+
    i;                   4  a  0 t $ R t^ RIHt ^ RIt^ RIHt ^ RIHtH	t	 ^ RI
Ht ^ RIHt ^ RIHt ^ RIHt ^ R	IHt ^ R
IHtHtHtHtHt ^ RIHt ^ RIHt ^ RIHtH t  ^ RI!H"t"H#t#H$t$ ^ RI%H&t&H't' ^ RI(H)t)H*t*H+t+H,t, ^ RI-H.t. ^ RI/H0t0 ^ RI1H2t2H3t3H4t4H5t5H6t6H7t7H8t8H9t9H:t:H;t;H<t< ^ RI=H>t>H?t?H@t@ ^ RIAHBtB ^ RICHDtDHEtEHFtFHGtG ^ RIHHItI ^ RIJHKtKHLtL ^ RIMHNtNHOtOHPtPHQtQ ^ RIRHStSHTtTHUtUHVtV ^ RIWHXtXHYtY ^ RIZH[t[H\t\ ^ RI]H^t^H_t_H`t`HataHbtbHctcHdtdHeteHftfHgtgHhth ^ RIiHjtj ^ RIkHltlHmtm ^ R InHoto ^R!IpHqtq ^ R"IrHstsHtttHutuHvtvHwtw ^ R#IxHytyHztz ^ R$I{H|t| ^ R%I}H~t ^ R&I}Ht ])! R'4      tR( tR) t^ R*IHt ]! R+4      tR, R- lt ! R. R/]4      tR0 tR1 tR2 tR3 tR4 tR5 tR6 tR7 tR8 tR9 t/ tR:]R;&   R< tR= tR> tRWR? ltR@ tRXRA ltRB tRXRC ltRD tRXRE ltRF tRG tRH tRI tRJ tRs]]RWRK l4       4       tRWRL ltRM tRN tRO t]RP 4       tRQ tRR tRS tRT t]RXRU l4       tRV tR# )Ya  
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
)annotationsN)SYMPY_DEBUG)SExpr)Add)Basic)cacheit)Tuple)factor_terms)expand
expand_mulexpand_power_baseexpand_trigFunction)Mul)ilcm)Rationalpi)EqNe_canonical_coeff)default_sort_keyordered)DummysymbolsWildSymbol)sympify)	factorial)reimargAbssign
unpolarifypolarify
polar_liftprincipal_branchunbranched_argumentperiodic_argument)exp	exp_polarlog)ceiling)coshsinh_rewrite_hyperbolics_as_expHyperbolicFunctionsqrt)	Piecewisepiecewise_fold)cossinsincTrigonometricFunction)besseljbesselybesselibesselk)
DiracDelta	Heaviside)
elliptic_k
elliptic_e)erferfcerfiEiexpintSiCiShiChifresnelsfresnelc)gamma)hypermeijerg)SingularityFunction)Integral)AndOrBooleanAtomNotBooleanFunction)cancelfactor)multiset_partitions)debug)debugfzc                   a \        V 4      p \        V R R4      '       dO   \        ;QJ d)    V3R lV P                   4       F  '       d   K   R# 	  R# ! V3R lV P                   4       4      # V P                  ! S!  # )is_PiecewiseFc              3  >   <"   T F  p\        V.SO5!  x  K  	  R # 5iN)_has).0ifs   & y/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/integrals/meijerint.py	<genexpr>_has.<locals>.<genexpr>T   s     114;A;;s   T)r5   getattrallargshas)resrd   s   &jre   ra   ra   O   s]     
CsNE**s11ss1s1s111177A;    c                  a aa	a
aaaaaaa R p\        \        VR4      4      w  ooo	op\        RR .R7      oS\        S,          ,          oS\        P
                  RR3V 3R llo
RV 3R llpR	 pS	V! S	4      RR3.S R&    ! R
 R\        4      pS
! \        SS,
          4      SS,
          S	^,
          ,          ,          S	.. . ^ .SS,          \        S	4      SS	^,
          ,          ,          \        S^ 8  4      4       S
! \        SS,
          4      SS,
          S	^,
          ,          ,          . S	.^ .. SS,          \        S	4      SS	^,
          ,          ,          \        S^ 8  4      4       S
! \        \        SS,          ^S,          ,          ,
          4      SS,
          S	^,
          ,          ,          S	.. . ^ .SS,          \        S	4      SS	^,
          ,          ,          \        S^ 8  4      4       S
! \        SS,          ^S,          ,          \        ,
          4      SS,
          S	^,
          ,          ,          . S	.^ .. SS,          \        S	4      SS	^,
          ,          ,          \        S^ 8  4      4       S
! SS,           S	) ,          ^S	,
          .. ^ .. SS,          SS	) ,          \        S	4      ,          \        V! S	4      4      R7       S
! \        SS,
          4      S	) ,          ^S	,
          .^S	,
          ^,          .^ .^S	,
          ^,          .SS,          ^\        \        S	,          ^,          4      ,          \        ^S	,
          4      ,          \        S4      S	) ,          ,          \        S	4      ^8  4       S
! SS	,          SS	,          ,
          SS,
          ,          ^ S	.. ^ S	.. SS,          SS	^,
          ,          \        S	\        ,          4      ,          \        ,          4       R oVV	V
VV3R lpV! ^ ^4       V! ^ R4       V! \        P                  ^4       V! \        P                  R4       VV	V
VVV3R lpV! ^ ^4       V! ^ R4       V! \        P                  ^4       V! \        P                  R4       S
! \!        \#        R4      S,          4      . . ^ .. 4       S
! \%        S4      . ^.\        P                  .^^ .S^,          ^,          \        \'        ^^4      ,          4       S
! \)        S4      . \        P                  .^ .\        P                  \        P                  .S^,          ^,          \        \'        ^^4      ,          4       S
! \        S4      . . \        P                  .^ .S^,          ^,          \+        \        4      4       S
! \-        S4      . . ^ .\        P                  .S^,          ^,          \+        \        4      4       S
! \/        S4      . . ^ .\'        R^4      .S^,          ^,          \+        \        4      ^,          4       VV3R loVV3R loV! \1        S4      S,          \        ^S,
          4      ,          SR4       V! \1        S4      S,          \        S^,
          4      ,          SR4       VV3R lpV! \1        S4      S,          VR4       V! \1        SS	,           4      V! \1        S	4      4      \        P
                  \3        ^^.. ^.^ .SS	,          4      3.,           R4       V! \1        \        SS	,
          4      4      V! \1        \        S	4      4      4      \        \3        ^^.\        P                  .^.^ \        P                  .SS	,          4      3.,           R4       V! \5        S4      V! \        P6                  ) \        ,          4      \        P8                  \3        . ^.^ ^ .. S\#        R4      ,          4      3.,           R4       S
! \;        S4      ^.. \        P                  .^ ^ .S^,          ^,          \+        \        4      ^,          4       S
! \=        S4      . ^.^ ^ .\        P                  .S^,          ^,          \+        \        4      ) ^,          4       S
! \?        S4      \        P                  .. ^ .\'        R^4      \'        R^4      .\#        R4      S^,          ,          ^,          S\+        \        4      ,          ^,          4       S
! \A        S4      . \        P                  ^.^ ^ .\        P                  \        P                  .S^,          ^,          \        \	        R4      ,          ) ^,          4       S
! \C        S	S4      . S	.S	^,
          ^ .. S4       S
! \E        S4      ^.. \        P                  .^ .S^,          ^\+        \        4      ,          4       S
! \G        S4      . ^.^ \        P                  .. S^,          ^\+        \        4      ,          4       S
! \I        S4      \        P                  .. ^ .\'        R^4      .S^,          ) S\+        \        4      ,          4       S
! \K        S4      ^.. \'        ^^4      .^ \'        ^^4      .\        ^,          S^,          ,          ^,          \        P                  4       S
! \M        S4      ^.. \'        ^^4      .^ \'        ^^4      .\        ^,          S^,          ,          ^,          \        P                  4       S
! \O        S	S4      . . S	^,          .S	) ^,          .S^,          ^,          4       S
! \Q        S	S4      . S	^,           ) ^,          .S	^,          S	) ^,          .S	^,           ) ^,          .S^,          ^,          4       S
! \S        S	S4      . ^S	,           ^,          .S	^,          .S	) ^,          ^S	,           ^,          .S^,          ^,          \        4       S
! \U        S	S4      . . S	^,          S	) ^,          .. S^,          ^,          \        P                  4       S
! \W        S4      \        P                  \        P                  .. ^ .^ .S) \        P                  4       S
! \Y        S4      \        P                  ^\        P                  ,          .. ^ .^ .S) \'        R^4      ^,          4       R# )z7Add formulae for the function -> meijerg lookup table. c                &    \        V \        .R 7      # )exclude)r   r\   )ns   &re   wild"_create_lookup_table.<locals>.wildZ   s    As##rm   pqabcrr   c                4    V P                   ;'       d    V ^ 8  #     )
is_Integerxs   &re   <lambda>&_create_lookup_table.<locals>.<lambda>]   s    (>(>Q(>rm   )
propertiesTc	                   < S	P                  \        V \        4      . 4      P                  V V\	        WW4V4      3.Wx34       R # r`   )
setdefault_mytyper\   appendrO   )
formulaanapbmbqr!   faccondhinttables
   &&&&&&&&&re   add!_create_lookup_table.<locals>.add`   sD    !,b188''*GBBC,H&I%JD:X 	Yrm   c                j   < SP                  \        V \        4      . 4      P                  WW#34       R # r`   )r   r   r\   r   )r   instr   r   r   s   &&&&re   addi"_create_lookup_table.<locals>.addid   s,    GQ	%%+VWD,G%Hrm   c           	     ^    V \        ^.. . ^ .\        4      3V \        . ^.^ .. \        4      3.#    )rO   r\   )as   &re   constant&_create_lookup_table.<locals>.constanth   s>    GQCR!a01GBaS"a013 	3rm   c                  &    ] tR t^nt]R 4       tRtR# )2_create_lookup_table.<locals>.IsNonPositiveIntegerc                	F    \        V4      pVP                  R J d   V^ 8*  # R# )TN)r$   ry   )clsr!   s   &&re   eval7_create_lookup_table.<locals>.IsNonPositiveInteger.evalp   s%    S/C~~%ax &rm    N)__name__
__module____qualname____firstlineno__classmethodr   __static_attributes__r   rm   re   IsNonPositiveIntegerr   n   s    		  
	 rm   r   )r   c                    \         \        R^4      ,          V) V,          ^,          ^^V ,          ,
          ,          ,          # )r   )r   r   )rr#   nus   &&&re   A1 _create_lookup_table.<locals>.A1   s/    8B?"TE"HQJ!ac'#:::rm   c                  < S! \        S^,          S,           4      VS,          ,           S,          S^,          S,           V ,          ,          ^S,           ^,          ^^V ,          ,
          S^,          ,           .. SVS,          ,
          ^,          .SVS,          ,           ^,          .SS^,          ,          SS^V ,          ,
          ,          S! WS4      ,          4       R#    Nr2   )r   sgnr   r   r   bts   &&re   tmpadd$_create_lookup_table.<locals>.tmpadd   s    T!Q$(^c!e#a'AA5!eQYAaC!A#&#a%i]Oq3q5y!m_a1fAaCLAA&	(rm   c                0  < S! \        SS\        S,          ,          ,           4      V\        S4      ,          \        S^,          ,          ,          ,           S,          SS\        S,          ,          ,           V ,          ,          ^V ,
          VS,          ^,          ,           .^V ,
          VS,          ^,          ,
          .^ \        P                  .. S\        S,          ,          S,          SS^,          V ,
          ,          S! WS4      ,          4       R# r   )r3   r\   r   Half)r   r   r   r   r   r   pqs   &&re   r   r      s    T!a1f*DGA!H 44q8!a1f*qHUSU1W_AAa01aff+radF1Ha!A#'l2aa=0	2rm   c                   < V S,          p\         P                  V,          \        V4      ,          \        . ^.V^,           ,          ^ .V^,           ,          . S4      3.# r   )r   NegativeOner   rO   subsNrr   r   s   & re   	make_log1'_create_lookup_table.<locals>.make_log1   sV    G!)A,.aS!a%[1#q1u+r1=? @ 	@rm   c           	        < V S,          p\        V4      \        ^.V^,           ,          . . ^ .V^,           ,          S4      3.# r   )r   rO   r   s   & re   	make_log2'_create_lookup_table.<locals>.make_log2   sH    G1!a!eb"qc1q5k1=? @ 	@rm   c                .   < S! V 4      S! V 4      ,           # r`   r   )r   r   r   s   &re   	make_log3'_create_lookup_table.<locals>.make_log3   s    400rm   z3/2NTr   r   )-listmapr   r\   r   Oner   r?   rM   rR   rU   r"   r7   r   r   r   r*   r&   r/   r   r.   r3   r6   r8   r,   rO   rE   ImaginaryUnitr   rG   rH   rI   rJ   rF   rB   rC   rD   rK   rL   r:   r;   r<   r=   r@   rA   )r   rs   cr   r   r   r   r   r   r   r   r   r   r   rr   r   r   r   s   f       @@@@@@@@@@re   _create_lookup_tabler   X   s	   $T7+,MAq!QS>?@A	!Q$A)*Dt YI3 Xa[$-.E"I x   	!a%!a%1q5))A3BQqSaQUSQZ)	!a%!a%1q5))2sQCQqSaQUSQZ)	!qsacl"#QUa!e$44qc2rA3!aQUSQZ)	1Q3!A#,"#QUa!e$44b1#sB!aQUSQZ)Q1"AwQCQqS!qb'%(2B%a()+AE
aR1q5'QUAI;q1uai[!A#	#bd1f+eAEl"3q6QB</A<A1q1u1vrAq62qs	AE
3qt9R!;( ( 1aL
1bM
1661
16622 2 1aL
1bM
1661
1662 JrN1r2sB/ QaS166(QFAqDFBA4FGQaffXsQVVQVV$4ad1fb(1a.>PQ AB1#q!tAvtBx8ABaffXq!tAvtBx8 QR!xA/Aab!D@
@ 	Q9QU#	#Y5Q9QU#	#Y51QIt$QU	#a&	aeeWaVR!qc1Q3%GHI	I	 	SQZ(3s1v;/w1vx!q!&&k1Q3?
@	AB	 	A	1??"2%	&1=='"qcAq6SUJrN";$ +% *& 
&	 1sB1a&!Q$q&$r(1*=1rA3A!Q$q&48)A+> A"qcHROXb!_#EzRT~VWYZVZGZ[\G\	$r(
1AQVVQK!Q!&&!&&)91a46
AeHDD  q!b1#Aqz2q1 AR!&&A31aRj9QaS1aff+r1a448< Q!&&2sXb!_$51uaRjI aS"x1~.HQN0CRU1a4ZPR]TUTZTZ[aS"x1~.HQN0CRU1a4ZPR]TUTZTZ[ 1r2!ur!tfad1f5 1ra!eHQJ<!A#r!tQxzlAqDFK6 1rQUAI;11q1uai/@!Q$q&"M 1r2!aRT{B1Q? 
1'aS1#r166B
1!&&)2sQC!Xb!_Q=NOrm   )timethisrO   c               $    V ^8  d   QhRRRRRR/# )r   rd   r   r{   r   returnztuple[type[Basic], ...]r   )formats   "re   __annotate__r   -  s'     	N 	Nu 	N 	N$; 	Nrm   c                   a R R lpSV P                   9  d   R# V P                  '       d   \        V 4      3# \        \	        V3R lV P
                   4       VR7      4      # )z3Create a hashable entity describing the type of f. c                    V ^8  d   QhRRRR/# )r   r{   ztype[Basic]r   ztuple[int, int, str]r   )r   s   "re   r   _mytype.<locals>.__annotate__/  s      { 3 rm   c                "    V P                  4       # r`   )	class_keyrz   s   &re   key_mytype.<locals>.key/  s    {{}rm   c              3  L   <"   T F  p\        VS4       F  q"x  K  	  K  	  R # 5ir`   )r   )rb   r   r   r{   s   &  re   rf   _mytype.<locals>.<genexpr>6  s     BFqGAqMqMFs   !$r   r   )free_symbolsis_Functiontypetuplesortedrj   )rd   r{   r   s   &f re   r   r   -  sK     			
AwxBAFFBLMMrm   c                      ] tR tRtRtRtR# )_CoeffExpValueErrori9  z<
Exception raised by _get_coeff_exp, for internal use only.
r   N)r   r   r   r   __doc__r   r   rm   re   r   r   9  s     	rm   r   c                X   ^ RI Hp \        V! V 4      4      P                  V4      w  r4V'       g   V\        P
                  3# Vw  pVP                  '       d*   VP                  V8w  d   \        R4      hW4P                  3# WA8X  d   V\        P                  3# \        RV ,          4      h)aP  
When expr is known to be of the form c*x**b, with c and/or b possibly 1,
return c, b.

