        \  }}t          |j        ‰¦  «        \  }}	t          |j        ‰¦  «        \  }}
|	dk     dk    r|	 }	t          |¦  «        }|
dk     dk    r|
 }
t          |¦  «        }|	j        r|
j        sdS |	j        |	j        }}|
j        |
j        }}t          ||z  ||z  ¦  «        }|||z  z  }|||z  z  }t          ||¦  «        \  }}t          ||¦  «        \  }} ||¦  «        } ||¦  «        }| ||z  z  } t          |j        ‰¦  «        \  }}t          |j        ‰¦  «        \  }}|dz   |z  dz
  Š| t          |¦  «        |‰z  z  z  } ˆfd„}t           ||j
        ¦  «         ||j        ¦  «         ||j        ¦  «         ||j        ¦  «        |‰z  ¦  «        }t          |j
        |j        |j        |j        |‰z  ¦  «        }ddlm}  || d¬¦  «        ||fS )	aá  
    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                óÔ   •— t          | j        ‰¦  «        \  }}|                      ¦   «         }t          | j        | j        | j        | j        t          ||‰¦  «        ‰|z  z  ¦  «        S rc   )	rÝ   râ   ro  rQ   rƒ   r  r…   r  ri  )rý   r‘   r¦   Úperrh  r{   s       €€rh   Úpbz_rewrite_saxena.<locals>.pb«  sb   ø€ Ý˜aœj¨!Ñ,Ô,‰ˆˆ1Ø�lŠl‰nŒnˆÝ�q”t˜QœX q¤t¨Q¬XÝ+¨A¨s°GÑ<Ô<¸QÀ¹TÑAñCô Cð 	Crj   r   TNrS   c                ó    •— ˆfd„| D ¦   «         S )Nc                ó   •— g | ]}|‰z   ‘ŒS r“   r“   )re   r‘   r,   s     €rh   rð   z/_rewrite_saxena.<locals>.tr.<locals>.<listcomp>Ñ  s   ø€ Ð#Ð#Ð#˜A��C‘Ð#Ð#Ð#rj   r“   )r  r,   s    €rh   r  z_rewrite_saxena.<locals>.trÐ  s   ø€ Ø#Ð#Ð#Ð# Ð#Ñ#Ô#Ð#rj   ©Ú	powdenest©Úpolar)rÝ   râ   r  Úis_Rationalr«   r¬   r   r  r$   rQ   rƒ   r  r…   r  rÖ   rž  )r‡   rü   Úg1Úg2r{   rh  rš  r6  rm  Úb1Úb2Úm1Ún1Úm2Ún2ÚtauÚr1Úr2ÚC1ÚC2Úa1r¦   Úa2r  rž  r,   s       ``                   @rh   Ú_rewrite_saxenar±  �  sY  øøø€ ð6Cð Cð Cð Cð Cð Cõ ˜"˜aÑ Ô �D€A€qÝ˜2œ;¨Ñ*Ô*�E€A€rÝ˜2œ;¨Ñ*Ô*�E€A€rØ
ˆQŠ�4ÒÐØˆSˆÝ�R‰[Œ[ˆØ
ˆQŠ�4ÒÐØˆSˆÝ�R‰[Œ[ˆØŒ>ð  ¤ð ØˆØŒT�2”4ˆ€BØŒT�2”4ˆ€BÝ
ˆr�"‰u�b˜‘eÑ
Ô
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�s�1‰vŒv˜˜C™ÑÑ
 €Cð$ð $ð $ð $ð $å	���B”E‘”˜B˜B˜rœy™MœM¨2¨2¨b¬e©9¬9°b°b¸¼±m´mÀRÈÁTÑ	JÔ	J€BÝ	�”˜œ	 2¤5¨"¬)°R¸±TÑ	:Ô	:€Bà(Ð(Ð(Ð(Ð(Ð(Øˆ9�S Ð%Ñ%Ô% r¨2Ð-Ð-rj   c                óæ2  ‡ ‡‡+‡,‡-‡.‡/‡0‡1‡2‡3‡4‡5— t          ‰ j        |¦  «        \  Š2}t          ‰j        |¦  «        \  Š-}t          t          ‰ j        ¦  «        t          ‰ j        ¦  «        t          ‰ j        ¦  «        t          ‰ j        ¦  «        g¦  «        \  }}Š4Š5t          t          ‰j        ¦  «        t          ‰j        ¦  «        t          ‰j        ¦  «        t          ‰j        ¦  «        g¦  «        \  }}Š.Š0||z   ‰4‰5z   dz  z
  }||z   ‰.‰0z   dz  z
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  z  t          t          ‰2¦  «        ¦  «        z   ‰5‰4z
  z  Š3t          d¦  «         t          d‰2||‰4‰5|‰1f¦  «         t          d‰-||‰.‰0|	‰,f¦  «         t          d|
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  dk    }t          t          ‰2¦  «        ¦  «        |t          z  k     }t!          t          t          ‰2¦  «        ¦  «        |t          z  ¦  «        }t          t          ‰-¦  «        ¦  «        |	t          z  k     }t!          t          t          ‰-¦  «        ¦  «        |	t          z  ¦  «        }t#          ||	z    t          z  t          j        z  ¦  «        }t'          |‰-z  ‰2z  ¦  «        }t'          |‰2z  ‰-z  ¦  «        }|d|z  k    ryt          t!          |
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  ¦  «        dk     ¦  «        ¦  «        }�n7d„ }t          t!          |
d¦  «        |dz
  |	z   dk    t)          t          t+          |d¦  «         ||¦  «        ¦  «        t          t          ‰,‰1z   ‰5z   ‰4z
  ¦  «        dk     t!          |d¦  «        ¦  «        ¦  «        ¦  «        }t          t!          |
d¦  «        |	dz
  |z   dk    t)          t          t+          |d¦  «         ||¦  «        ¦  «        t          t          ‰,‰1z   ‰0z   ‰.z
  ¦  «        dk     t!          |d¦  «        ¦  «        ¦  «        ¦  «        }t)          ||¦  «        }	 	 ‰0‰.z
  t          ‰-¦  «        d‰0‰.z
  z  z  z  t-          ‰/¦  «        z  ‰5‰4z
  t          ‰2¦  «        d‰5‰4z
  z  z  z  t-          ‰3¦  «        z  z   } t/          | dk    ¦  «        dk    r| dk    }!�ngˆ-ˆ.ˆ/ˆ0ˆ2ˆ3ˆ4ˆ5fd„}"t1           |"dd¦  «         |"dd¦  «        z  t          t!          t          ‰2¦  «        d¦  «        t!          t          ‰-¦  «        d¦  «        ¦  «        f |"t3          t          ‰-¦  «        ¦  «        d¦  «         |"t3          t          ‰-¦  «        ¦  «        d¦  «        z  t          t!          t          ‰2¦  «        d¦  «        t+          t          ‰-¦  «        d¦  «        ¦  «        f |"dt3          t          ‰2¦  «        ¦  «        ¦  «         |"dt3          t          ‰2¦  «        ¦  «        ¦  «        z  t          t+          t          ‰2¦  «        d¦  «        t!          t          ‰-¦  «        d¦  «        ¦  «        f |"t3          t          ‰-¦  «        ¦  «        t3          t          ‰2¦  «        ¦  «        ¦  «        df¦  «        }#| dk    t          t!          | d¦  «        t+          |#d¦  «        t          |¦  «        dk    ¦  «        t          t!          | d¦  «        t!          |#d¦  «        t          |¦  «        dk    ¦  «        g}$t)          |$Ž }!n# t4          $ r d}!Y nw xY w|df|df|df|df|df|df|df|df|df|df|df|df|df|df|!d ffD ]\  }%}&t          d!|&|%f¦  «         Œg Š+ˆ+fd"„}'‰+t          ||z  |z  |z  dk    |j        du |	j        du |||||¦  «        gz  Š+ |'d¦  «         ‰+t          t!          ‰4‰5¦  «        t!          |d¦  «        |	j        du ‰2j        du t          ‰1¦  «        dk     ||||¦	  «	        gz  Š+ |'d¦  «         ‰+t          t!          ‰.‰0¦  «        t!          |	d¦  «        |j        du ‰-j        du t          ‰,¦  «        dk     ||||¦	  «	        gz  Š+ |'d¦  «         ‰+t          t!          ‰.‰0¦  «        t!          ‰4‰5¦  «        t!          |d¦  «        t!          |	d¦  «        ‰2j        du ‰-j        du t          ‰,¦  «        dk     t          ‰1¦  «        dk     t+          ‰2‰-¦  «        |||¦  «        gz  Š+ |'d¦  «         ‰+t          t!          ‰.‰0¦  «        t!          ‰4‰5¦  «        t!          |d¦  «        t!          |	d¦  «        ‰2j        du ‰-j        du t          ‰,‰1z   ¦  «        dk     t+          ‰-‰2¦  «        |||¦  «        gz  Š+ |'d¦  «         ‰+t          ‰.‰0k    |j        du |j        du |	dk    ||||||¦
  «
        gz  Š+ |'d¦  «         ‰+t          ‰.‰0k     |j        du |j        du |	dk    ||||||¦
  «
        gz  Š+ |'d¦  «         ‰+t          ‰4‰5k    |j        du |	j        du |dk    ||||||¦
  «
        gz  Š+ |'d¦  «         ‰+t          ‰4‰5k     |j        du |	j        du |dk    ||||||¦
  «
        gz  Š+ |'d¦  «         ‰+t          ‰.‰0k    t!          ‰4‰5¦  «        t!          |d¦  «        |	dk    ‰2j        du t          ‰1¦  «        dk     |||||¦  «        gz  Š+ |'d¦  «         ‰+t          ‰.‰0k     t!          ‰4‰5¦  «        t!          |d¦  «        |	dk    ‰2j        du t          ‰1¦  «        dk     |||||¦  «        gz  Š+ |'d¦  «         ‰+t          t!          ‰.‰0¦  «        ‰4‰5k    |dk    t!          |	d¦  «        ‰-j        du t          ‰,¦  «        dk     |||||¦  «        gz  Š+ |'d¦  «         ‰+t          t!          ‰.‰0¦  «        ‰4‰5k     |dk    t!          |	d¦  «        ‰-j        du t          ‰,¦  «        dk     |||||¦  «        gz  Š+ |'d¦  «         ‰+t          ‰.‰0k     ‰4‰5k    |dk    |	dk    |||||||¦  «        gz  Š+ |'d¦  «         ‰+t          ‰.‰0k    ‰4‰5k     |dk    |	dk    |||||||¦  «        gz  Š+ |'d ¦  «         ‰+t          ‰.‰0k    ‰4‰5k    |dk    |	dk    |||||||||¦  «        gz  Š+ |'d#¦  «         ‰+t          ‰.‰0k     ‰4‰5k     |dk    |	dk    |||||||||¦  «        gz  Š+ |'d$¦  «         ‰+t          t!          |d¦  «        |j        du |j        du |
j        du |||¦  «        gz  Š+ |'d%¦  «         ‰+t          t!          |d¦  «        |j        du |j        du |
j        du |||¦  «        gz  Š+ |'d&¦  «         ‰+t          t!          |d¦  «        |j        du |	j        du |
j        du |||¦  «        gz  Š+ |'d'¦  «         ‰+t          t!          |d¦  «        |j        du |	j        du |
j        du |||¦  «        gz  Š+ |'d(¦  «         ‰+t          t!          ||z  d¦  «        |j        du |	j        du |||||¦  «        gz  Š+ |'d)¦  «         ‰+t          t!          ||z  d¦  «        |j        du |	j        du |||||¦  «        gz  Š+ |'d*¦  «         t;          ‰ |d¬+¦  «        }(t;          ‰|d¬+¦  «        })‰+t          |)t!          |d¦  «        ‰4|k     |j        du ||||¦  «        gz  Š+ |'d,¦  «         ‰+t          |)t!          |d¦  «        ‰5|k     |j        du ||||¦  «        gz  Š+ |'d-¦  «         ‰+t          |(t!          |d¦  «        ‰.|k     |	j        du ||||¦  «        gz  Š+ |'d.¦  «         ‰+t          |(t!          |d¦  «        ‰0|k     |	j        du ||||¦  «        gz  Š+ |'d/¦  «         t)          ‰+Ž }*t/          |*¦  «        dk    r|*S ‰+t          ||z   ‰.k    t!          |d¦  «        t!          |
d¦  «        |j        du |j        du |	j        du t          t          ‰-¦  «        ¦  «        ||z   ‰.z
  dz   t          z  k     |||||!¦  «        gz  Š+ |'d0¦  «         ‰+t          ||z   ‰0k    t!          |d¦  «        t!          |
d¦  «        |j        du |j        du |	j        du t          t          ‰-¦  «        ¦  «        ||z   ‰0z
  dz   t          z  k     |||||!¦  «        gz  Š+ |'d1¦  «         ‰+t          t!          ‰.‰0dz
  ¦  «        t!          |d¦  «        t!          |
d¦  «        |j        du |j        du |	dk    |	t          z  t          t          ‰-¦  «        ¦  «        k     |||||!¦  «        gz  Š+ |'d2¦  «         ‰+t          t!          ‰.‰0dz   ¦  «        t!          |d¦  «        t!          |
d¦  «        |j        du |j        du |	dk    |	t          z  t          t          ‰-¦  «        ¦  «        k     |||||!¦  «        gz  Š+ |'d3¦  «         ‰+t          ‰.‰0dz
  k     t!          |d¦  «        t!          |
d¦  «        |j        du |j        du |	dk    |	t          z  t          t          ‰-¦  «        ¦  «        k     t          t          ‰-¦  «        ¦  «        ||z   ‰.z
  dz   t          z  k     |||||!¦  «        gz  Š+ |'d4¦  «         ‰+t          ‰.‰0dz   k    t!          |d¦  «        t!          |
d¦  «        |j        du |j        du |	dk    |	t          z  t          t          ‰-¦  «        ¦  «        k     t          t          ‰-¦  «        ¦  «        ||z   ‰0z
  dz   t          z  k     |||||!¦  «        gz  Š+ |'d5¦  «         ‰+t          t!          |d¦  «        t!          |
d¦  «        ||z   dk    |j        du |	j        du |j        du t          t          ‰2¦  «        ¦  «        ||z   ‰4z
  dz   t          z  k     |||||!¦  «        gz  Š+ |'d6¦  «         ‰+t          t!          |d¦  «        t!          |
d¦  «        ||z   ‰5k    |j        du |	j        du |j        du t          t          ‰2¦  «        ¦  «        ||z   ‰5z
  dz   t          z  k     |||||!¦  «        gz  Š+ |'d7¦  «         ‰+t          t!          |d¦  «        t!          |
d¦  «        t!          ‰4‰5dz
  ¦  «        |j        du |	j        du |dk    |t          z  t          t          ‰2¦  «        ¦  «        k     t          t          ‰2¦  «        ¦  «        |dz   t          z  k     |||||!¦  «        gz  Š+ |'d8¦  «         ‰+t          t!          |d¦  «        t!          |
d¦  «        t!          ‰4‰5dz   ¦  «        |j        du |	j        du |dk    |t          z  t          t          ‰2¦  «        ¦  «        k     t          t          ‰2¦  «        ¦  «        |dz   t          z  k     |||||!¦  «        gz  Š+ |'d9¦  «         ‰+t          t!          |d¦  «        t!          |
d¦  «        ‰4‰5dz
  k     |j        du |	j        du |dk    |t          z  t          t          ‰2¦  «        ¦  «        k     t          t          ‰2¦  «        ¦  «        ||z   ‰4z
  dz   t          z  k     |||||!¦  «        gz  Š+ |'d:¦  «         ‰+t          t!          |d¦  «        t!          |
d¦  «        ‰4‰5dz   k    |j        du |	j        du |dk    |t          z  t          t          ‰2¦  «        ¦  «        k     t          t          ‰2¦  «        ¦  «        ||z   ‰5z
  dz   t          z  k     |||||!¦  «        gz  Š+ |'d;¦  «         t)          ‰+Ž S )<z> Return a condition under which the integral theorem applies. rž   rS   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                 óŽ   •— ‰‰fD ]>} t          j        | j        | j        ¦  «        D ]\  }}||z
  }|j        r|j        r  dS ŒŒ?dS )NFT)r
  r  rƒ   r…   Ú
is_integerÚis_positive)rý   rf   ÚjÚdiffr¢  r£  s       €€rh   Ú_c1z_check_antecedents.<locals>._c1ú  sm   ø€ Ø�b�ð 	!ð 	!ˆAÝ!Ô)¨!¬$°´Ñ5Ô5ð !ð !‘��1Ø˜1‘u�Ø”?ð ! tÔ'7ð !Ø ˜5˜5˜5øð!ð ˆtrj   c                óV   •— g | ]%}‰j         D ]}t          d |z   |z   ¦  «        dk    ‘ŒŒ&S ©rS   r   )r…   r!   ©re   rf   r¶  r£  s      €rh   rð   z&_check_antecedents.<locals>.<listcomp>  s;   ø€ Ð?Ð?Ð? Q¸¼Ð?Ð?°A�r�!�a‘%˜!‘)‰}Œ}˜qÒ Ð?Ð?Ð?Ð?rj   c                óV   •— g | ]%}‰j         D ]}t          d |z   |z   ¦  «        dk     ‘ŒŒ&S )rS   rž   )rƒ   r!   r»  s      €rh   rð   z&_check_antecedents.<locals>.<listcomp>  s;   ø€ ÐCÐCÐC¨¸R¼UÐCÐC¸�r�!�a‘%˜!‘)‰}Œ}˜uÒ$ÐCÐCÐCÐCrj   c                óŠ   •— g | ]?}‰‰z
  t          d |z   d z
  ¦  «        z  t          ‰¦  «        z
  t          dd¦  «        k    ‘Œ@S ©rS   éýÿÿÿrž   ©r!   r   ©re   rf   Úmur«   r¬   s     €€€rh   rð   z&_check_antecedents.<locals>.<listcomp>  sH   ø€ ÐOÐOÐOÀA��A‘•r˜!˜a™% !™)‘}”}Ñ$¥r¨"¡v¤vÑ-µ¸¸Q±´Ò?ÐOÐOÐOrj   c                ó„   •— g | ]<}‰‰z
  t          d |z   ¦  «        z  t          ‰¦  «        z
  t          dd¦  «        k    ‘Œ=S r¾  rÀ  rÁ  s     €€€rh   rð   z&_check_antecedents.<locals>.<listcomp>  sD   ø€ ÐKÐKÐKÀ��A‘•r˜!˜a™%‘y”yÑ ¥2 b¡6¤6Ñ)­H°R¸©O¬OÒ;ÐKÐKÐKrj   c                óŠ   •— g | ]?}‰‰z
  t          d |z   d z
  ¦  «        z  t          ‰¦  «        z