Examples
========

>>> from sympy.abc import x, a, b
>>> from sympy.integrals.meijerint import _get_coeff_exp
>>> _get_coeff_exp(a*x**b, x)
(a, b)
>>> _get_coeff_exp(x, x)
(1, 1)
>>> _get_coeff_exp(2*x, x)
(2, 1)
>>> _get_coeff_exp(x**3, x)
(1, 3)
)powsimpzexpr not of form a*x**bzexpr not of form a*x**b: %s)sympy.simplifyr   r   as_coeff_mulr   Zerois_Powbaser   r*   r   )exprr{   r   r   ms   &&   re   _get_coeff_expr   @  s    & 'wt}-::1=FQ!&&y
CQxxx66Q;%&?@@%%x	
!%%x!"?$"FGGrm   c                :   a V3R lo\        4       pS! WV4       V# )aS  
Find the exponents of ``x`` (not including zero) in ``expr``.

Examples
========

>>> from sympy.integrals.meijerint import _exponents
>>> from sympy.abc import x, y
>>> from sympy import sin
>>> _exponents(x, x)
{1}
>>> _exponents(x**2, x)
{2}
>>> _exponents(x**2 + x, x)
{1, 2}
>>> _exponents(x**3*sin(x + x**y) + 1/x, x)
{-1, 1, 3, y}
c                   < W8X  d   VP                  ^.4       R# V P                  '       d0   V P                  V8X  d   VP                  V P                  .4       R# V P                   F  pS! W1V4       K  	  R# )r   N)updater   r   r*   rj   )r   r{   rl   argument_exponents_s   &&& re   r   _exponents.<locals>._exponents_u  sX    9JJsO;;;499>JJz"		HS) "rm   )set)r   r{   rl   r   s   && @re   
_exponentsr   b  s     &* %CJrm   c                    V P                  \        4       Uu0 uF!  q!VP                  9   g   K  VP                  kK#  	  up# u upi )zAFind the types of functions in expr, to estimate the complexity. )atomsr   r   func)r   r{   es   && re   
_functionsr     s4     JJx0H0q4GFAFF0HHHs
   AAc                   aaaa R Uu. uF  p\        VS.R7      NK  	  upw  ooVVVV3R lo\        4       pS! W4       V# u upi )a8  
Find numbers a such that a linear substitution x -> x + a would
(hopefully) simplify expr.

Examples
========

>>> from sympy.integrals.meijerint import _find_splitting_points as fsp
>>> from sympy import sin
>>> from sympy.abc import x
>>> fsp(x, x)
{0}
>>> fsp((x-1)**3, x)
{1}
>>> fsp(sin(x+3)*x, x)
{-3, 0}
pqrp   c                R  < \        V \        4      '       g   R # V P                  SS,          S,           4      pV'       d8   VS,          ^ 8w  d*   VP                  VS,          ) VS,          ,          4       R # V P                  '       d   R # V P
                   F  pS! W14       K  	  R # r`   )
isinstancer   matchr   is_Atomrj   )r   rl   r   r   compute_innermostr   r   r{   s   &&  re   r   1_find_splitting_points.<locals>.compute_innermost  su    $%%JJqsQw1GGQqTE!A$J<<<		Hh, "rm   )r   r   )r   r{   rr   	innermostr   r   r   s   &f  @@@re   _find_splitting_pointsr    sL    $ +//$QDQC $/DAq
- 
- Id& 0s   Ac           	        \         P                  p\         P                  p\         P                  p\        V 4      p \        P                  ! V 4      pV F  pWa8X  d   W1,          pK  WP
                  9  d   W&,          pK-  VP                  '       d   WP                  P
                  9  d   VP                  P                  V4      w  rxW38w  d'   \        VP                  4      P                  V4      w  rxW38X  dI   W1VP                  ,          ,          pV\        \        WvP                  ,          RR7      4      ,          pK  WF,          pK  	  W#V3# )aI  
Split expression ``f`` into fac, po, g, where fac is a constant factor,
po = x**s for some s independent of s, and g is "the rest".

Examples
========

>>> from sympy.integrals.meijerint import _split_mul
>>> from sympy import sin
>>> from sympy.abc import s, x
>>> _split_mul((3*x)**s*sin(x**2)*x, x)
(3**s, x*x**s, sin(x**2))
F)r   )r   r   r   r   	make_argsr   r   r*   r   r   r   r$   r%   )	rd   r{   r   pogrj   r   r   r   s	   &&       re   
_split_mulr    s     %%C	
B	A!A==D6GBnn$HCxxxAUU%7%77vv**1-9%aff-::1=DA9QUU(NB:hq%%xe&DEECFA   A:rm   c                @   \         P                  ! V 4      p. pV F  pVP                  '       dZ   VP                  P                  '       d>   VP                  pVP
                  pV^ 8  d   V) p^V,          pW%.V,          ,          pKn  VP                  V4       K  	  V# )z
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   r  r   r*   ry   r   r   )rd   rj   gsr  rr   r   s   &     re   	_mul_argsr    s     ==D	B888(((A66D1uBv&(NBIIaL  Irm   c           	         \        V 4      p\        V4      ^8  d   R# \        V4      ^8X  d   \        V4      .# \        V^4       UUu. uF  w  r#\	        V!  \	        V!  3NK  	  upp# u uppi )a_  
Find all the ways to split ``f`` into a product of two terms.
Return None on failure.