  t          dd¦  «        k    ‘Œ@S r¾  rÀ  ©re   rf   ÚrhoÚur  s     €€€rh   rð   z&_check_antecedents.<locals>.<listcomp>  sH   ø€ ÐPÐPÐPÀQ��A‘•r˜!˜a™% !™)‘}”}Ñ$¥r¨#¡w¤wÑ.µ¸"¸a±´Ò@ÐPÐPÐPrj   c                ó„   •— g | ]<}‰‰z
  t          d |z   ¦  «        z  t          ‰¦  «        z
  t          dd¦  «        k    ‘Œ=S r¾  rÀ  rÅ  s     €€€rh   rð   z&_check_antecedents.<locals>.<listcomp>  sD   ø€ ÐLÐLÐLÀ��A‘•r˜!˜a™%‘y”yÑ ¥2 c¡7¤7Ñ*­X°b¸!©_¬_Ò<ÐLÐLÐLrj   r   c                ó^   — | dk    o't          t          d| z
  ¦  «        ¦  «        t          k     S )aã  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)`
            rS   )r$   r#   r   )r_   s    rh   Ú_condz!_check_antecedents.<locals>._cond$  s(   € ð ˜’6Ð2�c¥# a¨!¡e¡*¤*™oœoµÒ2Ð2rj   Fc                óÊ   •— | ‰‰z
  z  t          ‰¦  «        d‰‰z
  z  z  z  t          ‰¦  «        z  |‰	‰z
  z  t          ‰¦  «        d‰	‰z
  z  z  z  t          ‰¦  «        z  z   S rà   )r$   r9   )
Úc1Úc2Úomegar«   Úpsir¬   ÚsigmaÚthetarÇ  r  s
     €€€€€€€€rh   Ú	lambda_s0z%_check_antecedents.<locals>.lambda_s0S  sk   ø€ Ø˜1˜q™5‘z¥# e¡*¤*¨q°!°a±%©yÑ"9Ñ9½#¸c¹(¼(ÑBØ˜!˜a™%‘j¥ U¡¤¨a°°Q±©iÑ!8Ñ8½¸U¹¼ÑCñDð Drj   r    Tr®   r­   r@  rA  rB  é   é	   é
   rC  rD  rE  rF  é   z	  c%s: %sc                ó8   •— t          d| ‰d         f¦  «         d S )Nz  case %s: %sr    r  )ÚcountrŠ  s    €rh   Úprz_check_antecedents.<locals>.prl  s"   ø€ Ý� %¨¨r¬Ð!3Ñ4Ô4Ð4Ð4Ð4rj   r¹   é   é   é   é   é   é   é   )r„  ÚE1ÚE2ÚE3ÚE4é   é   é   é   é   é   é   é   é    é!   é"   é#   )rÝ   râ   r   r  r…   rƒ   r„   r†   r¢   r   r$   r*   r|  r€  rU   r!   r   r,   r¾   r&   rV   r   r8   rd  r6   r%   Ú	TypeErrorrµ  Úis_negativer‚  )6r¢  r£  r{   r6  rm  r§   rÜ   rt   ÚbstarÚcstarÚphir…  r¸  rÌ  rÍ  Úc3Úc4Úc5Úc6Úc7Úc8Úc9Úc10Úc11Úc12Úc13Úz0ÚzosÚzsoÚc14rÊ  Úc14_altÚlambda_cÚc15rÒ  Úlambda_sr†  rˆ   rf   rÙ  Ú
mt1_existsÚ
mt2_existsr¡   rŠ  rÂ  rÎ  r«   rÏ  r¬   rÆ  rÐ  rÑ  rÇ  r  s6   ``                                         @@@@@@@@@@@rh   Ú_check_antecedentsr  Ù  sÝ  øøøøøøøøøøøøø€ õ ˜bœk¨1Ñ-Ô-�H€Eˆ1Ý˜bœk¨1Ñ-Ô-�H€Eˆ1Ý•C˜œ‘J”J¥ B¤E¡
¤
­C°´©J¬J½¸B¼E¹
¼
ÐCÑDÔD�J€A€qˆ!ˆQÝ•C˜œ‘J”J¥ B¤E¡
¤
­C°´©J¬J½¸B¼E¹
¼
ÐCÑDÔD�J€A€qˆ!ˆQØ�‰E�Q˜‘U˜A‘IÑ€EØ�‰E�Q˜‘U˜A‘IÑ€EØ
Œ%�1�q‘5˜!‘)Ñ
˜aÑ
€CØ	Œ�!�a‘%˜‘Ñ	˜QÑ	€BØ
ˆa‰%�1�q‘5‰/€CØ
ˆq�1‰u‰+˜Ñ
˜SÑ
 €CÝˆq�1‰u�q‰y‰>�CÕ 3°EÑ :Ô :Ñ;Ô;Ñ;¸aÀ!¹eÑ
D€CÝ��Q‘˜‘‰^�cÕ"5°eÑ"<Ô"<Ñ=Ô=Ñ=ÀÀAÁÑF€Eå
Ð"Ñ#Ô#Ð#ÝÐ?Ø�A�q˜!˜Q  sÐ+ñ-ô -ð -åÐ?Ø�A�q˜!˜Q  rÐ*ñ,ô ,ð ,åÐ0°3¸¸SÀ%Ð2HÑIÔIÐIðð ð ð ð ð ð 
ˆ‰Œ€BÝ	Ð?Ð?Ð?Ð?¨"¬%Ð?Ñ?Ô?Ð	@€BÝ	ÐCÐCÐCÐC¨b¬eÐCÑCÔCÐ	D€BÝ	ÐOÐOÐOÐOÐOÐOÈÌÐOÑOÔOÐ	P€BÝ	ÐKÐKÐKÐKÐKÐKÀRÄUÐKÑKÔKÐ	L€BÝ	ÐPÐPÐPÐPÐPÐPÈ"Ì%ÐPÑPÔPÐ	Q€BÝ	ÐLÐLÐLÐLÐLÐLÀbÄeÐLÑLÔLÐ	M€BÝ
ˆc‰(Œ(�Q•r˜3 ™7 Q¨¡UÑ+¨q°1©u°q¸1±u©oÑ=ÀØ
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||z  ¦  «        |z  S )aà  
    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                ó   — d„ | D ¦   «         S )Nc                ó   — g | ]}| ‘ŒS r“   r“   rt  s     rh   rð   z(_int0oo.<locals>.neg.<locals>.<listcomp>  s   € ˆˆˆ�q��ˆˆˆrj   r“   r  s    rh   Únegz_int0oo.<locals>.neg  s   € Øˆ˜Aˆ‰ŒÐrj   )rÝ   râ   r…   r»   rƒ   r  r  rQ   )r¢  r£  r{   r…  r6  rÎ  r  r¯  r°  r¤  r¥  s              rh   Ú_int0oor  	  sÌ   € õ" ˜BœK¨Ñ+Ô+�F€CˆÝ˜bœk¨1Ñ-Ô-�H€Eˆ1ðð ð à	ˆˆRŒU‰Œ•d˜2œ5‘k”kÑ	!€BÝ	ˆbŒi‰Œ˜3˜3˜rœy™>œ>Ñ	)€BØ	ˆˆRŒU‰Œ•d˜2œ5‘k”kÑ	!€BÝ	ˆbŒi‰Œ˜3˜3˜rœy™>œ>Ñ	)€BÝ�2�r˜2˜r 5¨¡9Ñ-Ô-¨cÑ1Ð1rj   c           	     ó@  ‡‡	— t          ||¦  «        \  }Š	t          |j        |¦  «        \  }Šˆˆ	fd„}ddlm}  || |‰	‰z  z  z  d¬¦  «        t	           ||j        ¦  «         ||j        ¦  «         ||j        ¦  «         ||j        ¦  «        |j        ¦  «        fS )z Absorb ``po`` == x**s into g. c                ó"   •— ˆˆfd„| D ¦   «         S )Nc                ó    •— g | ]
}|‰‰z  z   ‘ŒS r“   r“   )re   r§   r¦   rm  s     €€rh   rð   z2_rewrite_inversion.<locals>.tr.<locals>.<listcomp>,  s!   ø€ Ð#Ð#Ð#˜A��A�a‘C‘Ð#Ð#Ð#rj   r“   rn  s    €€rh   r  z_rewrite_inversion.<locals>.tr+  s    ø€ Ø#Ð#Ð#Ð#Ð# Ð#Ñ#Ô#Ð#rj   r   r�  TrŸ  )	rÝ   râ   rÖ   rž  rQ   rƒ   r  r…   r  )
r‡   rü   rý   r{   r6  r‘   r  rž  r¦   rm  s
           @@rh   Ú_rewrite_inversionr  &  sÃ   øø€ å˜"˜aÑ Ô �D€A€qÝ˜!œ* aÑ(Ô(�D€A€qð$ð $ð $ð $ð $ð $à(Ð(Ð(Ð(Ð(Ð(ØˆI�c˜!˜a ™c™(‘l¨$Ð/Ñ/Ô/Ý�B�B�q”t‘H”H˜b˜b ¤™lœl¨B¨B¨q¬t©H¬H°b°b¸¼±l´lÀAÄJÑOÔOðQð Qrj   c                ó¤
  ‡ ‡‡‡‡‡‡— t          d¦  «         ‰ j        Št          ‰‰¦  «        \  }}|dk     r,t          d¦  «         t          t	          ‰ ¦  «        ‰¦  «        S ˆfd„Šˆfd„Št          t          ‰ j        ¦  «        t          ‰ j        ¦  «        t          ‰ j	        ¦  «        t          ‰ j
        ¦  «        g¦  «        \  }}}}||z   |z
  }||z
  |z
  }	||	z
  dz  }
||z
  Š‰dk    rt
          j        }n‰dk    rd}nt
          j        }d‰z
  dz  t          ‰ j
        Ž z   t          ‰ j	        Ž z
  ‰z  Š‰ j        }t          d||||||	|
‰f¦  «         t          d	|‰|f¦  «         ‰ j        |dz  k    s|dk    r||k    st          d
¦  «         dS t!          j        ‰ j        ‰ j        ¦  «        D ]'\  }}||z
  j        r||k    rt          d¦  «          dS Œ(||k    r*t          d¦  «         t'          ˆˆfd„‰ j        D ¦   «         Ž S ˆ ˆfd„}ˆˆˆfd„}ˆˆˆfd„}ˆˆˆfd„}g }|t'          d|k    d|k    |
t(          z  |z
  t(          dz  k    |dk     |‰t+          t
          j        t(          z  |	dz   z  ¦  «        z  ¦  «        ¦  «        gz  }|t'          |dz   |k    |dz   |k    |dk    |t(          dz  k     |dk    ||z
  dz   t(          z  |z
  t(          dz  k     |‰t+          t
          j        t(          z  ||z
  z  ¦  «        z  ¦  «         |‰t+          t
          j         t(          z  ||z
  z  ¦  «        z  ¦  «        ¦  «        gz  }|t'          ||k    |dk    |dk    ‰|z   t(          z  |z
  t(          dz  k     |‰¦  «        ¦  «        gz  }|t'          t/          t'          ||dz
  k    d|k    |‰dz  k    ¦  «        t'          |dz   ||z   k    ||z   ||z   dz  k    ¦  «        ¦  «        |dk    |t(          dz  k     |dz   t(          z  |z
  t(          dz  k     |‰t+          t
          j        t(          z  |	z  ¦  «        z  ¦  «         |‰t+          t
          j         t(          z  |	z  ¦  «        z  ¦  «        ¦  «        gz  }|t'          d|k    |
dk    |dk    ||
t(          z  z   t(          dz  k     ||z   t(          z  |z
  t(          dz  k     |‰t+          t
          j        t(          z  |	z  ¦  «        z  ¦  «         |‰t+          t
          j         t(          z  |	z  ¦  «        z  ¦  «        ¦  «        gz  }||dk    gz  }t/          |Ž S )z7 Check antecedents for the laplace inversion integral. z#Checking antecedents for inversion:r   z  Flipping G.c           
     óœ  •— t          |‰¦  «        \  }}| |z  } |||z  z  }||z  }g }|t          t          j        t	          |¦  «        z  t
          z  dz  ¦  «        z  }|t          t          j         t	          |¦  «        z  t
          z  dz  ¦  «        z  }	|r|}
n|	}