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

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

Examples
========

>>> from sympy.integrals.meijerint import _mul_as_two_parts
>>> from sympy import sin, exp, ordered
>>> from sympy.abc import x
>>> list(ordered(_mul_as_two_parts(x*sin(x)*exp(x))))
[(x, exp(x)*sin(x)), (x*exp(x), sin(x)), (x*sin(x), exp(x))]
N)r  lenr   rY   r   )rd   r
  r{   ys   &   re   _mul_as_two_partsr    sb    . 
1B
2w{
2w!|b	{-@Q-GH-G6AS!Wc1g-GHHHs   	A*c                "   R p\        \        V P                  4      \        V P                  4      ,
          4      pV^V P                  ,           V^,          ,           ,          pV^\
        ,          V^,
          V P                  ,          ,          ,          pV\        V! V P                  V4      V! V P                  V4      V! V P                  V4      V! V P                  V4      V P                  V,          WV,          ,          ,          4      3# )zFReturn C, h such that h is a G function of argument z**n and
g = C*h. c                    \         P                  ! V \        V4      4       UUu. uF  w  r#W#,           V,          NK  	  upp# u uppi )z4(a1, .., ak) -> (a1/n, (a1+1)/n, ..., (ak + n-1)/n) )	itertoolsproductrange)paramsrr   r   rc   s   &&  re   inflate_inflate_g.<locals>.inflate  s7    &/&7&7a&IJ&Ida		&IJJJs   A)r   r  r   r   r   r   deltarO   r   aotherr   botherr   )r  rr   r  vCs   &&   re   
_inflate_gr  	  s    
K 	
#add)c!$$i
 A	AHqsNA!B$1q5!''/	""AggaddA&!(<addA&!(<jj!maA#h.0 0 0rm   c                    R p\        V! V P                  4      V! V P                  4      V! V P                  4      V! V P                  4      ^V P
                  ,          4      # )zHTurn the G function into one of inverse argument
(i.e. G(1/x) -> G'(x)) c                >    V  Uu. uF  p^V,
          NK  	  up# u upi r   r   )lr   s   & re   tr_flip_g.<locals>.tr  s      !q!Aq!!!   )rO   r   r  r   r  r   )r  r!  s   & re   _flip_gr$    sB    "2add8R\2add8R\1QZZ<PPrm   c           	     L   V^ 8  d   \        \        V 4      V) 4      # \        VP                  4      p\        VP                  4      p\        W4      w  r@V P                  pV^\        ,          ^V,
          ^,          ,          V\        R^4      ,          ,          ,          pWRV,          ,          p\        V4       Uu. uF  qf^,           V,          NK  	  ppV\        V P                  V P                  V P                  \        V P                  4      V,           V4      3# u upi )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   )_inflate_fox_hr$  r   r   r   r  r   r   r   r  rO   r   r  r   r   r  )r  r   r   r   Dr\   rr   bss   &&      re   r&  r&  "  s     	1ugaj1"--	!##A	!##A aDA	

A!B$1q5!)	QQ/	//AAIA"1X	&Xq5!))XB	&gaddAHHaddDNR,?CCC 
's   >D!zdict[tuple[str, str], Dummy]_dummiesc                V    \        W3/ VB pWBP                  9   d   \        V 3/ VB # V# )z
Return a dummy. This will return the same dummy if the same token+name is
requested more than once, and it is not already in expr.
This is for being cache-friendly.
)_dummy_r   r   )nametokenr   kwargsds   &&&, re   _dummyr0  ?  s4     	&v&AT$V$$Hrm   c                ^    W3\         9   g   \        V 3/ VB \         W3&   \         W3,          # )zT
Return a dummy associated to name and token. Same effect as declaring
it globally.
)r)  r   )r,  r-  r.  s   &&,re   r+  r+  K  s1    
 =H$"'"7"7$TM""rm   c                   a \         ;QJ dA    V3R lV P                  \        \        4       4       F  '       g   K   R'       * # 	  R'       * # ! V3R lV P                  \        \        4       4       4      '       * # )zCheck if f(x), when expressed using G functions on the positive reals,
will in fact agree with the G functions almost everywhere c              3  B   <"   T F  pSVP                   9   x  K  	  R # 5ir`   )r   rb   r   r{   s   & re   rf   _is_analytic.<locals>.<genexpr>X  s     N6Md1)))6M   TF)anyr   r?   r"   )rd   r{   s   &fre   _is_analyticr8  U  sM     sNaggi6MNssNNsNNsNaggi6MNNNNrm   c                  aa V'       d   V P                  R \        4      p Rp\        V \        4      '       g   V # \	        R\
        R7      w  oop\        SS8  \        SS4      4      SS8*  3\        \        \        S4      4      \        8*  \        \        S4      ^\        ,          ,
          4      \        8*  4      \        \        S4      \        ,
          ^ 4      3\        \        ^\        S4      ,          \        ,           4      \        8*  \        ^\        S4      ,          \        ,
          4      \        8*  4      \        \        S4      ^ 4      3\        \        ^\        S4      ,          \        ,           4      \        8  \        ^\        S4      ,          \        ,
          4      \        8*  4      \        P                  3\        \        \        S4      \        ^,          ,
          4      \        ^,          8*  \        \        S4      \        ^,          ,           4      \        ^,          8*  4      \        \        S4      ^ 4      3\        \        \        S4      \        ^,          ,
          4      \        ^,          8*  \        \        S4      \        ^,          ,           4      \        ^,          8  4      \        P                  3\        \        \        S^,          ^,          ^,           4      4      \        8  \        \        \        S^,          ^,          ^,           4      4      \        4      4      \        P                  3\        \        \        S^,          ^,          ^,           4      4      \        8  \        ^S^,          ^,          ^,           ,          ^ 4      4      \        P                  3\        \        \!        S4      4      \        8*  \        \!        \#        R\        ,          \        P$                  ,          4      S,          4      4      \        8*  4      \        \!        \#        \        P$                  ) \        ,          4      S,          4      ^ 4      3\        \        \!        S4      4      \        ^,          8*  \        \!        \#        \        ) \        P$                  ,          4      S,          4      4      \        ^,          8*  4      \        \!        \#        \        P$                  ) \        ,          ^,          4      S,          4      ^ 4      3\        SS8*  \        SS8  V4      4      SS8*  3\        S^,          ^4      S^,          ^8  ,          S^,          ^8  3\        ^S,          ^4      \'        \        \        S4      4      4      \        S4      ,          ^8  ,          \        S4      ^8  3\        S^4      \'        \        \        S4      4      4      \        S4      ,          ^8  ,          \        S4      ^8  3\        \        S4      4      \        ^,          8  \'        \        \        S4      4      4      \)        \        S^,          4      4      ,          ^8  ,          S^,          ^8  3.pV P*                  ! V P,                   Uu. uF  p\/        WA4      NK  	  up!  p RpV'       Ed   Rp\1        V4       EF  w  pw  rxVP*                  V P*                  8w  d   K&  \1        V P,                  4       EF  w  rW'P,                  ^ ,          P2                  9   d&   V
P5                  VP,                  ^,          4      p^pM$^ pV
P5                  VP,                  ^ ,          4      pV'       g   Kz  VP,                  RV VP,                  V^,           R ,            Uu. uF  qP7                  V4      NK  	  ppV	.pV EF/  p\1        V P,                  4       EF  w  ppVV9   d   K  VV8X  d   VV.,          p K=  \        V\        4      '       d^   VP,                  ^,          V8X  dF   \        V\        4      '       d0   VP,                  ^ ,          VP,                  9   d   VV.,          p K  \        V\        4      '       g   K  VP,                  ^ ,          V8X  g   K  \        V\        4      '       g   K  VP,                  ^,          VP,                  9   g   EK  VV.,          p EK-  	  EK2  	  \9        V4      \9        V4      ^,           8w  d   EK  \1        V P,                  4       UUu. uF  w  ppVV9  g   K  VNK  	  uppVP7                  V4      .,           p\:        '       d   VR9  d   \=        RV4       V P*                  ! V!  p Rp EK  	  EK  	  EK  VV3R lpV P                  R	 V4      p \:        '       d   \=        R
V 4       V # u upi u upi u uppi )az  
Do naive simplifications on ``cond``.

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

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

Examples
========

>>> from sympy.integrals.meijerint import _condsimp as simp
>>> from sympy import Or, Eq
>>> from sympy.abc import x, y
>>> simp(Or(x < y, Eq(x, y)))
x <= y
c                    V P                   # r`   is_Relational_s   &re   r|   _condsimp.<locals>.<lambda>o  s    aoorm   Fzp q r)r   TNzused new rule:c                  < V P                   R 8w  g   V P                  ^ 8w  d   V # V P                  pVP                  \	        S4      S,          4      pV'       g+   VP                  \        \        S4      S,          4      4      pV'       gw   \        V\        4      '       d_   VP                  ^ ,          P                  '       g<   VP                  ^,          \        P                  J d   VP                  ^ ,          ^ 8  # V # VS,          ^ 8  # )z==)rel_oprhslhsr   r!   r(   r&   r   r)   rj   is_polarr   Infinity)relLHSr   r   r   s   &  re   rel_touchup_condsimp.<locals>.rel_touchup  s    ::AJ ggIIc!fai 		-jmQ.>?@A#011#((1+:N:N:Nqzz1a(J!qrm   c                    V P                   # r`   r;  r=  s   &re   r|   r?    s    !//rm   z_condsimp: )
rx   r                           )replacer   r   rV   r   r   rS   r   rR   r"   r!   r   r   falser   truer(   r+   r   r6   r3   r   rj   	_condsimp	enumerater   r   r   r  r   print)r   firstr   rulesr>  changeirulefrotorr   arg1r   numr{   	otherargs	otherlistarg2karg3arg_newargsrH  r   r   s   &&                    @@re   rW  rW  [  s   & ||57GHdO,,g4(GAq!
 
AE2a8	a1f% 
SQ[BCFQrTM 2b 8	9	CFRK			S3q6B2%s1SV8b='9R'?	@	CFA		S3q6B"$c!CF(R-&8B&>	?	
		SQ"Q$2a4'SVbd]);r!t)C	D	CFA		SQ"Q$2a4'SVbd]);bd)B	C	
		SQT!VaZ!B&3s1a46A:+?(D	E	
		CAqDFQJ 2%r!QT!VaZ.!'<	=	
		S$Q'(B."9RU1??-B#CA#EFG2M
O		1??*:2*= >q @A1	E	G 
S$Q'(BqD0"9bS-@#A!#CDEAM
O		1??*:2*=a*? @ BCQ	G	I 
AFCAqM	"AF+	AqD!1q	!1a4!8,	AaCs3s1v;'A.2	3SVaZ@	AqSSV%c!f,q0	13q6A:>
c!f+1
SQ[!1$s1a4y/!AA!E	F1qQ9E< 99DII>Iqy*I>?DF
& )% 0E9Cxx499$$TYY/000

388A;/ACC

388A;/A03##PQ'(AS0ST0S1VVAY0S	TC	%D#,TYY#74	>$4<%!,I!%dC00TYYq\Q5F *4 5 5$))A,$)):S%!,I!%dC00TYYq\Q5F *4 5 5$))A,$)):S%!,I! $8 & y>S^a%771:4991E 21EIQy0  41E 257WWQZLA;$FF.6yy'*G 0 !1R <<1;?D{mT"K ?  U&2s   g!-g&<g+
g+
c                d    \        V \        4      '       d   V # \        V P                  4       4      # )zRe-evaluate the conditions. )r   boolrW  doit)r   s   &re   
_eval_condrl    s%    $TYY[!!rm   c                Z    \        W4      pV'       g   VP                  \         R 4      pV# )zBring 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                    V # r`   r   )r{   r  s   &&re   r|   &_my_principal_branch.<locals>.<lambda>  s    rm   )r'   rT  )r   periodfull_pbrl   s   &&& re   _my_principal_branchrr    s'     4
(Ckk*N;Jrm   c           	       a	a
 \        W4      w  po
\        VP                  V4      w  po	VP                  4       p\        WV4      pV \	        S	4      VS
^,           S	,          ^,
          ,          ,          ,          pV	V
3R lpV\        V! VP                  4      V! VP                  4      V! VP                  4      V! VP                  4      WS,          4      3# )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.
c                h   < V  Uu. uF   q^S,           S,          ,           ^,
          NK"  	  up# u upi r   r   )r   r   r   ss   & re   r!  _rewrite_saxena_1.<locals>.tr  s-    +,-1aQUAI!!1---s   &/)
r   r   
get_periodrr  r"   rO   r   r  r   r  )r   r  r  r{   r>  r   rp  r  r!  r   ru  s   &&&&     @@re   _rewrite_saxena_1rx    s     " DAq!**a(DAq\\^FQ'A 	SVAQ	A&&'A.gbh188bh188c  rm   c                8   V P                   p\        V P                  V4      w  rE\        \	        V P
                  4      \	        V P                  4      \	        V P                  4      \	        V P                  4      .4      w  rgrW8  dc   R p
\        \        V
! V P
                  4      V
! V P                  4      V
! V P                  4      V
! V P                  4      W,          4      V4      # V P
                   Uu. uF  p\        V4      ) ^8  NK  	  upV P                   Uu. uF  p^^\        V4      ,
          8  NK  	  up,           p\        V!  pYP                   Uu. uF  p\        V4      ) ^8  NK  	  up,          pYP                   Uu. uF  p^^\        V4      ,
          8  NK  	  up,          p\        V!  p\        V P                  4      ) V	^,           V,
          ^,          ,           W,
          8  pR pR pV! R4       V! RW4WgW34       V! R\!        V P                  4      \!        V P                  4      34       V! R\!        V P
                  4      \!        V P                  4      34       V! RWV34       . p. p^V8*  W8  ^V8*  .p^V8*  ^V8*  \#        W^,           4      \%        \        \#        V^ 4      \#        Wh^,           4      4      4      .p^V8*  \#        W4      .p\'        \)        V^,          4      ^,           4       FB  pV\+        \-        \/        V4      4      V^V,          ,
          \0        ,          4      .,          pKD  	  V^ 8  \-        \/        V4      4      V\0        ,          8  .p\+        V^ 4      V.pV'       d   . pVVV3 F"  pV\        VV,           V,           !  .,          pK$  	  VV,          pV! R	V4       V.pV'       d   . p\        \#        V^ 4      V^,           V8*  Wi8*  \-        \/        V4      4      V\0        ,          8  .VO5!  .pVV,          pV! R
V4       VV.pV'       d   . p\        W8  ^V8*  V^ 8  \#        \-        \/        V4      4      V\0        ,          4      .VO5!  .pV\        W^,
          8*  \#        V^ 4      \#        \-        \/        V4      4      ^ 4      .VO5!  .,          pVV,          pV! RV4       . pV\#        W4      \#        V^ 4      \#        \/        V4      ^ 4      \+        V^ 4      .,          pV'       g   VV.,          p. p\3        V P                  V P                  4       F  w  rVW,
          .,          pK  	  V\        \5        V!  4      ^ 8  .,          p\        V!  pVV.,          pV! RV.4       \        V^ 8  \-        \/        V4      4      V\0        ,          8  4      .pV'       g   VV.,          p\        V!  pVV.,          pV! RV.4       \7        V!  # u upi u upi u upi u upi )a:  
Return a condition under which the mellin transform of g exists.
Any power of x has already been absorbed into the G function,
so this is just $\int_0^\infty g\, dx$.

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

If ``helper`` is True, only check if the MT exists at infinity, i.e. if
$\int_1^\infty g\, dx$ exists.
c                >    V  Uu. uF  p^V,
          NK  	  up# u upi r   r   r   r{   s   & re   r!   _check_antecedents_1.<locals>.tr  s     #$%1aAEE1%%%r#  c                     \        V !   R # r`   )_debug)msgs   *re   rZ   #_check_antecedents_1.<locals>.debug"  s	    rm   c                    \        W4       R # r`   _debugf)stringr!   s   &&re   r[   $_check_antecedents_1.<locals>.debugf%  s
    rm   z$Checking antecedents for 1 function:z*  delta=%s, eta=%s, m=%s, n=%s, p=%s, q=%sz  ap = %s, %sz  bq = %s, %sz"  cond_3=%s, cond_3*=%s, cond_4=%sz	  case 1:z	  case 2:z	  case 3:z  extra case:z  second extra case:)r  r   r   r   r  r   r   r   r   _check_antecedents_1rO   r  r  r   rR   r   r   r   rU   r  r-   r   r"   r(   r   zipr   rS   ) r  r{   helperr  etar>  r   rr   r   r   r!  r   r   tmpcond_3cond_3_starcond_4rZ   r[   condscase1tmp1tmp2tmp3re  extrar   case2case3
case_extraru  case_extra_2s    &&&                             re   r  r    sG    GGEAJJ*FCCIs144y#add)SY?@JA!u	&#GBqttHbl,.qttHblAE%K$%' 	'  !tt
$t!BqE6A:t
$qtt'Dt!A1It'D
DC#YF)1RUFQJ))C88,8aABqE	M8,,Cs)K!$$xi1q519a-'!%/F 

01
7a#%
?T!$$Zahh89
?T!$$Zahh89
/&v1NOE EFAE16"DFAFBqa%L#c"Q(Bqa%L.I*JKDFBqHD757#a'(C+C01EAaCK3CDEE )19c-c23eBh>
?CQZ ED$#C%)**  	UNE	+u HEAq1q5A:qv(-.r9C<AC DE	UNE	+u VEQ	2c2Ec2J.KUSUX+V  E	c!1u*blBs3Fs3K/La,PYSXYZZE	UNE	+u J2a8Rq\2.A#.F+JBsTUJWWJvh

AADD!$$	aeW  2c1g;?##Jj!J	j\E	/J<(	3':3'?#@58#KLML %L	l^E	
 <.1
 u:m %'D *,s   9X!XXXc                   ^ RI Hp \        V P                  V4      w  r4^V,          pV P                   F  pV\        V^,           4      ,          pK  	  V P                   F#  pV\        ^V,
          ^,
          4      ,          pK%  	  V P                   F#  pV\        ^V,
          ^,
          4      ,          pK%  	  V P                   F  pV\        V^,           4      ,          pK  	  V! \        V4      4      # )a[  
Evaluate $\int_0^\infty g\, dx$ using G functions,
assuming the necessary conditions are fulfilled.

Examples
========

>>> from sympy.abc import a, b, c, d, x, y
>>> from sympy import meijerg
>>> from sympy.integrals.meijerint import _int0oo_1
>>> _int0oo_1(meijerg([a], [b], [c], [d], x*y), x)
gamma(-a)*gamma(c + 1)/(y*gamma(-d)*gamma(b + 1))
)	gammasimp)
r   r  r   r   r   rM   r   r  r  r$   )r  r{   r  r  r>  rl   r   r   s   &&      re   	_int0oo_1r  r  s     )AJJ*FC
C%CTTuQU| TTuQUQY XXuQUQY XXuQU| Z_%%rm   c                  aaa VV3R lp\        VS4      w  rx\        VP                  S4      w  ry\        VP                  S4      w  rzV	^ 8  R8X  d   V	) p	\        V4      pV
^ 8  R8X  d   V
) p
\        V4      pV	P                  '       d   V
P                  '       g   R# T	P                  V	P
                  rV
P                  V
P
                  r\        W,          W,          4      pWV,          ,          pWV,          ,          p\        VV4      w  pp\        VV4      w  ppV! V4      pV! V4      pV VV,          ,          p \        VP                  S4      w  pp\        VP                  S4      w  ppV^,           V,          ^,
          oV \        V4      VS,          ,          ,          p V3R lp\        V! VP                  4      V! VP                  4      V! VP                  4      V! VP                  4      VS,          4      p\        VP                  VP                  VP                  VP                  VS,          4      p^ RIHp V! V RR7      W#3# )a  
Rewrite the integral ``fac*po*g1*g2`` from 0 to oo in terms of G
functions with argument ``c*x``.