|t          t          t          |d¦  «        t	          |¦  «        dk    ¦  «        t	          | ¦  «        dk    ¦  «        gz  }|t          t          |d¦  «        t          t          |¦  «        d¦  «        t	          |¦  «        dk    t	          |
¦  «        dk     ¦  «        gz  }|t          t          |d¦  «        t          t          |¦  «        d¦  «        t	          |¦  «        dk    t	          |
¦  «        dk    t	          | ¦  «        dk    ¦  «        gz  }t          |Ž S )Nrž   r   r    )rÝ   r,   r   r¾   r!   r   rU   rV   r   r   r"   )r‘   r¦   r¿   r_   ÚplusÚcoeffÚexponentrŠ  ÚwpÚwmÚwr{   s              €rh   Ústatement_halfz4_check_antecedents_inversion.<locals>.statement_half<  s‰  ø€ Ý(¨¨AÑ.Ô.‰ˆˆxØ	ˆX‰ˆØ	ˆU�A‰X‰ˆØ	ˆX‰ˆØˆØ�s•1”?¥2 a¡5¤5Ñ(­Ñ+¨AÑ-Ñ.Ô.Ñ.ˆØ�s•A”OÐ#¥B q¡E¤EÑ)­"Ñ,¨QÑ.Ñ/Ô/Ñ/ˆØð 	ØˆAˆAàˆAØ•#•b�˜A˜q™œ¥2 a¡5¤5¨A¢:Ñ.Ô.µ°1±´¸²Ñ<Ô<Ð=Ñ=ˆØ•#•b˜˜A‘h”h¥¥2 a¡5¤5¨!¡¤­b°©e¬e°aªi½¸A¹¼ÀºÑCÔCÐDÑDˆØ•#•b˜˜A‘h”h¥¥2 a¡5¤5¨!¡¤­b°©e¬e°aªi½¸A¹¼À!ºÝ˜‘e”e˜r’kñ#ô #ð $ñ 	$ˆå�5ˆzÐrj   c           
     óX   •— t           ‰| |||d¦  «         ‰| |||d¦  «        ¦  «        S )zW Provide a convergence statement for z**a * exp(b*z**c),
             c/f sphinx docs. TF)rU   )r‘   r¦   r¿   r_   r  s       €rh   Ú	statementz/_check_antecedents_inversion.<locals>.statementN  s@   ø€ õ �>�> ! Q¨¨1¨dÑ3Ô3Ø!�> ! Q¨¨1¨eÑ4Ô4ñ6ô 6ð 	6rj   rž   rS   z9  m=%s, n=%s, p=%s, q=%s, tau=%s, nu=%s, rho=%s, sigma=%sz   epsilon=%s, theta=%s, delta=%sz-  Computation not valid for these parameters.Fz  Not a valid G function.z$  Using asymptotic Slater expansion.c                ó2   •— g | ]} ‰|d z
  dd‰¦  «        ‘ŒS rº  r“   ©re   r‘   r  r_   s     €€rh   rð   z0_check_antecedents_inversion.<locals>.<listcomp>|  ó-   ø€ Ð=Ð=Ð=°1�Y�Y˜q 1™u a¨¨AÑ.Ô.Ð=Ð=Ð=rj   c                ó<   •‡ — t          ˆˆ fd„‰j        D ¦   «         Ž S )Nc                ó2   •— g | ]} ‰|d z
  dd‰¦  «        ‘ŒS rº  r“   r!  s     €€rh   rð   z;_check_antecedents_inversion.<locals>.E.<locals>.<listcomp>  r"  rj   )rU   rƒ   )r_   rý   r  s   `€€rh   ÚEz'_check_antecedents_inversion.<locals>.E~  s)   øø€ ÝÐ=Ð=Ð=Ð=Ð=¸¼Ð=Ñ=Ô=Ð>Ð>rj   c                ó(   •—  ‰‰‰ d‰z  | ¦  «        S rà   r“   )r_   rÐ  r  rÑ  s    €€€rh   ÚHz'_check_antecedents_inversion.<locals>.H�  s   ø€ Øˆy˜  ¨¨%©°Ñ3Ô3Ð3rj   c                ó*   •—  ‰‰‰ d‰z  | d¦  «        S )NrS   Tr“   ©r_   rÐ  r  rÑ  s    €€€rh   ÚHpz(_check_antecedents_inversion.<locals>.Hp„  s!   ø€ Øˆ~˜e e V¨Q¨u©W°a¸Ñ>Ô>Ð>rj   c                ó*   •—  ‰‰‰ d‰z  | d¦  «        S )NrS   Fr“   r)  s    €€€rh   ÚHmz(_check_antecedents_inversion.<locals>.Hm‡  s!   ø€ Øˆ~˜e e V¨Q¨u©W°a¸Ñ?Ô?Ð?rj   )r|  râ   rÝ   Ú_check_antecedents_inversionr  r   r  r…   rƒ   r„   r†   rª   ÚNaNr   r  r€  r
  r  r´  rU   r   r,   r¾   rV   )rý   r{   r6  rê   rÜ   rt   r«   r¬   rª  r¢   rÆ  Úepsilonr  r‘   r¦   r%  r'  r*  r,  rŠ  rÐ  r  r  rÑ  r_   s   ``                  @@@@@rh   r-  r-  2  s%  øøøøøøø€ å
Ð0Ñ1Ô1Ð1Ø	Œ
€AÝ˜!˜QÑÔ�D€A€qØˆ1‚u€uÝˆÑÔÐå+­G°A©J¬J¸Ñ:Ô:Ð:ðð ð ð ð ð$6ð 6ð 6ð 6ð 6õ •C˜œ‘I”I�s 1¤4™yœy­#¨a¬d©)¬)µS¸¼±Y´YÐ?Ñ@Ô@�J€A€qˆ!ˆQØ
ˆa‰%�!‰)€CØ	
ˆQ‰�‰€BØ�‰8�Q‰,€CØ�‰E€EØ�‚z€zÝ”&ˆˆØ	�ŠˆØˆˆå”%ˆØ�%‰i˜‰]�S !¤$˜ZÑ'­#¨q¬t¨*Ñ4°eÑ;€EØŒG€EÝÐGØ��1�a˜˜b # uÐ-ñ/ô /ð /åÐ.°¸%ÀÐ0GÑHÔHÐHð ŒG�q˜‘sŠNˆN˜q Ašv˜v¨!¨qª&¨&ÝÐ>Ñ?Ô?Ð?Øˆuõ
 Ô! !¤$¨¬Ñ-Ô-ð ð ‰ˆˆ1Ø�‰EÔð 	 ! a¢% %ÝÐ.Ñ/Ô/Ð/Ø�5�5øð 	ˆA‚v€vÝÐ5Ñ6Ô6Ð6ÝÐ=Ð=Ð=Ð=Ð=¸¼Ð=Ñ=Ô=Ð>Ð>ð?ð ?ð ?ð ?ð ?ð ?ð4ð 4ð 4ð 4ð 4ð 4ð 4ð?ð ?ð ?ð ?ð ?ð ?ð ?ð@ð @ð @ð @ð @ð @ð @ð €Eà	�c�!�q’&˜!˜qš& #¥b¡&¨5¡.µB°q±DÒ"8¸%À!º)Ø�!�A•c�!œ/­"Ñ,¨b°1©fÑ5Ñ6Ô6Ñ6Ñ7Ô7ñ9ô 9ð :ñ :€Eð 
�c�!�a‘%˜1’*˜a !™e qšj¨%°!ª)°U½RÀ¹T²\À1ÈÂ6Ø�q‘5˜1‘9�b‘. 5Ñ(­B¨q©DÒ0Ø�"�Q•s�1œ?­2Ñ-¨q°1©uÑ5Ñ6Ô6Ñ6Ñ7Ô7Ø�"�Q•s�AœOÐ+­BÑ.°°A±Ñ6Ñ7Ô7Ñ7Ñ8Ô8ñ:ô :ð ;ñ ;€Eð
 
�c�!�q’&˜!˜qš& %¨!¢)Ø˜7‘?¥BÑ&¨Ñ.µ"°Q±$Ò6¸¸¸!¹¼ñ>ô >ð ?ñ ?€Eð 
�c•"•S˜˜a !™eš Q¨#¢X¨s°e¸A±gª~Ñ>Ô>Ý˜˜Q™ ! a¡%š¨¨Q©°1°q±5¸!±)Ò);Ñ<Ô<ñ>ô >à˜!’)˜U¥R¨¡Tš\¨C°!©GµR©<¸%Ñ+?Å2ÀaÁ4Ò+GØ�"�Q•s�1œ?­2Ñ-¨bÑ0Ñ1Ô1Ñ1Ñ2Ô2Ø�"�Q•s�AœOÐ+­BÑ.¨rÑ1Ñ2Ô2Ñ2Ñ3Ô3ñ	5ô 5ð 6ñ 6€Eð 
�c�!�q’&˜# š' 5¨1¢9¨e°c½"±f©n½rÀ!¹tÒ.CØ˜‘=¥"Ñ$ uÑ,µ°1±Ò4Ø�"�Q•s�1œ?­2Ñ-¨bÑ0Ñ1Ô1Ñ1Ñ2Ô2Ø�"�Q•s�AœOÐ+­BÑ.¨rÑ1Ñ2Ô2Ñ2Ñ3Ô3ñ5ô 5ð 6ñ 6€Eð
 
ˆa�1ŠfˆXÑ€Eõ ˆuˆ:Ðrj   c                óÀ   — t          | j        |¦  «        \  }}t          t          | j        | j        | j        | j        |||z  z  ¦  «        | ¦  «        \  }} ||z  | z  S )zO
    Compute the laplace inversion integral, assuming the formula applies.
    )rÝ   râ   r   rQ   rƒ   r  r…   r  )rý   r{   r§   r¦   r‘   r  s         rh   Ú_int_inversionr1  ¬  s[   € õ ˜!œ* aÑ(Ô(�D€A€qÝ�' !¤$¨¬°!´$¸¼À!ÀAÀqÁDÁ&ÑIÔIÈAÈ2ÑNÔN�D€A€qØˆQ‰3ˆq‰5€Lrj   c                óê
  ‡ ‡!— ddl m}mŠ!m}mŠ  t
          si at          t
          ¦  «         t          | t          ¦  «        r¡t          | j
        |¦  «                             |¦  «        \  }}t          |¦  «        dk    rdS |d         }|j        r|j        |k    s|j        j        sdS n||k    rdS ddt          | j        | j        | j        | j        ||z  ¦  «        fgdfS | }|                      |t,          ¦  «        } t/          | t,          ¦  «        }|t
          v �rKt
          |         }	|	D �]:\  }
}}}|                      |
d¬¦  «        }|�ri }|                     ¦   «         D ](\  }}t5          t7          |d¬¦  «        d¬¦  «        ||<   Œ)|}t          |t8          ¦  «        s|                     |¦  «        }|d	k    rŒ”t          |t8          t:          f¦  «        s"t5          |                     |¦  «        ¦  «        }t=          |¦  «        d	k    rŒæt          |t>          ¦  «        s ||¦  «        }g }|D �]&\  }}tA          t5          |                     |¦  «                             t,          |¦  «        d¬¦  «        |¦  «        }	 |                     |¦  «                             t,          |¦  «        }n# tB          $ r Y Œ�w xY wtE          ||fz   Ž  #                    tH          j%        tH          j&        tH          j'        ¦  «        rŒÒt          |j        |j        |j        |j        t5          |j
        d¬¦  «        ¦  «        }| (                    ||fz   ¦  «         �Œ(|r||fc S �Œ<|sdS tS          d
¦  «         ˆ ˆ!fd„}|} tU          dd| ¦  «        }d„ }	  || |||d	d¬¦  «        \  }}} |||||¦  «        }n# |$ r d}Y nw xY w|€†tW          dd¦  «        }|| j,        vrmt[          | |¦  «        r]	  ||                      |||z  ¦  «        |||dd	¬¦  «        \  }}} |||||¦  «                             |d¦  «        }n# |$ r d}Y nw xY w|�5| #                    tH          j%        tH          j.        tH          j&        ¦  «        rtS          d¦  «         dS t_          j0        |¦  «        }g }|D ]®} |                      |¦  «        \  }}t          |¦  «        dk    rtc          d¦  «        ‚|d         }tA          |j
        |¦  «        \  }}||dt          |j        |j        |j        |j        t5          t7          |d¬¦  «        d¬¦  «        ||z  z  ¦  «        fgz  }Œ¯tS          d|¦  «         |dfS )aH  
    Try to rewrite f as a sum of single G functions of the form
    C*x**s*G(a*x**b), where b is a rational number and C is independent of x.
    We guarantee that result.argument.as_coeff_mul(x) returns (a, (x**b,))
    or (a, ()).
    Returns a list of tuples (C, s, G) and a condition cond.