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

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

Examples
========

>>> from sympy.integrals.meijerint import _rewrite_saxena
>>> from sympy.abc import s, t, m
>>> from sympy import meijerg
>>> g1 = meijerg([], [], [0], [], s*t)
>>> g2 = meijerg([], [], [m/2], [-m/2], t**2/4)
>>> r = _rewrite_saxena(1, t**0, g1, g2, t)
>>> r[0]
s/(4*sqrt(pi))
>>> r[1]
meijerg(((), ()), ((-1/2, 0), ()), s**2*t/4)
>>> r[2]
meijerg(((), ()), ((m/2,), (-m/2,)), t/4)
c                   < \        V P                  S4      w  rV P                  4       p\        V P                  V P
                  V P                  V P                  \        WS4      SV,          ,          4      # r`   )	r   r   rw  rO   r   r  r   r  rr  )r  r   r   perrq  r{   s   &   re   pb_rewrite_saxena.<locals>.pb  sZ    ajj!,llnqttQXXqttQXX+AG<QTAC 	Crm   TNc                >   < V  Uu. uF  qS,           NK  	  up# u upi r`   r   )r   r   r*   s   & re   r!  _rewrite_saxena.<locals>.tr  s    !"#AC###   	powdenestpolar)r   r   r$  is_Rationalr   r   r   r  r"   rO   r   r  r   r  r   r  )r   r  g1g2r{   rq  r  r>  ru  b1b2m1n1m2n2taur1r2C1C2a1r   a2r!  r  r*   s   &&&&ff                   @re   _rewrite_saxenar    s   6C "a DA2;;*EA2;;*EA
Q4SR[
Q4SR[>>>TT244TT244
rube
C	"uB	"uBBFBBFB	BB	BB2b5LC2;;*EB2;;*EB q5!)a-C
s1vC
 C$	BEEBryyM2bee9bmRT	JB			255"))RT	:B(S%r--rm   c                5  a aa.a/a0a1a2a3a4a5a6 \        S P                  V4      w  o3p\        SP                  V4      w  o/p\        \        S P                  4      \        S P
                  4      \        S P                  4      \        S P                  4      .4      w  rEo5o6\        \        SP                  4      \        SP
                  4      \        SP                  4      \        SP                  4      .4      w  rgo0o2WE,           S5S6,           ^,          ,
          pWg,           S0S2,           ^,          ,
          p	S P                  S5S6,
          ^,          ,           ^,           p
SP                  S0S2,
          ^,          ,           ^,           pS2S0,
          S6S5,
          ,
          p^S6S5,
          ,
          V,
          V
,
          p\        S2V,
          V,
          ,          \        \        S/4      4      ,           S2S0,
          ,          o1\        S6V,
          V,
          ,          \        \        S34      4      ,           S6S5,
          ,          o4\        R4       \        RS3WES5S6W34       \        RS/WgS0S2W34       \        RWS1S434       V V3R lpV! 4       p\        S P                   UUu. uF2  pSP                   F  p\        ^V,           V,           4      ^ 8  NK!  	  K4  	  upp!  p\        S P
                   UUu. uF2  pSP
                   F  p\        ^V,           V,           4      ^8  NK!  	  K4  	  upp!  p\        S P
                   Uu. uFG  pS0S2,
          \        ^V,           ^,
          4      ,          \        V4      ,
          \!        R^4      8  NKI  	  up!  p\        S P                   Uu. uF@  pS0S2,
          \        ^V,           4      ,          \        V4      ,
          \!        R^4      8  NKB  	  up!  p\        SP
                   Uu. uFG  pS5S6,
          \        ^V,           ^,
          4      ,          \        V
4      ,
          \!        R^4      8  NKI  	  up!  p\        SP                   Uu. uF@  pS5S6,
          \        ^V,           4      ,          \        V
4      ,
          \!        R^4      8  NKB  	  up!  p\        V4      ^\        V
^,
          S2S0,
          ,          S6S5,
          S2S0,
          ,          ,           V^,
          S6S5,
          ,          ,           4      ,          ,           ^ 8  p\        V4      ^\        V
^,
          S2S0,
          ,          S6S5,
          S2S0,
          ,          ,           V^,
          S6S5,
          ,          ,           4      ,          ,
          ^ 8  p\        \        S34      4      V\        ,          8  p\#        \        \        S34      4      V\        ,          4      p\        \        S/4      4      V	\        ,          8  p\#        \        \        S/4      4      V	\        ,          4      p\%        W,           ) \        ,          \        P&                  ,          4      p\)        VS/,          S3,          4      p\)        VS3,          S/,          4      p V^V ,          8X  dx   \        \#        V^ 4      W,           ^8*  \+        \-        V^4      \        W,           S6,           S5,
          4      ^8  \        W,           S2,           S0,
          4      ^8  4      4      p!EMR p"\        \#        V^ 4      V^,
          V	,           ^ 8*  \+        \        \-        V^4      V"! V4      4      \        \        W,           S6,           S5,
          4      ^8  \#        V^4      4      4      4      p!\        \#        V^ 4      V	^,
          V,           ^ 8*  \+        \        \-        V ^4      V"! V 4      4      \        \        W,           S2,           S0,
          4      ^8  \#        V ^4      4      4      4      p#\+        V!V#4      p!  S2S0,
          \        S/4      ^S2S0,
          ,          ,          ,          \/        S14      ,          S6S5,
          \        S34      ^S6S5,
          ,          ,          ,          \/        S44      ,          ,           p$\1        V$^ 8  4      R8w  d   V$^ 8  p%EMV/V0V1V2V3V4V5V63R lp&\3        V&! ^^4      V&! RR4      ,          \        \#        \        S34      ^ 4      \#        \        S/4      ^ 4      4      3V&! \5        \        S/4      4      ^4      V&! \5        \        S/4      4      R4      ,          \        \#        \        S34      ^ 4      \-        \        S/4      ^ 4      4      3V&! ^\5        \        S34      4      4      V&! R\5        \        S34      4      4      ,          \        \-        \        S34      ^ 4      \#        \        S/4      ^ 4      4      3V&! \5        \        S/4      4      \5        \        S34      4      4      R	34      p'V$^ 8  \        \#        V$^ 4      \-        V'^ 4      \        V4      R8  4      \        \#        V$^ 4      \#        V'^ 4      \        V4      ^ 8  4      .p(\+        V(!  p% V^3V^3V^3V^3V^3V^3V^3V^3V^	3V^
3V^3V^3V^3V!^3V%^33 F  w  p)p\        R
VV)34       K  	  . o.V.3R lp*S.\        Wg,          V,          V,          ^ 8g  VP8                  R	J V	P8                  R	J VVVVV4      .,          o.V*! ^4       S.\        \#        S5S64      \#        V^ 4      V	P8                  R	J S3P8                  R	J \        V
4      ^8  VVVV4	      .,          o.V*! ^4       S.\        \#        S0S24      \#        V	^ 4      VP8                  R	J S/P8                  R	J \        V4      ^8  VVVV4	      .,          o.V*! ^4       S.\        \#        S0S24      \#        S5S64      \#        V^ 4      \#        V	^ 4      S3P8                  R	J S/P8                  R	J \        V4      ^8  \        V
4      ^8  \-        S3S/4      VVV4      .,          o.V*! ^4       S.\        \#        S0S24      \#        S5S64      \#        V^ 4      \#        V	^ 4      S3P8                  R	J S/P8                  R	J \        W,           4      ^8  \-        S/S34      VVV4      .,          o.V*! ^4       S.\        S0S28  VP8                  R	J VP8                  R	J V	^ 8  VVVVVV4
      .,          o.V*! ^4       S.\        S0S28  VP8                  R	J VP8                  R	J V	^ 8  VVVVVV4
      .,          o.V*! ^4       S.\        S5S68  VP8                  R	J V	P8                  R	J V^ 8  VVVVVV4
      .,          o.V*! ^4       S.\        S5S68  VP8                  R	J V	P8                  R	J V^ 8  VVVVVV4
      .,          o.V*! ^	4       S.\        S0S28  \#        S5S64      \#        V^ 4      V	^ 8  S3P8                  R	J \        V
4      ^8  VVVVV4      .,          o.V*! ^
4       S.\        S0S28  \#        S5S64      \#        V^ 4      V	^ 8  S3P8                  R	J \        V
4      ^8  VVVVV4      .,          o.V*! ^4       S.\        \#        S0S24      S5S68  V^ 8  \#        V	^ 4      S/P8                  R	J \        V4      ^8  VVVVV4      .,          o.V*! ^4       S.\        \#        S0S24      S5S68  V^ 8  \#        V	^ 4      S/P8                  R	J \        V4      ^8  VVVVV4      .,          o.V*! ^4       S.\        S0S28  S5S68  V^ 8  V	^ 8  VVVVVVV4      .,          o.V*! ^4       S.\        S0S28  S5S68  V^ 8  V	^ 8  VVVVVVV4      .,          o.V*! ^4       S.\        S0S28  S5S68  V^ 8  V	^ 8  VVVVVVVVV!4      .,          o.V*! ^4       S.\        S0S28  S5S68  V^ 8  V	^ 8  VVVVVVVVV!4      .,          o.V*! ^4       S.\        \#        V^ 4      VP8                  R	J VP8                  R	J VP8                  R	J VVV4      .,          o.V*! ^4       S.\        \#        V^ 4      VP8                  R	J VP8                  R	J VP:                  R	J VVV4      .,          o.V*! ^4       S.\        \#        V^ 4      VP8                  R	J V	P8                  R	J VP:                  R	J VVV4      .,          o.V*! ^4       S.\        \#        V^ 4      VP8                  R	J V	P8                  R	J VP8                  R	J VVV4      .,          o.V*! ^4       S.\        \#        WE,          ^ 4      VP8                  R	J V	P8                  R	J VVVVV4      .,          o.V*! ^4       S.\        \#        Wg,          ^ 4      VP8                  R	J V	P8                  R	J VVVVV4      .,          o.V*! ^4       \=        S VR	R7      p+\=        SVR	R7      p,S.\        V,\#        V^ 4      S5V8  VP8                  R	J VVVV4      .,          o.V*! R4       S.\        V,\#        V^ 4      S6V8  VP8                  R	J VVVV4      .,          o.V*! R4       S.\        V+\#        V^ 4      S0V8  V	P8                  R	J VVVV4      .,          o.V*! R4       S.\        V+\#        V^ 4      S2V8  V	P8                  R	J VVVV4      .,          o.V*! R4       \+        S.!  p-\1        V-4      R8w  d   V-# S.\        Wg,           S08  \#        V^ 4      \#        V^ 4      VP8                  R	J VP8                  R	J V	P:                  R	J \        \        S/4      4      Wg,           S0,
          ^,           \        ,          8  VVVV!V%4      .,          o.V*! ^4       S.\        Wg,           S28  \#        V^ 4      \#        V^ 4      VP8                  R	J VP8                  R	J V	P:                  R	J \        \        S/4      4      Wg,           S2,
          ^,           \        ,          8  VVVV!V%4      .,          o.V*! ^4       S.\        \#        S0S2^,
          4      \#        V^ 4      \#        V^ 4      VP8                  R	J VP8                  R	J V	^ 8  V	\        ,          \        \        S/4      4      8  VVVV!V%4      .,          o.V*! ^4       S.\        \#        S0S2^,           4      \#        V^ 4      \#        V^ 4      VP8                  R	J VP8                  R	J V	^ 8  V	\        ,          \        \        S/4      4      8  VVVV!V%4      .,          o.V*! ^4       S.\        S0S2^,
          8  \#        V^ 4      \#        V^ 4      VP8                  R	J VP8                  R	J V	^ 8  V	\        ,          \        \        S/4      4      8  \        \        S/4      4      Wg,           S0,
          ^,           \        ,          8  VVVV!V%4      .,          o.V*! ^4       S.\        S0S2^,           8  \#        V^ 4      \#        V^ 4      VP8                  R	J VP8                  R	J V	^ 8  V	\        ,          \        \        S/4      4      8  \        \        S/4      4      Wg,           S2,
          ^,           \        ,          8  VVVV!V%4      .,          o.V*! ^4       S.\        \#        V^ 4      \#        V^ 4      WE,           ^ 8  VP8                  R	J V	P8                  R	J VP:                  R	J \        \        S34      4      WE,           S5,
          ^,           \        ,          8  VVVV!V%4      .,          o.V*! ^4       S.\        \#        V^ 4      \#        V^ 4      WE,           S68  VP8                  R	J V	P8                  R	J VP:                  R	J \        \        S34      4      WE,           S6,
          ^,           \        ,          8  VVVV!V%4      .,          o.V*! ^4       S.\        \#        V^ 4      \#        V^ 4      \#        S5S6^,
          4      VP8                  R	J V	P8                  R	J V^ 8  V\        ,          \        \        S34      4      8  \        \        S34      4      V^,           \        ,          8  VVVV!V%4      .,          o.V*! ^ 4       S.\        \#        V^ 4      \#        V^ 4      \#        S5S6^,           4      VP8                  R	J V	P8                  R	J V^ 8  V\        ,          \        \        S34      4      8  \        \        S34      4      V^,           \        ,          8  VVVV!V%4      .,          o.V*! ^!4       S.\        \#        V^ 4      \#        V^ 4      S5S6^,
          8  VP8                  R	J V	P8                  R	J V^ 8  V\        ,          \        \        S34      4      8  \        \        S34      4      WE,           S5,
          ^,           \        ,          8  VVVV!V%4      .,          o.V*! ^"4       S.\        \#        V^ 4      \#        V^ 4      S5S6^,           8  VP8                  R	J V	P8                  R	J V^ 8  V\        ,          \        \        S34      4      8  \        \        S34      4      WE,           S6,
          ^,           \        ,          8  VVVV!V%4      .,          o.V*! ^#4       \+        S.!  # u uppi u uppi u upi u upi u upi u upi   \6         d    Rp% EL[i ; i)z=Return a condition under which the integral theorem applies. 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                    < SS3 Fi  p \         P                  ! V P                  V P                  4       F7  w  rW,
          pVP                  '       g   K!  VP
                  '       g   K5    R # 	  Kk  	  R# FT)r  r  r   r   
is_integeris_positive)r  rc   jdiffr  r  s       re   _c1_check_antecedents.<locals>._c1  sU    bA!))!$$5u???t'7'7'7  6 
 rm   c                `    V ^8g  ;'       d#    \        \        ^V ,
          4      4      \        8  # )aw  Returns True if abs(arg(1-z)) < pi, avoiding arg(0).