    Returns None on failure.
    rS   )Úmellin_transformÚinverse_mellin_transformÚIntegralTransformErrorÚMellinTransformStripErrorNr   T)Úold)Úlift)Úexponents_onlyFz)Trying recursive Mellin transform method.c           
     ó²   •— 	  ‰| |||dd¬¦  «        S # ‰$ r= ddl m}  ‰ |t          t          | ¦  «        ¦  «        ¦  «        |||dd¬¦  «        cY S w xY w)zÔ 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)rÖ   r=  rZ   r   )ÚFrm  r{   Ústripr=  r6  r4  s        €€rh   Úmy_imtz_rewrite_single.<locals>.my_imt
  sª   ø€ ð
	0Ø+Ð+¨A¨q°!°UØ7;ÀdðLñ Lô Lð Løà(ð 	0ð 	0ð 	0Ø/Ð/Ð/Ð/Ð/Ð/Ø+Ð+Ø��¥ q¡	¤	Ñ*Ô*Ñ+Ô+¨Q°°5Ø¨$ð0ñ 0ô 0ð 0ð 0ð 0ð	0øøøs   ƒ ”?AÁArm  zrewrite-singlec           	     ó4  — t          | |d¬¦  «        }|�^ddlm} |\  }}t           ||d¬¦  «        ¦  «        }t	          ||ft          | |t          j        t          j        f¦  «        df¦  «        S t          | |t          j        t          j        f¦  «        S )NT)Úonly_doubler   ©ÚhyperexpandÚnonrepsmall)Úrewrite)	Ú_meijerint_definite_4rÖ   rD  Ú_my_unpolarifyr6   rT   r   rØ   rL  )rg   r{   r¡   rD  ro   rˆ   s         rh   Úmy_integratorz&_rewrite_single.<locals>.my_integrator  s¥   € Ý! ! Q°DÐ9Ñ9Ô9ˆØˆ=Ø2Ð2Ð2Ð2Ð2Ð2Ø‰IˆC�Ý   ¨S¸-Ð!HÑ!HÔ!HÑIÔIˆCÝ˜c 4˜[Ý& q¨1­a¬fµa´jÐ*AÑBÔBÀDÐIñKô Kð Kå˜˜A�qœv¥q¤zÐ2Ñ3Ô3Ð3rj   )Ú
integratorr=  r<  r‘   )rJ  r<  r=  z"Recursive Mellin transform failed.zUnexpected form...z"Recursive Mellin transform worked:)2Ú
transformsr3  r4  r5  r6  Ú_lookup_tablerÀ   ró   rQ   r[   râ   r×   r  rÙ   rÚ   r,   r¡  rƒ   r  r…   r  r²   r_   r€   rô   Úitemsr&   r'   rb  rW   rd  r»   rÝ   Ú
ValueErrorr
   rn   r   rL  ÚComplexInfinityÚNegativeInfinityr�   r|  r*  r%  rÌ   r0  r.  r   rû   ÚNotImplementedError)"rg   r{   Ú	recursiver3  r5  r  rÜ   Úf_r§   r  r‚   Útermsrˆ   r‰   r²   Úsubs_rY  rZ  ro   r‡   rý   r«  r@  rm  rI  r>  r?  r6  r‘   rm   r¿   r¦   r6  r4  s"                                   @@rh   Ú_rewrite_singlerV  ¼  sq  øø€ ð;ð ;ð ;ð ;ð ;ð ;ð ;ð ;ð ;ð ;ð ;ð ;õ ð ,ØˆÝ�]Ñ+Ô+Ð+å�!•WÑÔð 
PÝ˜!œ* aÑ(Ô(×5Ò5°aÑ8Ô8‰ˆˆqÝˆq‰6Œ6�AŠ:ˆ:Ø�4ØˆaŒDˆØŒ8ð 	ØŒv˜Š{ˆ{ !¤%Ô"3ˆ{Ø�tð à�!ŠVˆVØ�4Ø�A•w˜qœt Q¤X¨q¬t°Q´X¸uÀQ¹wÑGÔGÐHÐIÈ4ÐOÐOà	
€BØ	�Šˆq•!‰Œ€AÝ�•1‰Œ€AØ�MÐÑÝ˜!ÔˆØ*+ð #	%ñ #	%Ñ&ˆG�U˜D $Ø—7’7˜7¨�7Ñ-Ô-ˆDØñ !%Ø�Ø#Ÿzšz™|œ|ð Að A‘G�C˜Ý!+­H°R¸dÐ,CÑ,CÔ,CØ;?ð"Añ "Aô "A�E˜#‘J�Jà�Ý! $­Ñ-Ô-ð +ØŸ9š9 T™?œ?�DØ˜5’=�=ØÝ! $­­{Ð(;Ñ<Ô<ð 7Ý% d§i¢i°¡o¤oÑ6Ô6�DÝ˜dÑ#Ô# uÒ,Ð,ØÝ! %­Ñ.Ô.ð (Ø!˜E $™KœK�EØ�Ø#ð *ñ *‘F�C˜Ý'­
°3·8²8¸D±>´>×3FÒ3FÅqÈ!Ñ3LÔ3LØBFð)Hñ )Hô )HØIJñLô L�Bð!ØŸFšF 4™LœL×-Ò-­a°Ñ3Ô3˜˜øÝ%ð !ð !ð !Ø ˜ð!øøøõ
 ˜r Q D™yÐ*×.Ò.­q¬z½1Ô;LÍaÔN`ÑaÔað !Ø Ý ¤ a¤h°´°a´hÝ *¨1¬:ÀdÐ KÑ KÔ KñMô M�Aà—J’J˜r Q D™yÑ)Ô)Ð)Ñ)Øð %Ø ˜9Ð$Ð$Ð$ùð ð ØˆtÝ
Ð6Ñ7Ô7Ð7ð0ð 0ð 0ð 0ð 0ð 0ð 	€AÝˆsÐ$ aÑ(Ô(€Að4ð 4ð 4ðØ&Ð& q¨!¨Q¸=Ø05ÀðFñ Fô F‰ˆˆ5�!àˆF�1�a˜˜EÑ"Ô"ˆˆøØ!ð ð ð Øˆˆˆðøøøà€yõ �CÐ)Ñ*Ô*ˆØ�A”NÐ"Ð"¥|°A°qÑ'9Ô'9Ð"ðØ.Ð.¨q¯vªv°a¸¸1¹©~¬~¸qÀ!Ø:GØ8<ÀuðNñ Nô N‘��5˜!ð �F˜1˜a  EÑ*Ô*×/Ò/°°1Ñ5Ô5��øØ)ð ð ð Ø���ðøøøà€y�A—E’E�!œ*¥a¤e­QÔ->Ñ?Ô?€yÝÐ3Ñ4Ô4Ð4ØˆtÝŒ=˜ÑÔ€DØ
€CØð 	(ð 	(ˆØ�~Š~˜aÑ Ô ‰ˆˆ1Ýˆq‰6Œ6�AŠ:ˆ:Ý%Ð&:Ñ;Ô;Ð;ØˆaŒDˆÝ˜aœj¨!Ñ,Ô,‰ˆˆ1Ø��A•w˜qœt Q¤X¨q¬t°Q´XÝ)­(Ø#$¨4ð+1ñ +1ô +1ØAEð Gñ  Gô  Gà ! 1¡ñ %ñ&ô &ð 'ð (ñ 	(ˆˆõ Ð/°Ñ3Ô3Ð3Ø�ˆ9Ðs7   Ê	.J8Ê8
KËKÎ#N= Î=OÏOÏ6AQ ÑQÑQc                óx   — t          | |¦  «        \  }}}t          |||¦  «        }|r|||d         |d         fS dS )zÿ
    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   rS   N)rþ   rV  )rg   r{   rR  r‡   rü   rý   s         rh   Ú	_rewrite1rX  J  sS   € õ ˜A˜qÑ!Ô!�J€CˆˆQÝ˜˜1˜iÑ(Ô(€AØð #Ø�B˜˜!œ˜a œdÐ"Ð"ð#ð #rj   c           	     óò  ‡— t          | ‰¦  «        \  }}}t          ˆfd„t          |¦  «        D ¦   «         ¦  «        rdS t          |¦  «        }|sdS t	          t          |ˆfd„ˆfd„ˆfd„g¦  «        ¦  «        }t          j        d|¦  «        D ]e\  }\  }}t          |‰|¦  «        }	t          |‰|¦  «        }
|	r9|
r7t          |	d         |
d         ¦  «        }|dk    r|||	d	         |
d	         |fc S ŒfdS )
a  
    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  ó>   •K  — | ]}t          |‰d ¦  «        du V — ŒdS )FN)rV  r.  s     €rh   ri   z_rewrite2.<locals>.<genexpr>a  s4   øè è € Ð
LÐ
L°t�?˜4  EÑ*Ô*¨dÐ2Ð
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Lrj   Nc           	     ó¨   •— t          t          t          | d         ‰¦  «        ¦  «        t          t          | d         ‰¦  «        ¦  «        ¦  «        S ©Nr   rS   )Úmaxr  ræ   ©r«   r{   s    €rh   r|   z_rewrite2.<locals>.<lambda>g  ó=   ø€ •#•c�* Q q¤T¨1Ñ-Ô-Ñ.Ô.µµJ¸qÀ¼tÀQÑ4GÔ4GÑ0HÔ0HÑIÔI€ rj   c           	     ó¨   •— t          t          t          | d         ‰¦  «        ¦  «        t          t          | d         ‰¦  «        ¦  «        ¦  «        S r\  )r]  r  rí   r^  s    €rh   r|   z_rewrite2.<locals>.<lambda>h  r_  rj   c           	     ó¨   •— t          t          t          | d         ‰¦  «        ¦  «        t          t          | d         ‰¦  «        ¦  «        ¦  «        S r\  )r]  r  rø   r^  s    €rh   r|   z_rewrite2.<locals>.<lambda>i  sD   ø€ •#•cÕ0°°1´°qÑ9Ô9Ñ:Ô:ÝÕ0°°1´°qÑ9Ô9Ñ:Ô:ñ<ô <€ rj   ©FTrS   Fr   )
rþ   r/  r  r  r»   r   r
  r  rV  rU   )rg   r{   r‡   rü   rý   r  rR  Úfac1Úfac2r¢  r£  rˆ   s    `          rh   Ú	_rewrite2re  X  sT  ø€ õ ˜A˜qÑ!Ô!�J€CˆˆQÝ
Ð
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L½yÈ¹|¼|Ð
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LÔ
LÑLÔLð ØˆtÝ˜!ÑÔ€AØð ØˆtÝ�W�QØIÐIÐIÐIØIÐIÐIÐIð	<ð 	<ð 	<ð 	<ð=ñ >ô >ñ 	?ô 	?€Aõ $-Ô#4°]ÀAÑ#FÔ#Fð 3ð 3Ñˆ	‘<�D˜$Ý˜T 1 iÑ0Ô0ˆÝ˜T 1 iÑ0Ô0ˆØð 	3�"ð 	3Ý�r˜!”u˜b œeÑ$Ô$ˆDØ�uŠ}ˆ}Ø˜B  1¤ r¨!¤u¨dÐ2Ð2Ð2Ð2øð3ð 3rj   c                ó  — t          | ¦  «        } g }t          t          | |¦  «        t          j        hz  t
          ¬¦  «        D ]y}t          |                      |||z   ¦  «        |¦  «        }|sŒ,|                     |||z
  ¦  «        }t          |t          t          ¦  «        r|                     |¦  «         Œv|c S |                      t          ¦  «        r�t          d¦  «         t          t!          | ¦  «        |¦  «        }|rat#          |t$          ¦  «        s7ddlm}  |t+          |¦  «        |                     t.          ¦  «        ¦  «        S |                     |¦  «         |rt3          t5          |¦  «        ¦  «        S dS )a#  
    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)
    rË   ú*Try rewriting hyperbolics in terms of exp.r   ©ÚcollectN)r   rÐ   rø   r   rØ   r   Ú_meijerint_indefinite_1r²   rd   rP   rQ   r�   rn   r3   r|  Úmeijerint_indefiniter2   ró   r»   Úsympy.simplify.radsimpri  r   rì   r,   ÚextendÚnextr   )rg   r{   Úresultsr‘   ro   Úrvri  s          rh   rk  rk  u  s…  € õ 	�‰
Œ
€AØ€GÝÕ*¨1¨aÑ0Ô0µA´F°8Ñ;ÕAQÐRÑRÔRð ð ˆÝ% a§f¢f¨Q°°A±Ñ&6Ô&6¸Ñ:Ô:ˆØð 	ØØ�hŠh�q˜!˜a™%Ñ Ô ˆÝ�•U�GÑ$Ô$ð 	Ø�NŠN˜3ÑÔÐÐàˆJˆJˆJØ‡u‚uÕÑ Ô ð ÝÐ;Ñ<Ô<Ð<Ý!Ý'¨Ñ*Ô*¨Añ/ô /ˆàð 	Ý˜b¥$Ñ'Ô'ð @Ø:Ð:Ð:Ð:Ð:Ð:Ø�w�|¨BÑ/Ô/°·²½#±´Ñ?Ô?Ð?Ø�NŠN˜2ÑÔÐØð &Ý•G˜GÑ$Ô$Ñ%Ô%Ð%ð&ð &rj   c           	     óÆ  ‡‡— t          d| d‰¦  «         ddlm}m} t	          | ‰¦  «        }|€dS |\  }}}}t          d|¦  «         t
          j        }	|D �]ë\  }
}}t          |j        ‰¦  «        \  }}t          |‰¦  «        \  }}||z  }||
z  ‰d|z   z  z  |z  }|dz   |z  Št          dd	t
          j
        ¦  «        }ˆfd
„}t          d„  ||j        ¦  «        D ¦   «         ¦  «        rit          t          |j        ¦  «        t          |j        ¦  «        d‰z
  gz   t          |j        ¦  «        ‰ gz   t          |j        ¦  «        |¦  «         }ngt          t          |j        ¦  «        d‰z
  gz   t          |j        ¦  «        t          |j        ¦  «        t          |j        ¦  «        ‰ gz   |¦  «        }|j        rA|                      ‰d¦  «                             t
          j        t
          j        ¦  «        sd}nd} ||                     ||‰|z  z  ¦  «        |¬¦  «        }|	 |||z  d¬¦  «        z  }	�Œíˆfd„}t/          |	d¬¦  «        }	|	j        r<g }|	j        D ]%\  }}t5           ||¦  «        ¦  «        }|||fgz  }Œ&t7          |ddiŽ}	nt5           ||	¦  «        ¦  «        }	t7          |	t5          |¦  «        ft9          | ‰¦  «        df¦  «        S )z0 Helper that does not attempt any substitution. z,Trying to compute the indefinite integral ofÚwrtr   )rD  rž  Nz could rewrite:rS   r§   zmeijerint-indefinitec                ó    •— ˆfd„| D ¦   «         S )Nc                ó   •— g | ]}|‰z   ‘ŒS r“   r“   )re   r‘   rÆ  s     €rh   rð   z7_meijerint_indefinite_1.<locals>.tr.<locals>.<listcomp>»  s   ø€ Ð'Ð'Ð' �A˜‘GÐ'Ð'Ð'rj   r“   )r«   rÆ  s    €rh   r  z#_meijerint_indefinite_1.<locals>.trº  s   ø€ Ø'Ð'Ð'Ð' QÐ'Ñ'Ô'Ð'rj   c              3  ó8   K  — | ]}|j         o	|d k    dk    V — ŒdS )r   TN)r´  rw  s     rh   ri   z*_meijerint_indefinite_1.<locals>.<genexpr>¼  s2   è è € ÐCÐC°QˆqŒ|Ð0  a¢¨DÒ 0ÐCÐCÐCÐCÐCÐCrj   )ÚplaceTrŸ  c                óš   •— t          t          | ¦  «        d¬¦  «        } t          j        |                      ‰¦  «        d         ¦  «        S )aÁ  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deeprS   )r   rZ   r   Ú
_from_argsÚas_coeff_add)ro   r{   s    €rh   Ú_cleanz'_meijerint_indefinite_1.<locals>._cleanÎ  s@   ø€ õ. � ™œ¨5Ð1Ñ1Ô1ˆÝŒ~˜c×.Ò.¨qÑ1Ô1°!Ô4Ñ5Ô5Ð5rj   )Úevaluater|  F)r|  rÖ   rD  rž  rX  r   rØ   rÝ   râ   r*  r½   r/  r…   rQ   r»   rƒ   r  r  Úis_extended_nonnegativer²   rn   r.  rO  r7   ra   rm   rH  r6   rT   )rg   r{   rD  rž  r   r‡   rü   Úglrˆ   ro   r  rm  rý   r‘   r¦   r6  r¿   Úfac_r§   r  r¡   rv  r{  r`  rê   rÆ  s    `                       @rh   rj  rj  š  s@  øø€ å
Ð9¸1¸eÀQÑGÔGÐGØ5Ð5Ð5Ð5Ð5Ð5Ð5Ð5å	�1�a‰Œ€BØ	€zàˆtàÑ€CˆˆR�Ý
Ð˜bÑ!Ô!Ð!Ý
Œ&€CØð %-ñ %-‰ˆˆ1ˆaÝ˜aœj¨!Ñ,Ô,‰ˆˆ1Ý˜b !Ñ$Ô$‰ˆˆ1Ø	ˆQ‰ˆð �Q‰w˜˜Q ™U™Ñ# aÑ'ˆØ�1‰u�a‰iˆõ �3Ð.µ´Ñ6Ô6ˆð	(ð 	(ð 	(ð 	(ð 	(åÐCÐC¸"¸"¸Q¼T¹(¼(ÐCÑCÔCÑCÔCð 	^ÝÝ�Q”T‘
”
�D ¤™NœN¨a°©e¨WÑ4µd¸1¼4±j´jÀSÀDÀ6Ñ6IÍ4ÐPQÔPXÉ>Ì>Ð[\ñ^ô ^ð ^ˆAˆAõ Ý�Q”T‘
”
˜a ™e˜WÑ$¥d¨1¬8¡n¤nµd¸1¼4±j´jÅ$ÀqÄxÁ.Ä.ÐUXÐTXÐSYÑBYÐ[\ñ^ô ^ˆAð Ô$ð 	¨Q¯VªV°A°q©\¬\×-=Ò-=½a¼eÅQÔEVÑ-WÔ-Wð 	ØˆEˆEàˆEØˆK˜Ÿš˜q ! A q¡D¡&Ñ)Ô)°Ð7Ñ7Ô7ˆð 	ˆyˆy˜˜a™ tÐ,Ñ,Ô,Ñ,ˆ‰ð6ð 6ð 6ð 6ð 6õ4 ˜ tÐ
,Ñ
,Ô
,€CØ
Ôð *ØˆØ”Hð 	 ð 	 ‰DˆAˆqÝ˜v˜v a™yœyÑ)Ô)ˆAØ˜˜A˜�xÑˆGˆGÝ˜Ð1¨5Ð1Ð1ˆˆå˜V˜V C™[œ[Ñ)Ô)ˆÝ�c�>¨$Ñ/Ô/Ð0µ8¸A¸q±>´>À4Ð2HÑIÔIÐIrj   c                ó   — t          d| |||f¦  «         t          | ¦  «        } |                      t          ¦  «        rt	          d¦  «         dS |                      t
          ¦  «        rt	          d¦  «         dS | |||f\  }}}}t          d¦  «        }|                      ||¦  «        } |}||k    rt          j	        dfS g }	|t          j
        u r7|t          j        ur)t          |                      || ¦  «        || | ¦  «        S |t          j
        u �r9t	          d¦  «         t          | |¦  «        }
t	          d|
¦  «         t          |
t          d¬	¦  «        t          j	        gz   D ]â}t	          d
|¦  «         |j        st	          d¦  «         Œ)t#          |                      |||z   ¦  «        |¦  «        }|€t	          d¦  «         Œbt#          |                      |||z
  ¦  «        |¦  «        }|€t	          d¦  «         Œ›|\  }}|\  }}t%          t'          ||¦  «        ¦  «        }|dk    rt	          d¦  «         ŒØ||z   }||fc S �n\|t          j        u r-t          | ||t          j        ¦  «        }|d          |d         fS ||ft          j	        t          j        fk    rHt#          | |¦  «        }|r4t)          |d         t*          ¦  «        r|	                     |¦  «         �n¿|S �n»|t          j        u r¤t          | |¦  «        D ]“}||z
  dk    dk    r„t          d|¦  «         t#          |                      |||z   ¦  «        t/          ||z   |z
  ¦  «        z  |¦  «        }|r5t)          |d         t*          ¦  «        r|	                     |¦  «         Œ�|c S Œ”|                      |||z   ¦  «        } ||z
  }d}|t          j        urut1          t          j        t5          |¦  «        z  ¦  «        }t7          |¦  «        }|                      |||z  ¦  «        } | t/          ||z
  ¦  «        |z  z  } t          j        }t	          d||¦  «         t	          d| ¦  «         t#          | |¦  «        }|r3t)          |d         t*          ¦  «        r|	                     |¦  «         n|S |                     t8          ¦  «        r«t	          d¦  «         t          t;          |¦  «        |||¦  «        }|r{t=          |t>          ¦  «        sQddl m!}  |tE          |d         ¦  «        |d          #                    t0          ¦  «        ¦  «        f|dd…         z   }|S |	 $                    |¦  «         |	rtK          tM          |	¦  «        ¦  «        S dS )aà  