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

If ``z`` is 1 then arg is NaN. This raises a
TypeError on `NaN < pi`. Previously this gave `False` so
this behavior has been hardcoded here but someone should
check if this NaN is more serious! This NaN is triggered by
test_meijerint() in test_meijerint.py:
`meijerint_definite(exp(x), x, 0, I)`
)r"   r!   r   )r\   s   &re   _cond!_check_antecedents.<locals>._cond$  s'     622c#a!e*o22rm   Fc                "  < V SS,
          ,          \        S4      ^SS,
          ,          ,          ,          \        S4      ,          VS	S,
          ,          \        S4      ^S	S,
          ,          ,          ,          \        S4      ,          ,           # r   )r"   r7   )
c1c2omegar   psir   sigmathetaur  s
   &&re   	lambda_s0%_check_antecedents.<locals>.lambda_s0S  sc    1q5z#e*q!a%y"99#c(B!a%jUaQi!88UCD Drm   Tz	  c%s: %sc                2   < \        R V SR,          34       R# )z  case %s: %sNr   r  )countr  s   &re   pr_check_antecedents.<locals>.prl  s    %r!34rm   )r  E1E2E3E4r   )r   r   r   r  r   r   r   r   r   r   r"   r(   r~  r  rR   r   r   r   r*   r   r$   rS   r   r6   rl  r4   r#   	TypeErrorr  is_negativer  )7r  r  r{   r>  ru  r   r   rr   bstarcstarrhomuphir  r  r  rc   r  r  c3c4c5c6c7c8c9c10c11c12c13z0zoszsoc14r  c14_altlambda_cc15r  lambda_sr  r   r  
mt1_exists
mt2_existsr   r  r  r   r  r   r  r  r  r  s7   ff&                                           @@@@@@@@@re   _check_antecedentsr    sz    bkk1-HE1bkk1-HE1CJBEE
CJBEE
CDJA!QCJBEE
CJBEE
CDJA!QEQUAIEEQUAIE
%%1q5!)
a
C	!a%	Q	B
a%1q5/C
q1u+
S
 Cq1uqy>C 3E :;;a!e
DCQ^c"5e"<==AFE
"#?A!Q+-?A!Q*,03S%2HI 
B	"%%?%QAr!a%!)}q  %?	@B	beeCeRUUr!a%!)}u$U$eC	DB	OAAr!a%!)}$r"v-Q?O	PB	RUUKUAr!a%y 2b6)HRO;UK	LB	"%%P%QAr!a%!)}$r#w."a@%P	QB	beeLeAr!a%y 2c7*Xb!_<eL	MB
c(Qr37QU+q1uq1uo=
B!eA    B
c(Qr37QU+q1uq1uo=
B!eA    B"5)*U2X5C
S$U+,eBh
7C"5)*U2X5C
S$U+,eBh
7C 
u}b 0	1B
RXe^
$C
RXe^
$C
ae|"S!*emq0RQZBHqL1$4!5!91q()A-/0	3 "S!*eai%/14SCU3Z0BHqL1,-12c1:>@A baj%!)e"3q"8SCU3Z0BHqL1,-12c1:>@A gE3u:1q5	223s8;1uc%j1a!e9--c%j89 hl#u,a<CD D !B#Ib"$55R+E2A6;Nu;UWX8YZ\4 3E :;R@4PcdiPjKkmoAppR+E2A6;Nu;UWX8YZ\2t$7$>?@2tTghmTnOoAppR+E2A6;Nu;UWX8YZ\4 3E :;TBUV[B\=]^`degH a<r(A8QC2Fr(A8QC1EGC c(C Gb!Wr1gAwQ"aGb!Wr1gRy3)"IRy3)c2Y@a 	aY'@ E5	c!#a%'Q, 1 1T 95;L;LPT;TVXZ\^`be  EqE	c"Q(BuaL%*;*;t*CUEVEVZ^E^`bcf`gjk`kb"c# $ $EqE	c"Q(BuaL%*;*;t*CUEVEVZ^E^`bce`fij`jb"c# $ $EqE	c"Q(Bq!HblBuaL##t+U->->$-F2QR
TVWZT[^_T_UE"BB0 1 1E qE	c"Q(Bq!HblBuaL##t+U->->$-F28WXHXUE"BB0 1 1E qE	c!a%$.0A0AT0I5TU:b"b#s, - -EqE	c!a%$.0A0AT0I5TU:b"b#s, - -EqE	c!a%$.0A0AT0I5TU:b"b#s, - -EqE	c!a%$.0A0AT0I5TU:b"b#s, - -EqE	c!a%Aq2eQ<!U=N=NRV=VS'A+r2r2s4 5 5ErF	c!a%Aq2eQ<!U=N=NRV=VS'A+r2r2s4 5 5ErF	c"Q(AE5A:r%|U=N=NRV=VR&1*b"b"c3 4 4ErF	c"Q(AE5A:r%|U=N=NRV=VR&1*b"b"c3 4 4ErF	c!a%Q
EQJb"b"c30 1 1ErF	c!a%Q
EQJb"b"c30 1 1ErF	c!a%Q
EQJb"b"b#sC9 : :ErF	c!a%Q
EQJb"b"b#sC9 : :ErF	c"Q(AMMT153D3D3LcooaeNegikmorsttErF	c"Q(AMMT153D3D3LcooaeNegikmorsttErF	c"Q(AMMT153D3D3LcooaeNegikmorsttErF	c"Q(AMMT153D3D3LcooaeNegikmorsttErF	c"QS!*e//479J9Jd9Rb"c3( ) )ErF	c"QS!*e//479J9Jd9Rb"c3( ) )ErF &b!D9J%b!D9J	c*bAhAu/@/@D/H#rSUWYZ[[EtH	c*bAhAu/@/@D/H#rSUWYZ[[EtH	c*bAhAu/@/@D/H#rSUWYZ[[EtH	c*bAhAu/@/@D/H#rSUWYZ[[EtH 	E
A!}	c!%!)R1Xr#qz1==D3H%J[J[_cJcejevevz~e~)%01QUQY]B4FFb#sC) * *E rF	c!%!)R1Xr#qz1==D3H%J[J[_cJcejevevz~e~)%01QUQY]B4FFb#sC) * *E rF	c"QA,1a"S!*ammt6KUM^M^bfMf1*eBh-@-G)HHb#sC) * *E rF	c"QA,1a"S!*ammt6KUM^M^bfMf1*eBh-@-G)HHb#sC) * *E rF	c!a!e)R1Xr#qz1==D3H%J[J[_cJc1*eBh-@-G)HH)%01QUQY]B4FFb#sC) * *E rF	c	AE	2a8RQZ$)>@Q@QUY@Y[`de[e(S!4U!;<<)%01QUQY]B4FFb#sC	) * *E
 rF	c"Q(BsAJ	1==D3H%J[J[_cJcejevevz~e~)%01QUQY]B4FFb#sC) * *E rF	c"Q(BsAJ	1==D3H%J[J[_cJcejevevz~e~)%01QUQY]B4FFb#sC) * *E rF	c"Q(BsAJ1a!eammt6KUM^M^bfMf1*eBh-@-G)HH)%01UQYNBb#sC) * *E rF	c"Q(BsAJ1a!eammt6KUM^M^bfMf1*eBh-@-G)HH)%01UQYNBb#sC) * *E rF	c
1a"S!*a!a%i$)>@Q@QUY@Y[`de[eb3*5122&'1519q="*<<
BS#	    E
 rF	c
1a"S!*a!a%i$)>@Q@QUY@Y[`de[eb3*5122&'1519q="*<<
BS#	    E
 rFu:C @COKPLv  sN   8Aj:
/8Ak 
 AAk%AAkAAk(AAk'BAk GAk kAk+k*Ak+c                   \        V P                  V4      w  r4\        VP                  V4      w  rTR pV! V P                  4      \        VP                  4      ,           p\        VP
                  4      V! V P                  4      ,           pV! V P                  4      \        VP                  4      ,           p	\        VP                  4      V! V P
                  4      ,           p
\        WxWWS,          4      V,          # )a  
Express integral from zero to infinity g1*g2 using a G function,
assuming the necessary conditions are fulfilled.

Examples
========

>>> from sympy.integrals.meijerint import _int0oo
>>> from sympy.abc import s, t, m
>>> from sympy import meijerg, S
>>> g1 = meijerg([], [], [-S(1)/2, 0], [], s**2*t/4)
>>> g2 = meijerg([], [], [m/2], [-m/2], t/4)
>>> _int0oo(g1, g2, t)
4*meijerg(((0, 1/2), ()), ((m/2,), (-m/2,)), s**(-2))/s**2
c                0    V  Uu. uF  q) NK  	  up# u upi r`   r   r{  s   & re   neg_int0oo.<locals>.neg  s    AqAs   )r   r   r   r   r   r  r  rO   )r  r  r{   r  r>  r  r  r  r  r  r  s   &&&        re   _int0oor  	  s    " BKK+FCbkk1-HE	RUUd255k	!B	bii3ryy>	)B	RUUd255k	!B	bii3ryy>	)B2259-c11rm   c           	     \  aa	 \        W4      w  po	\        VP                  V4      w  poVV	3R lp^ RIHp V! WS	S,          ,          ,          RR7      \	        V! VP
                  4      V! VP                  4      V! VP                  4      V! VP                  4      VP                  4      3# )zAbsorb ``po`` == x**s into g. c                L   < V  Uu. uF  qSS,          ,           NK  	  up# u upi r`   r   )r   r   r   ru  s   & re   r!  _rewrite_inversion.<locals>.tr+  s#    !"#AAaC###s   !r  Tr  )	r   r   r   r  rO   r   r  r   r  )
r   r  r  r{   r>  r   r!  r  r   ru  s
   &&&&    @@re   _rewrite_inversionr  &  s    " DAq!**a(DAq$(cac(l$/BqttHblBqttHblAJJOQ Qrm   c                  a aaaaa \        R4       S P                  p\        VS4      w  r4V^ 8  d!   \        R4       \        \	        S 4      S4      # V3R loV3R lo\        \        S P                  4      \        S P                  4      \        S P                  4      \        S P                  4      .4      w  rVrxWV,           V,
          p	W,
          V,
          p
W,
          ^,          pW,
          oS^8X  d   \
        P                  pMS^8  d   ^pM\
        P                  p^S,
          ^,          \        S P                  !  ,           \        S P                  !  ,
          S,          oS P                  p\        RWVWxWVS34       \        RVSV34       S P                  V^,          8  g   V^8  d   Wx8  g   \        R4       R# \         P"                  ! S P                  S P                  4       F3  w  rW,
          P$                  '       g   K  W8  g   K'  \        R	4        R# 	  Wx8  d?   \        R
4       \'        S P                   Uu. uF  pS! V^,
          ^ ^ V4      NK  	  up!  # V V3R lpVVV3R lpVVV3R lpVVV3R lp. pV\'        ^V8*  ^V8*  V\(        ,          V,
          \(        ^,          8  V^ 8  V! V\+        \
        P,                  \(        ,          V
^,           ,          4      ,          4      4      .,          pV\'        V^,           V8*  V^,           V8*  V^ 8  V\(        ^,          8  V^ 8H  WW,
          ^,           \(        ,          V,
          \(        ^,          8  V! V\+        \
        P,                  \(        ,          W,
          ,          4      ,          4      V! V\+        \
        P,                  ) \(        ,          W,
          ,          4      ,          4      4      .,          pV\'        WX8H  V^ 8H  V^ 8  SV,           \(        ,          V,
          \(        ^,          8  V! V4      4      .,          pV\'        \/        \'        Wx^,
          8*  ^V	8*  V	S^,          8*  4      \'        V^,           WV,           8*  WV,           Wx,           ^,          8*  4      4      V^ 8  V\(        ^,          8  V	^,           \(        ,          V,
          \(        ^,          8  V! V\+        \
        P,                  \(        ,          V
,          4      ,          4      V! V\+        \
        P,                  ) \(        ,          V
,          4      ,          4      4      .,          pV\'        ^V8*  V^ 8  V^ 8  W\(        ,          ,           \(        ^,          8  W,           \(        ,          V,
          \(        ^,          8  V! V\+        \
        P,                  \(        ,          V
,          4      ,          4      V! V\+        \
        P,                  ) \(        ,          V
,          4      ,          4      4      .,          pVV^ 8H  .,          p\/        V!  # u upi )z6Check antecedents for the laplace inversion integral. z#Checking antecedents for inversion:z  Flipping G.c           
     ^  < \        VS4      w  rVW,          p WV,          ,          pW&,          p. pV\        \        P                  \	        V4      ,          \
        ,          ^,          4      ,          pV\        \        P                  ) \	        V4      ,          \
        ,          ^,          4      ,          p	V'       d   Tp
MT	p
V\        \        \        V^ 4      \	        V4      ^ 8*  4      \	        V 4      R8*  4      .,          pV\        \        V^ 4      \        \        V4      ^ 4      \	        V4      ^ 8  \	        V
4      ^ 8  4      .,          pV\        \        V^ 4      \        \        V4      ^ 4      \	        V4      ^ 8  \	        V
4      ^ 8*  \	        V 4      R8*  4      .,          p\        V!  # )r   r   )r   r*   r   r   r   r   rR   rS   r   r   r    )r   r   r   r\   pluscoeffexponentr  wpwmwr{   s   &&&&&      re   statement_half4_check_antecedents_inversion.<locals>.statement_half<  sG   (A.		AX	s1??2a5(+A-..sAOO#BqE)",Q.//AA#bAq2a5A:.1<==#bAh2a5!beaiACDD#bAh2a5!beaiA!erk# $ 	$5zrm   c           
     <   < \        S! WW#R4      S! WW#R4      4      # )zIProvide a convergence statement for z**a * exp(b*z**c),
c/f sphinx docs. TF)rR   )r   r   r   r\   r  s   &&&&re   	statement/_check_antecedents_inversion.<locals>.statementN  s)     >!d3!!e46 	6rm   z9  m=%s, n=%s, p=%s, q=%s, tau=%s, nu=%s, rho=%s, sigma=%sz   epsilon=%s, theta=%s, delta=%sz-  Computation not valid for these parameters.Fz  Not a valid G function.z$  Using asymptotic Slater expansion.c                t   < \        SP                   Uu. uF  pS! V^,
          ^ ^ V 4      NK  	  up!  # u upi r   )rR   r   )r\   r   r  r  s   & re   E'_check_antecedents_inversion.<locals>.E~  s3    =1Yq1uaA.=>>=s   5c                *   < S! SS) ^S,          V 4      # r   r   )r\   r  r  r  s   &re   H'_check_antecedents_inversion.<locals>.H  s    %33rm   c                ,   < S! SS) ^S,          V R4      # )r   Tr   r\   r  r  r  s   &re   Hp(_check_antecedents_inversion.<locals>.Hp  s    eeVQuWa>>rm   c                ,   < S! SS) ^S,          V R4      # )r   Fr   r  s   &re   Hm(_check_antecedents_inversion.<locals>.Hm  s    eeVQuWa??rm   )r~  r   r   _check_antecedents_inversionr$  r   r  r   r   r   r   r   NaNr   r  r  r  r  r  rR   r   r*   r   rS   )r  r{   r\   r>  r   r   rr   r   r   r  r   r  epsilonr  r   r   r  r  r  r  r  r  r  r  r  s   ff                   @@@@re   r  r  2  s   
01	