    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$Integrating %s wrt %s from %s to %s.z+Integrand has DiracDelta terms - giving up.Nz5Integrand has Singularity Function terms - giving up.r{   Tz  Integrating -oo to +oo.z  Sensible splitting points:)rÉ   Úreversez  Trying to split atz  Non-real splitting point.z'  But could not compute first integral.z(  But could not compute second integral.Fz)  But combined condition is always false.r   rS   zTrying x -> x + %szChanged limits tozChanged function torg  rh  )'r€  r   rn   r@   r|  rR   r   r²   r   rØ   rP  rL  Úmeijerint_definiterø   rÐ   r   Úis_extended_realÚ_meijerint_definite_2r9  rU   rd   rQ   r�   rA   r,   r¾   r#   r$   r3   r2   ró   r»   rl  ri  r   rì   rm  rn  r   )rg   r{   r‘   r¦   rS  Úx_Úa_Úb_r)  ro  r÷   r¿   Úres1Úres2Úcond1Úcond2rˆ   ro   Úsplitrõ  rp  ri  s                         rh   r‚  r‚  ô  s¹  € õ< Ð2°Q¸¸1¸a°LÑAÔAÐAÝ�‰
Œ
€AØ‡u‚u�ZÑÔð ÝÐ<Ñ=Ô=Ð=Øˆtà‡u‚uÕ Ñ!Ô!ð ÝÐFÑGÔGÐGØˆtà˜˜1˜a�Z�N€BˆˆB�õ 	ˆc‰
Œ
€AØ	�Šˆq�!‰Œ€AØ	€AàˆA‚v€vÝ”˜ˆ~Ðà€GØ�AÔÐÐ 1­A¬JÐ#6Ð#6Ý! !§&¢&¨¨Q¨B¡-¤-°°Q°B¸¸Ñ;Ô;Ð;à	
�aÔ Ð	 Ñ	 åÐ*Ñ+Ô+Ð+Ý*¨1¨aÑ0Ô0ˆ	ÝÐ-¨yÑ9Ô9Ð9Ý˜	Õ'7ÀÐFÑFÔFÍ!Ì&ÈÑQð 	ð 	ˆAÝÐ)¨1Ñ-Ô-Ð-ØÔ%ð ÝÐ4Ñ5Ô5Ð5ØÝ(¨¯ª°°1°q±5Ñ)9Ô)9¸1Ñ=Ô=ˆDØˆ|ÝÐ@ÑAÔAÐAØÝ(¨¯ª°°1°q±5Ñ)9Ô)9¸1Ñ=Ô=ˆDØˆ|ÝÐAÑBÔBÐBØØ‰KˆD�%Ø‰KˆD�%Ý�S ¨Ñ.Ô.Ñ/Ô/ˆDØ�uŠ}ˆ}ÝÐBÑCÔCÐCØØ˜‘+ˆCØ˜�9ÐÐÐñ)	ð, 
�aŒjˆˆÝ   A q­!¬*Ñ5Ô5ˆØ�A”ˆw˜˜AœˆÐà
ˆQˆ•A”F�AœJÐ'Ò	'Ð	'å# A qÑ)Ô)ˆØð 	Ý�C˜”F�GÑ$Ô$ð Ø—’˜sÑ#Ô#Ð#Ñ#à�
ñ		ð •”
ˆ?ˆ?Ý/°°1Ñ5Ô5ð 	'ð 	'�Ø˜‘I ’N tÒ+Ð+ÝÐ0°%Ñ8Ô8Ð8Ý/°·²°q¸!¸e¹)Ñ0DÔ0DÝ1:¸1¸u¹9Àq¹=Ñ1IÔ1Iñ1JØKLñNô N�Càð 'Ý  A¤­Ñ0Ô0ð 'Ø#ŸNšN¨3Ñ/Ô/Ð/Ð/à#&˜J˜J˜Jøà�FŠF�1�a˜!‘eÑÔˆØ�‰EˆØˆØ•A”JÐÐÝ•a”o¥c¨!¡f¤fÑ,Ñ-Ô-ˆCÝ�A‘”ˆAØ—’�q˜#˜a™%Ñ Ô ˆAØ•˜1˜q™5Ñ!Ô! #Ñ%Ñ%ˆAÝ”
ˆAåÐ" A qÑ)Ô)Ð)ÝÐ$ aÑ(Ô(Ð(Ý# A qÑ)Ô)ˆØð 	Ý�C˜”F�GÑ$Ô$ð Ø—’˜sÑ#Ô#Ð#Ð#à�
Ø	‡v‚vÕ Ñ!Ô!ð 	ÝÐ;Ñ<Ô<Ð<ÝÝ'¨Ñ+Ô+¨R°°Rñ9ô 9ˆàð 	Ý˜b¥$Ñ'Ô'ð Ø:Ð:Ð:Ð:Ð:Ð:Ø�g�l¨2¨a¬5Ñ1Ô1°2°a´5·;²;½sÑ3CÔ3CÑDÔDÐFÈÈAÈBÈBÌÑO�Ø�	Ø�NŠN˜2ÑÔÐØð &Ý•G˜GÑ$Ô$Ñ%Ô%Ð%ð&ð &rj   c                óP  — | dfg}|d         d         }|h}t          |¦  «        }||vr||dfgz  }|                     |¦  «         t          |¦  «        }||vr||dfgz  }|                     |¦  «         |                     t          t
          ¦  «        r=t          t          |¦  «        ¦  «        }||vr||dfgz  }|                     |¦  «         |                     t          t          ¦  «        r2ddl	m
}  ||¦  «        }||vr||dfgz  }|                     |¦  «         |S )	z6 Try 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   rn   r;   r3   r   r8   r9   Úsympy.simplify.furŽ  )rg   r{   ro   ÚorigÚsawÚexpandedrŽ  Úreduceds           rh   Ú_guess_expansionr”    sc  € àÐ#Ð$Ð
%€CàˆrŒ7�1Œ:€DØˆ&€CÝ˜$ÑÔ€HØ�sÐÐØ�˜<Ð(Ð)Ñ)ˆØ�Š�ÑÔÐå�d‰|Œ|€HØ�sÐÐØ�˜8Ð$Ð%Ñ%ˆØ�Š�ÑÔÐà‡x‚xÕ%Õ'9Ñ:Ô:ð Ý�k¨$Ñ/Ô/Ñ0Ô0ˆØ˜3ÐÐØ�XÐ8Ð9Ð:Ñ:ˆCØ�GŠG�HÑÔÐà‡x‚x••SÑÔð Ø3Ð3Ð3Ð3Ð3Ð3Ø�- Ñ%Ô%ˆØ˜#ÐÐØ�WÐ4Ð5Ð6Ñ6ˆCØ�GŠG�GÑÔÐà€Jrj   c                óü   — t          dd| d¬¦  «        }|                      ||¦  «        } |}| dk    rt          j        dfS t	          | |¦  «        D ]+\  }}t          d|¦  «         t          ||¦  «        }|r|c S Œ,dS )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*  r²   r   rØ   r”  r|  Ú_meijerint_definite_3)rg   r{   Údummyrý   Úexplanationro   s         rh   r„  r„  Ÿ  s¤   € õ  �3Ð-¨q¸4Ð@Ñ@Ô@€EØ	�Šˆq�%ÑÔ€AØ€AàˆA‚v€vÝŒv�tˆ|Ðå*¨1¨aÑ0Ô0ð ð ‰ˆˆ;Ýˆx˜Ñ%Ô%Ð%Ý# A qÑ)Ô)ˆØð 	ØˆJˆJˆJð	ðð rj   c                ó<  ‡— t          | ‰¦  «        }|r|d         dk    r|S | j        rot          d¦  «         ˆfd„| j        D ¦   «         }t	          d„ |D ¦   «         ¦  «        r6g }t
          j        }|D ]\  }}||z  }||gz  }Œt          |Ž }|dk    r||fS dS dS dS )z²
    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.
    rS   Fz#Expanding and evaluating all terms.c                ó0   •— g | ]}t          |‰¦  «        ‘ŒS r“   )rG  )re   rý   r{   s     €rh   rð   z)_meijerint_definite_3.<locals>.<listcomp>É  s$   ø€ Ð<Ð<Ð<°Õ% a¨Ñ+Ô+Ð<Ð<Ð<rj   c              3  ó   K  — | ]}|d uV — Œ	d S rc   r“   )re   r¡   s     rh   ri   z(_meijerint_definite_3.<locals>.<genexpr>Ê  s&   è è € Ð+Ð+ ˆq˜ˆ}Ð+Ð+Ð+Ð+Ð+Ð+rj   N)rG  Úis_Addr|  rm   rl   r   rØ   rU   )rg   r{   ro   ÚressrŠ  r¡   r¿   s    `     rh   r˜  r˜  ½  së   ø€ õ    1Ñ
%Ô
%€CØ
ð ˆs�1Œv˜ŠˆØˆ
Ø„xð ÝÐ4Ñ5Ô5Ð5Ø<Ð<Ð<Ð<°Q´VÐ<Ñ<Ô<ˆÝÐ+Ð+ dÐ+Ñ+Ô+Ñ+Ô+ð 	ØˆEÝ”&ˆCØð ð ‘��1Ø�q‘�Ø˜!˜‘��Ý�U�ˆAØ�EŠzˆzØ˜A�v�ðð ð	ð 	ð ˆzrj   c                ó:   — t          t          | ¦  «        ¦  «        S rc   )rd  r&   )rg   s    rh   rH  rH  Õ  s   € Ý•j ‘m”mÑ$Ô$Ð$rj   c                ó–  — ddl m} t          d| ¦  «         |sòt          | |d¬¦  «        }|�Þ|\  }}}}t          d|||¦  «         t          j        }	|D ]g\  }
}} |
dk    rŒt          ||
z  |||z  z  | |¦  «        \  }
} |	|
t          | |¦  «        z  z  }	t          |t          | |¦  «        ¦  «        }|dk    r nŒht          |¦  «        }|dk    rt          d¦  «         n*t          d	|	¦  «         t           ||	¦  «        ¦  «        |fS t          | |¦  «        }|��)d
D �]'}|\  }}}}}t          d||||¦  «         t          j        }	|D ]Ÿ\  }}}|D ]•\  }}}t          ||z  |z  ||||z   z  z  ||||¦  «        }|€t          d¦  «            dS |\  }
}}t          d|
||¦  «         t          |t          |||¦  «        ¦  «        }|dk    r n|	|
t          |||¦  «        z  z  }	Œ–ŒŸ t          |¦  «        }|dk    rt          d|¦  «         Œòt          d|	f¦  «         |r|	|fc S t           ||	¦  «        ¦  «        |fc S dS dS )a�  
    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   rC  ÚIntegratingF)rR  Nú#Could rewrite as single G function:úBut cond is always False.z&Result before branch substitutions is:rb  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)rÖ   rD  r|  rX  r   rØ   rp  r–  rU   r‚  rH  re  r±  r  r  r€  )rg   r{   rB  rD  r   r‡   rü   rý   rˆ   ro   r  rm  rh  r¢  r£  r­  Ús1Úf1r®  Ús2Úf2r¡   Úf1_Úf2_s                           rh   rG  rG  Ù  s  € ð +Ð*Ð*Ð*Ð*Ð*å
ˆ=˜!ÑÔÐàð >Ý�q˜! uÐ-Ñ-Ô-ˆØˆ>Ø!ÑˆC��Q˜ÝÐ8¸#¸rÀ1ÑEÔEÐEÝ”&ˆCØð ð ‘��1�aØ˜’6�6ØÝ(¨¨Q©°°1°a±4±¸¸AÑ>Ô>‘��1Ø�q� 1 a™œÑ(Ñ(�Ý˜4Õ!5°a¸Ñ!;Ô!;Ñ<Ô<�Ø˜5’=�=Ø�Eð !å! $Ñ'Ô'ˆDØ�uŠ}ˆ}ÝÐ2Ñ3Ô3Ð3Ð3åÐ?ÀÑEÔEÐEÝ% k k°#Ñ&6Ô&6Ñ7Ô7¸Ð=Ð=õ 
�1�a‰Œ€BØ	�~Ø$ð 	>ñ 	>ˆGØ$&Ñ!ˆC��R˜˜TÝÐ6¸¸RÀÀRÑHÔHÐHÝ”&ˆCØ ð ð ‘
��B˜Ø"$ð ð ‘J�B˜˜BÝ'¨¨B©¨r©	°2°a¸"¸r¹'±l±?Ø(*¨B°°7ñ<ô <�Aà�yÝÐ8Ñ9Ô9Ð9Ø˜˜˜˜Ø"#‘K�A�s˜CÝÐ6¸¸3ÀÑDÔDÐDÝ˜tÕ%7¸¸SÀ!Ñ%DÔ%DÑEÔE�DØ˜u’}�}Ø˜Ø˜1�W S¨#¨qÑ1Ô1Ñ1Ñ1�C�CàØÝ! $Ñ'Ô'ˆDØ�uŠ}ˆ}ÝÐ@À'ÑJÔJÐJÐJåÐCÀcÀWÑMÔMÐMØð %Ø ˜9Ð$Ð$Ð$Ý% k k°#Ñ&6Ô&6Ñ7Ô7¸Ð=Ð=Ð=Ð=ð9 €~ð	>ð 	>rj   c           	     óì  — | }|}t          dd¬¦  «        }|                      ||¦  «        } t          d| ¦  «         t          | |¦  «        st          d¦  «         dS t          j        }| j        rt          | j        ¦  «        }nt          | t          ¦  «        r| g}nd}|�r‡g }g }|�rn|                     ¦   «         }	t          |	t          ¦  «        r…t          |	¦  «        }
|
j        r||
j        z  }ŒM	 t          |	j        d         |¦  «        \  }}n# t          $ r d}Y nw xY w|dk    r|                     |¦  «         nÓ|                     |	¦  «         n½|	j        r¡t          |	¦  «        }
|
j        r||
j        z  }ŒÙ||	j        j        vr\	 t          |	j
        |¦  «        \  }}n# t          $ r d}Y nw xY w|dk    r*|                     |t'          |	j        ¦  «        z  ¦  «         |                     |	¦  «         n|                     |	¦  «         |�°nt)          |Ž }t+          |Ž } || j        vr�t          d	| |¦  «         t-          t/          |¦  «        d¦  «        }|d
k    rt          d¦  «         dS | t1          ||z   ¦  «        z  }t          d||¦  «         t3          |                     ||¦  «        |f¦  «        S t5          | |¦  «        }|��ž|\  }}}}t          d|||¦  «         t          j        }|D ]a\  }}} t7          ||z  |||z  z  | |¦  «        \  }} ||t9          | ||¦  «        z  z  }t;          |t=          | |¦  «        ¦  «        }|d
k    r nŒbt?          |¦  «        }|d
k    rt          d¦  «         dS t          d|¦  «         ddl m!} t?           ||¦  «        ¦  «        }| "                    tF          ¦  «        s|tG          |¦  «        z  }|                     |||z   ¦  «        }t          |tH          ¦  «        s|                     |||z   ¦  «        }ddl%m&} t3          |                     ||¦  «        |f ||                     ||¦  «        ||d¦  «        df¦  «        S dS )aê  
    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§   TrŸ  zLaplace-invertingzBut expression is not analytic.Nr   rS   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:rC  )ÚInverseLaplaceTransform)'r   r²   r|  r0  r   rØ   Úis_Mulr»   rm   ró   r,   Úpopr   rÝ   rÒ   r�   rÙ   rÚ   rÌ   r.   r   r   r   r"   r@   r6   rX  r  r1  rU   r-  rH  rÖ   rD  rn   rA   rb  rK  r¬  )rg   r{   r§   rS  Út_Úshiftrm   r`  Úexponentialsr#   r^  r‘   r¦   rˆ   ro   r   r‡   rü   rý   r  rm  rD  r¬  s                          rh   Úmeijerint_inversionr²  !  só  € ð$ 
€BØ	
€BÝˆc˜ÐÑÔ€AØ	�Šˆr�1‰Œ€AÝ
Ð Ñ"Ô"Ð"Ý˜˜1ÑÔð ÝÐ0Ñ1Ô1Ð1Øˆtõ ŒF€Eà„xð Ý�A”F‰|Œ|ˆˆÝ	�A•sÑ	Ô	ð Øˆsˆˆàˆàñ "ØˆØˆØñ 	$Ø—(’(‘*”*ˆCÝ˜#�sÑ#Ô#ð $Ý˜c‘{”{�Ø”;ð Ø˜DœIÑ%�DØðÝ)¨#¬(°1¬+°qÑ9Ô9‘D�A�q�qøÝ*ð ð ð Ø�A�A�Aðøøøà˜’6�6Ø ×'Ò'¨Ñ*Ô*Ð*Ð*à—N’N 3Ñ'Ô'Ð'Ð'Ø”ð $Ý˜c‘{”{�Ø”;ð Ø˜DœIÑ%�DØØ˜CœHÔ1Ð1Ð1ðÝ-¨c¬g°qÑ9Ô9™˜˜1˜1øÝ.ð ð ð Ø˜˜˜ðøøøà˜A’v�vØ$×+Ò+¨A­c°#´(©m¬m©OÑ<Ô<Ð<Ø—’˜sÑ#Ô#Ð#Ð#à—’˜sÑ#Ô#Ð#ð; ñ 	$õ< �\Ð"ˆÝ�ˆMˆà�”ÐÐÝÐ?ÀÀEÑJÔJÐJÝ•"�U‘)”)˜QÑÔˆØ�5Š=ˆ=ÝÐHÑIÔIÐIØ�4Ø•
˜1˜u™9Ñ%Ô%Ñ%ˆÝÐBÀCÈÑNÔNÐNå˜#Ÿ(š( 1 b™/œ/¨4Ð0Ñ1Ô1Ð1å	�1�a‰Œ€BØ	�~ØÑˆˆR��DÝÐ4°c¸2¸qÑAÔAÐAÝŒfˆØð 	ð 	‰GˆAˆq�!Ý% c¨!¡e¨R°°1±©W°a¸Ñ;Ô;‰DˆAˆqØ�1•^ A q¨!Ñ,Ô,Ñ,Ñ,ˆCÝ�tÕ9¸!¸QÑ?Ô?Ñ@Ô@ˆDØ�uŠ}ˆ}Ø�ð å˜dÑ#Ô#ˆØ�5Š=ˆ=ÝÐ.Ñ/Ô/Ð/Ð/Ð/åÐ7¸Ñ=Ô=Ð=Ø2Ð2Ð2Ð2Ð2Ð2Ý   ¨SÑ!1Ô!1Ñ2Ô2ˆCØ—7’7�9Ñ%Ô%ð $Ø•y ‘|”|Ñ#�Ø—(’(˜1˜a %™iÑ(Ô(ˆCÝ˜d¥DÑ)Ô)ð /Ø—y’y  A¨¡IÑ.Ô.�Ø;Ð;Ð;Ð;Ð;Ð;Ý˜cŸhšh q¨"™oœo¨tÐ4Ø5Ð5°b·g²g¸aÀ±n´nÀaÈÈTÑRÔRÐTXÐYñ[ô [ð [ð/ €~s$   Ã5D ÄD#Ä"D#ÆF( Æ(F7Æ6F7)rg   r   r{   r   rÂ   rÃ   rº   )F)²rÓ   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.hyperbolicr0   r1   r2   r3   Ú(sympy.functions.elementary.miscellaneousr5   Ú$sympy.functions.elementary.piecewiser6   r7   Ú(sympy.functions.elementary.trigonometricr8   r9   r:   r;   Úsympy.functions.special.besselr<   r=   r>   r?   Ú'sympy.functions.special.delta_functionsr@   rA   Ú*sympy.functions.special.elliptic_integralsrB   rC   Ú'sympy.functions.special.error_functionsrD   rE   rF   rG   rH   rI   rJ   rK   rL   rM   rN   Ú'sympy.functions.special.gamma_functionsrO   Úsympy.functions.special.hyperrP   rQ   Ú-sympy.functions.special.singularity_functionsrR   Ú	integralsrT   Úsympy.logic.boolalgrU   rV   rW   rX   rY   Úsympy.polysrZ   r[   Úsympy.utilities.iterablesr\   Úsympy.utilities.miscr]   r|  r^   r€  r_   rd   rÀ   Úsympy.utilities.timeutilsrÁ   Útimeitr€   rN  rÒ   rÝ   ræ   rí   rø   rþ   r  r  r  r  r   r#  Ú__annotations__r*  r%  r0  r9  rd  ri  rp  r‚  r–  r±  r  r  r  r-  r1  rL  rV  rX  re  rk  rj  r‚  r”  r„  r˜  rH  rG  r²  r“   rj   rh   ú<module>rÚ     s5  ððð ð ð8 #Ð "Ð "Ð "Ð "Ð "Ø Ð Ð Ð à Ð Ð Ð Ð Ð Ø Ð Ð Ð Ð Ð Ð Ð Ø Ð Ð Ð Ð Ð Ø "Ð "Ð "Ð "Ð "Ð "Ø $Ð $Ð $Ð $Ð $Ð $Ø 'Ð 'Ð 'Ð 'Ð 'Ð 'Ø -Ð -Ð -Ð -Ð -Ð -ð8ð 8ð 8ð 8ð 8ð 8ð 8ð 8ð 8ð 8ð 8ð 8ð 8ð 8à Ð Ð Ð Ð Ð Ø #Ð #Ð #Ð #Ð #Ð #Ø +Ð +Ð +Ð +Ð +Ð +Ð +Ð +Ø :Ð :Ð :Ð :Ð :Ð :Ð :Ð :Ð :Ð :Ø 8Ð 8Ð 8Ð 8Ð 8Ð 8Ð 8Ð 8Ø :Ð :Ð :Ð :Ð :Ð :Ð :Ð :Ð :Ð :Ð :Ð :Ø &Ð &Ð &Ð &Ð &Ð &Ø >Ð >Ð >Ð >Ð >Ð >ðð ð ð ð ð ð ð ð ð ð ð ð ð ð ð ð ð ð ð ð ð ð ð ð ð ð GÐ FÐ FÐ FÐ FÐ FÐ FÐ FÐ FÐ FØ 7Ð 7Ð 7Ð 7Ð 7Ð 7ð9ð 9ð 9ð 9ð 9ð 9ð 9ð 9ð 9ð 9ð 9ð 9à 9Ð 9Ð 9Ð 9Ð 9Ð 9Ø JÐ JÐ JÐ JÐ JÐ JÐ JÐ Jðð ð ð ð ð ð ð ð ð ð ð à MÐ MÐ MÐ MÐ MÐ MÐ MÐ MÐ MÐ MÐ MÐ MØ IÐ IÐ IÐ IÐ IÐ IÐ IÐ IØ MÐ MÐ MÐ MÐ MÐ MÐ MÐ Mð6ð 6ð 6ð 6ð 6ð 6ð 6ð 6ð 6ð 6ð 6ð 6ð 6ð 6ð 6ð 6ð 6ð 6ð 6ð 6ð 6ð 6ð 6ð 6ð 6ð 6à 9Ð 9Ð 9Ð 9Ð 9Ð 9Ø 8Ð 8Ð 8Ð 8Ð 8Ð 8Ð 8Ð 8Ø MÐ MÐ MÐ MÐ MÐ MØ Ð Ð Ð Ð Ð Ø JÐ JÐ JÐ JÐ JÐ JÐ JÐ JÐ JÐ JÐ JÐ JÐ JÐ JØ &Ð &Ð &Ð &Ð &Ð &Ð &Ð &Ø 9Ð 9Ð 9Ð 9Ð 9Ð 9Ø 0Ð 0Ð 0Ð 0Ð 0Ð 0Ø 2Ð 2Ð 2Ð 2Ð 2Ð 2ð 
€Eˆ#�J„J€ðð ð ðJPð JPð JPðb /Ð .Ð .Ð .Ð .Ð .Ø	ˆ�)Ñ	Ô	€ð	Nð 	Nð 	Nð 	Nð	ð 	ð 	ð 	ð 	˜*ñ 	ô 	ð 	ðHð Hð HðDð ð ðBIð Ið Ið
!ð !ð !ðH$ð $ð $ðNð ð ð.Ið Ið Ið>0ð 0ð 0ð Qð Qð QðDð Dð Dð4 +-€Ð ,Ð ,Ð ,Ñ ,ð	ð 	ð 	ð#ð #ð #ðOð Oð Oðyð yð yð yðv"ð "ð "ð	ð 	ð 	ð 	ðð ð ð*mð mð mð mð`&ð &ð &ð<F.ð F.ð F.ð F.ðRjð jð jð`	2ð 2ð 2ð:	Qð 	Qð 	Qðwð wð wðtð ð ð €ð 	ØðIð Ið Iñ „ñ 	„ðIðX#ð #ð #ð #ð3ð 3ð 3ð:"&ð "&ð "&ðJWJð WJð WJðt ðG&ð G&ñ „ðG&ðTð ð ð@ð ð ð<ð ð ð0%ð %ð %ð ðD>ð D>ð D>ñ „ðD>ðNn[ð n[ð n[ð n[ð n[rj   