A!QDA1u+GAJ::$6 CIs144y#add)SY?@JA!
%!)C	
B8Q,CEEz&&	%%%i]S!$$Z'#qtt*4e;EGGEG1#u-/.%0GH GGqsNqAv!&>?
 !!!$$-E!%./ . 	v56=1Yq1uaA.=>>?4?@ E	c!q&!q&#b&5.BqD"8%!)Ac!//",b1f56679 : :E 
c!a%1*a!eqj%!)URT\16519b.5(BqD0Qs1??2-qu5667QsAOO+B.6778: ; ;E
 
c!&!q&%!)7?B&."Q$6!> ? ?E 
c"S!eQ#XseAg~>Q!%15!));<>!)URT\C!GR<%+?2a4+GQs1??2-b0112QsAOO+B.r1223	5 6 6E 
c!q&#'519e"fnr!t.C="$u,14Qs1??2-b0112QsAOO+B.r12235 6 6E
 
a1fXE u:[ >s   Yc                    \        V P                  V4      w  r4\        \        V P                  V P
                  V P                  V P                  W2V,          ,          4      V) 4      w  rPWR,          V ,          # )zG
Compute the laplace inversion integral, assuming the formula applies.
)r   r   r&  rO   r   r  r   r  )r  r{   r   r   r   r  s   &&&   re   _int_inversionr    sT     !**a(DA'!$$!$$!qD&IA2NDA3q5Lrm   c                h  a a! ^RI HpHo!HpHo  \
        '       g   / s\        \
        4       \        V \        4      '       d   \        V P                  V4      P                  V4      w  rV\        V4      ^8  d   R# V^ ,          pVP                  '       d1   VP                  V8w  g   VP                  P                   '       g   R# MWa8w  d   R# ^^ \        V P"                  V P$                  V P&                  V P(                  WV,          4      3.R3# T pV P+                  V\,        4      p \/        V \,        4      pV\
        9   Ed?   \
        V,          p	V	 EF*  w  rrV P1                  V
RR7      pV'       g   K$  / pVP3                  4        F!  w  pp\5        \7        VRR7      RR7      VV&   K#  	  Tp\        V\8        4      '       g   VP+                  V4      pVR8X  d   K  \        V\8        \:        34      '       g   \5        VP+                  V4      4      p\=        V4      R8X  d   K  \        V\>        4      '       g	   V! V4      p. pV EF   w  pp\A        \5        VP+                  V4      P+                  \,        V4      RR7      V4      p VP+                  V4      P+                  \,        V4      p\E        TT3,           !  PG                  \H        PJ                  \H        PL                  \H        PN                  4      '       d   K  \        TP"                  TP$                  TP&                  TP(                  \5        TP                  RR7      4      pTPQ                  TT3,           4       EK#  	  V'       g   EK'  VV3u # 	  V'       g   R# \S        R4       V V!3R	 lpTp \U        R
RV 4      pR p V! WVVRRR7      w  pppV! VVVV4      pVft   \W        RR4      pVV PX                  9  dW   \[        W4      '       dF    V! V P+                  VVV,          4      VVVRRR7      w  pppV! VVVV4      P+                  V^4      pVeD   VPG                  \H        PJ                  \H        P\                  \H        PL                  4      '       d   \S        R4       R# \^        P`                  ! V4      p. pV F  p V P                  V4      w  pp\        V4      ^8  d   \c        R4      hV^ ,          p\A        VP                  V4      w  ppVV^ \        VP"                  VP$                  VP&                  VP(                  \5        \7        VRR7      RR7      VV,          ,          4      3.,          pK  	  \S        RV4       VR3#   \B         d     EKE  i ; i  T d    Rp ELi ; i  T d    Rp ELki ; i)a,  
Try to rewrite f as a sum of single G functions of the form
C*x**s*G(a*x**b), where b is a rational number and C is independent of x.
We guarantee that result.argument.as_coeff_mul(x) returns (a, (x**b,))
or (a, ()).
Returns a list of tuples (C, s, G) and a condition cond.
Returns None on failure.
)mellin_transforminverse_mellin_transformIntegralTransformErrorMellinTransformStripErrorNT)old)lift)exponents_onlyFz)Trying recursive Mellin transform method.c           
        <  S! WW#RRR7      #   S d/    ^ RI Hp S! T! \        \        T 4      4      4      YTRRR7      u # i ; i)zCalling 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)
as_meijergneedeval)simplify)r   r'  rW   r   )Fru  r{   stripr'  r   r  s   &&&& re   my_imt_rewrite_single.<locals>.my_imt
  sY    
	0+A!7;dL L( 	0/+q	*+Q5$0 0	0s    2AAru  zrewrite-singlec           	         \        WR R7      pVeU   ^ RIHp Vw  rE\        V! VRR7      4      p\	        WE3\        W\        P                  \        P                  34      R 34      # \        W\        P                  \        P                  34      # )T)only_doublehyperexpandnonrepsmall)rewrite)	_meijerint_definite_4r   r/  _my_unpolarifyr4   rQ   r   r   rE  )rd   r{   r   r/  rl   r   s   &&    re   my_integrator&_rewrite_single.<locals>.my_integrator  sw    !!D9=2IC S-!HICc[&qaffajj*ABDIK Kqvvqzz233rm   )
integratorr'  r&  r   )r6  r&  r'  z"Recursive Mellin transform failed.zUnexpected form...z"Recursive Mellin transform worked:)2
transformsr  r  r  r   _lookup_tabler   r   rO   rX   r   r   r  r   r   r*   r  r   r  r   r  r   r\   r   r   itemsr$   r%   rj  rT   rl  r   r   
ValueErrorr	   rk   r   rE  ComplexInfinityNegativeInfinityr   r~  r0  r+  r   r8  r  r   r  NotImplementedError)"rd   r{   	recursiver  r  r   r   f_r   r   r   termsr   r   r   subs_r^  r_  rl   r   r  r  r*  ru  r4  r(  r)  r>  r   rj   r   r   r   r  s"   &&&                             @@re   _rewrite_singlerB    s   ; ; =]+!W!**a(55a8q6A:aD888vv{!%%"3"3"3 #4VAwqttQXXqttQXXuwGHI4OO	
B	q!A1AM!*+&GD7777-Dt#zz|GC!+HRd,C;?"AE#J  , !$--99T?D5=!${(;<<%diio6Dd#u,!%..!$KE#FC'
388D>3F3Fq!3LBF)HIJLB!FF4L--a3 rQDy*..qzz1;L;LaN`N`aa ahhahh *1::d KMAJJrQDy) $ 39$G +,L 
670 	As$a(A4&qQ=05F5!1aE" 	y C)*ANN"|A'9'9.qvva1~q!:G8<uN5! 1aE*//15 	yAEE!**aeeQ->->??34==D
C~~a 1q6A:%&:;;aDajj!,1AwqttQXXqttQXX)(#$4+1AE G !1 %& ' ( 	(  /39_ & ! !` "  * s7   %V2V AV$ VV	V! V!$	V10V1c                p    \        W4      w  r4p\        WQV4      pV'       d   W4V^ ,          V^,          3# R# )z
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.
N)r  rB  )rd   r{   r>  r   r  r  s   &&&   re   	_rewrite1rD  J  s;     A!JCQi(A!ad"" 	rm   c           	     P  a \        V S4      w  r#p\        ;QJ d(    V3R l\        V4       4       F  '       g   K   RM	  RM! V3R l\        V4       4       4      '       d   R# \        V4      pV'       g   R# \	        \        VV3R lV3R lV3R l.4      4      p\        P                  ! RV4       Fm  w  pw  rx\        VSV4      p	\        VSV4      p
V	'       g   K,  V
'       g   K6  \        V	^,          V
^,          4      pVR8w  g   KY  W#V	^ ,          V
^ ,          V3u # 	  R# )	z
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  B   <"   T F  p\        VSR 4      RJ x  K  	  R# 5i)FN)rB  r4  s   & re   rf   _rewrite2.<locals>.<genexpr>a  s     
L|t?4E*d2|r6  TFNc           	        < \        \        \        V ^ ,          S4      4      \        \        V ^,          S4      4      4      # rw   )maxr  r   r   r{   s   &re   r|   _rewrite2.<locals>.<lambda>g  ,    #c*QqT1-.JqtQ4G0HIrm   c           	        < \        \        \        V ^ ,          S4      4      \        \        V ^,          S4      4      4      # rw   )rI  r  r   rJ  s   &re   r|   rK  h  rL  rm   c           	        < \        \        \        V ^ ,          S4      4      \        \        V ^,          S4      4      4      # rw   )rI  r  r  rJ  s   &re   r|   rK  i  s1    #c01q9:01q9:<rm   r  )
r  r7  r  r  r   r   r  r  rB  rR   )rd   r{   r   r  r  r   r>  fac1fac2r  r  r   s   &f          re   	_rewrite2rQ  X  s     Aq!JCQ
s
Ly|
Lsss
Ly|
LLL!AWQII	<= > 	?A $-#4#4]A#F	<DT1i0T1i02""r!ube$Du}1r!ud22 $Grm   c                   \        V 4      p . p\        \        W4      \        P                  0,          \
        R7       Fw  p\        V P                  WV,           4      V4      pV'       g   K/  VP                  WV,
          4      p\        V\        \        4      '       d   VP                  V4       Ku  Vu # 	  V P                  \        4      '       d{   \        R4       \        \!        V 4      V4      pV'       dS   \#        V\$        4      '       g,   ^ RIHp V! \+        V4      VP-                  \.        4      4      # VP1                  V4       V'       d   \3        \5        V4      4      # R# )z
Compute an indefinite integral of ``f`` by rewriting it as a G function.

Examples
========

>>> from sympy.integrals.meijerint import meijerint_indefinite
>>> from sympy import sin
>>> from sympy.abc import x
>>> meijerint_indefinite(sin(x), x)
-cos(x)
r   *Try rewriting hyperbolics in terms of exp.collectN)r   r   r  r   r   r   _meijerint_indefinite_1r   ra   rN   rO   r   rk   r1   r~  meijerint_indefiniter0   r   r   sympy.simplify.radsimprU  r
   r   r*   extendnextr   )rd   r{   resultsr   rl   rvrU  s   &&     re   rW  rW  u  s    	
AG*10AFF8;AQR%affQA&6:hhqa% UG$$NN3J S 	uu  ;<!'*A/b$'':|B/#??NN2GG$%% rm   c           	       aa \        RV RS4       ^ RIHpHp \	        V S4      pVf   R# Vw  rVrx\        RV4       \
        P                  p	V EFh  w  rp\        VP                  S4      w  r\        VS4      w  ppVV,          pWZ,          S^V,           ,          ,          V,          pV^,           V,          o\        RR\
        P                  4      pV3R lp\        ;QJ d,    R	 V! VP                  4       4       F  '       g   K   R
M!	  RM! R	 V! VP                  4       4       4      '       dv   \        \        VP                  4      \        VP                   4      ^S,
          .,           \        VP                  4      S) .,           \        VP"                  4      V4      ) pMs\        \        VP                  4      ^S,
          .,           \        VP                   4      \        VP                  4      \        VP"                  4      S) .,           V4      pVP$                  '       dH   V P'                  S^ 4      P)                  \
        P*                  \
        P,                  4      '       g   ^ pMRpV! VP'                  VVSV,          ,          4      VR7      pW! VV,          R
R7      ,          p	EKk  	  V3R lp\/        V	RR7      p	V	P0                  '       dB   . pV	P2                   F#  w  pp\5        V! V4      4      pVVV3.,          pK%  	  \7        VRR/ p	M\5        V! V	4      4      p	\7        V	\5        V4      3\9        V S4      R
34      # )z/Helper that does not attempt any substitution. z,Trying to compute the indefinite integral ofwrt)r/  r  Nz could rewrite:r   zmeijerint-indefinitec                >   < V  Uu. uF  qS,           NK  	  up# u upi r`   r   )r   r   r  s   & re   r!  #_meijerint_indefinite_1.<locals>.tr  s    %&'QGGQ'''r  c              3  X   "   T F   qP                   ;'       d	    V^ 8*  R8H  x  K"  	  R# 5i)rx   TN)r  )rb   r   s   & re   rf   *_meijerint_indefinite_1.<locals>.<genexpr>  s&     C(Q||00aD 00(s   **TF)placer  c                   < \        \        V 4      RR7      p \        P                  ! V P	                  S4      ^,          4      # )a9  This multiplies out superfluous powers of x we created, and chops off
constants:

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

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

Examples
========

>>> from sympy import sqrt, cancel
>>> from sympy.abc import x
>>> a = sqrt(2*x + 1)
>>> bad = (3*x*a**5 + 2*x - a**5 + 1)/a**2
>>> bad.expand().as_independent(x)[0]
0
>>> cancel(bad).expand().as_independent(x)[0]
1
F)deep)r   rW   r   
_from_argsas_coeff_add)rl   r{   s   &re   _clean'_meijerint_indefinite_1.<locals>._clean  s4    . 51~~c..q1!455rm   )evaluaterj  )r~  r   r/  r  rD  r   r   r   r   r0  r   r7  r   rO   r   r   r  r  is_extended_nonnegativer   rk   r  r;  r5   r^   rj   r3  r4   rQ   )rd   r{   r/  r  r
  r   r  glr   rl   r  ru  r  r   r   r>  r   fac_r   r!  r   rc  rh  rh  r   r  s   &f                       @re   rV  rV    s   
91eQG5	1aB	zCR
b!
&&Caajj!,b!$1	Q wQU#a'1uai 3.6	(3C"QTT(C333C"QTT(CCCQTT
DNaeW4d144jSD66I4PQPXPX>[\^ ^A QTT
aeW$d188nd144j$qxx.UXTXSYBY[\^A $$$QVVAq\-=-=aeeQEVEV-W-WEEq!AqD&)7 	yat,,K N64 t
,C
HHDAqvay)AAxG  151VC[)c>$/08Aq>42HIIrm   c                
   \        RWW#34       \        V 4      p V P                  \        4      '       d   \	        R4       R# V P                  \
        4      '       d   \	        R4       R# WW#3w  rErg\        R4      pV P                  W4      p TpW#8X  d   \        P                  R3# . p	V\        P                  J d4   V\        P                  Jd    \        V P                  W) 4      W) V) 4      # V\        P                  J Ed!   \	        R4       \        W4      p
\	        RV
4       \        V
\        RR	7      \        P                  .,            F  p\	        R
V4       VP                   '       g   \	        R4       K.  \#        V P                  WV,           4      V4      pVf   \	        R4       Ka  \#        V P                  WV,
          4      V4      pVf   \	        R4       K  Vw  rVw  r\%        \'        W4      4      pVR8X  d   \	        R4       K  W,           pVV3u # 	  EM~V\        P                  J d/   \        WV\        P                  4      pV^ ,          ) V^,          3# W#3\        P                  \        P                  38X  dH   \#        W4      pV'       d3   \)        V^ ,          \*        4      '       d   V	P-                  V4       EMV# EMV\        P                  J d   \        W4       F  pVV,
          ^ 8  R8X  g   K  \        RV4       \#        V P                  WV,           4      \/        VV,           V,
          4      ,          V4      pV'       g   Kl  \)        V^ ,          \*        4      '       d   V	P-                  V4       K  Vu # 	  V P                  WV,           4      p W2,
          p^ pV\        P                  Jd}   \1        \        P2                  \5        V4      ,          4      p\7        V4      pV P                  VVV,          4      p V \/        W1,
          4      V,          ,          p \        P                  p\	        RW#4       \	        RV 4       \#        W4      pV'       d2   \)        V^ ,          \*        4      '       d   V	P-                  V4       MV# VP                  \8        4      '       d   \	        R4       \        \;        V4      WVV4      pV'       dr   \=        V\>        4      '       gK   ^ RI H!p V! \E        V^ ,          4      V^ ,          PG                  \0        4      4      3VR,          ,           pV# V	PI                  V4       V	'       d   \K        \M        V	4      4      # R# )a  
Integrate ``f`` over the interval [``a``, ``b``], by rewriting it as a product
of two G functions, or as a single G function.

Return res, cond, where cond are convergence conditions.

Examples
========

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

This function is implemented as a succession of functions
meijerint_definite, _meijerint_definite_2, _meijerint_definite_3,
_meijerint_definite_4. Each function in the list calls the next one
(presumably) several times. This means that calling meijerint_definite
can be very costly.
z$Integrating %s wrt %s from %s to %s.z+Integrand has DiracDelta terms - giving up.Nz5Integrand has Singularity Function terms - giving up.r{   Tz  Integrating -oo to +oo.z  Sensible splitting points:)r   reversez  Trying to split atz  Non-real splitting point.z'  But could not compute first integral.z(  But could not compute second integral.Fz)  But combined condition is always false.zTrying x -> x + %szChanged limits tozChanged function torS  rT  :r   NN)'r  r   rk   r>   r~  rP   r   r   r   r   r<  rE  meijerint_definiter  r   r   is_extended_real_meijerint_definite_2rW  rR   ra   rO   r   r?   r*   r   r!   r"   r1   r0   r   r   rX  rU  r
   r   rY  rZ  r   )rd   r{   r   r   r?  x_a_b_r/  r[  r  r   res1res2cond1cond2r   rl   splitr  r\  rU  s   &&&&                  re   rp  rp    s>   < 2Q1LA
AuuZ<=uu !!FG1ZNBB 	c
A	qA	Av~GA1AJJ#6!!&&B-B;;	
a  	 *+*10	-y9	'7F!&&QQA)1-%%%45(q5)91=D|@A(q5)91=D|ABKDKDS./Du}BC+C9) R, 
ajj q!**5AwA
AFFAJJ'	'#A)CFG$$s#
	  

?/5INt+0%8/qe)0D1:1u9q=1I1JKLNCsA00#NN3/#&J 6 FF1!eEAJJaooc!f,-CAAq#a% A15!#%%A

A"A)$a(#A)CFG$$s#
	vv !!;<'+RR9b$'':l2a512a5;;s3CDFBO	NN2GG$%% rm   c                6   V R3.pVR,          ^ ,          pV0p\        V4      pWT9  d   W%R3.,          pVP                  V4       \        V4      pWT9  d   W%R3.,          pVP                  V4       VP                  \        \
        4      '       d7   \        \        V4      4      pWT9  d   W%R3.,          pVP                  V4       VP                  \        \        4      '       d1   ^ RI	H
p V! V4      pWt9  d   W'R3.,          pVP                  V4       V# )z5Try to guess sensible rewritings for integrand f(x). zoriginal integrandr   r   zexpand_trig, expand_mul)sincos_to_sumztrig power reductionr   )r   r   r   rk   r9   r1   r   r6   r7   sympy.simplify.fur|  )rd   r{   rl   origsawexpandedr|  reduceds   &&      re   _guess_expansionr    s   #$
%Cr71:D&C$H<())d|H8$%%xx%'9::k$/089::CGGHxxS3%4566CGGGJrm   c                    \        RRV RR7      pV P                  W4      p TpV ^ 8X  d   \        P                  R3# \	        W4       F(  w  r4\        RV4       \        W14      pV'       g   K&  Vu # 	  R# )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)positiveTryingN)r0  r   r   r   r  r~  _meijerint_definite_3)rd   r{   dummyr  explanationrl   s   &&    re   rr  rr    sl      3-q4@E	qAAAvvvt|*10x%#A)3J	 1rm   c                   \        W4      pV'       d   V^,          R8w  d   V# V P                  '       d   \        R4       V P                   Uu. uF  p\        W14      NK  	  pp\        ;QJ d    R V 4       F  '       d   K   RM	  RM! R V 4       4      '       dD   . p\
        P                  pV F  w  rgW&,          pWW.,          pK  	  \        V!  pVR8w  d   W'3# R# R# R# u upi )z
Try to integrate f dx from zero to infinity.

This function calls _meijerint_definite_4 to try to compute the
integral. If this fails, it tries using linearity.
Fz#Expanding and evaluating all terms.c              3  (   "   T F  qR Jx  K
  	  R # 5ir`   r   )rb   r   s   & re   rf   (_meijerint_definite_3.<locals>.<genexpr>  s     +d}ds   TN)r2  is_Addr~  rj   ri   r   r   rR   )rd   r{   rl   r  ressr  r   r   s   &&      re   r  r    s      
%C
s1v
xxx4556VV<V%a+V<3+d+333+d+++E&&C  UAEzv  , <s   C&c                *    \        \        V 4      4      # r`   )rl  r$   )rd   s   &re   r3  r3    s    jm$$rm   c                h   ^ RI Hp \        RV 4       V'       g   \        WRR7      pVe   Vw  rVrx\        RWVV4       \        P
                  p	V Fg  w  rp V
^ 8X  d   K  \        WZ,          WaV,          ,          W4      w  rW\        W4      ,          ,          p	\        V\        W4      4      pVR8X  g   Kg   M	  \        V4      pVR8X  d   \        R4       M\        RV	4       \        V! V	4      4      V3# \        W4      pVEe+   R EF!  pVw  rVrp\        R	WVW4       \        P
                  p	V F  w  pppV F  w  ppp\        W_,          V,          WaVV,           ,          ,          VVW4      pVf   \        R
4          R# Vw  p
pp\        RV
VV4       \        V\        VVV4      4      pVR8X  d    M W\        VVV4      ,          ,          p	K  	  K   M	  \        V4      pVR8X  d   \        RV4       K  \        RV	34       V'       d   W3u # \        V! V	4      4      V3u # 	  R# R# )aa  
Try to integrate f dx from zero to infinity.

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

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

The parameter ``only_double`` is used internally in the recursive algorithm
to disable trying to rewrite f as a single G-function.
r.  IntegratingF)r>  N#Could rewrite as single G function:But cond is always False.z&Result before branch substitutions is: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: %sr  )r   r/  r~  rD  r   r   rx  r  rR   r  r3  rQ  r  r  r  r  )rd   r{   r-  r/  r
  r   r  r  r   rl   r  ru  rq  r  r  r  s1f1r  s2f2r   f1_f2_s   &&&                     re   r2  r2    s    +
=!qu->!CQ8#1E&&Ca6(a4>1((4!5a!;<5=  "$'Du}23?E%k#&67== 
1B	~$G$&!CRT6H&&C 
B"$JBB'r	2"r'l?(*B<Ay89"#KAsC63Dt%7S!%DEDu}WS#q111C #%  !  "$'Du}@'JCcWM9$%k#&67==7 % rm   c           	        T pTp\        RRR7      pV P                  WB4      p \        RV 4       \        W4      '       g   \        R4       R# \        P
                  pV P                  '       d   \        V P                  4      pM\        V \        4      '       d   V .pMRpV'       Ed   . p. pV'       Edu   VP                  4       p	\        V	\        4      '       d   \        V	4      p
V
P                  '       d   WjP                  ,          pK`   \        V	P                  ^ ,          V4      w  rV^8X  d   VP                  X4       K  VP                  V	4       K  V	P                   '       d   \        V	4      p
V
P                  '       d   WjP                  ,          pK  WP"                  P$                  9  dL    \        V	P                  V4      w  rV^8X  d,   VP                  X\'        V	P"                  4      ,          4       VP                  V	4       EKi  VP                  V	4       EK}  \)        V!  p\+        V!  p WP$                  9  dv   \        RW4       \-        \/        V4      ^ 4      pVR8X  d   \        R	4       R# V \1        W%,           4      ,          p\        R
W4       \3        VP                  W$4      V34      # \5        W4      pVEe   Vw  pppp\        RVVV4       \        P
                  pV Fd  w  ppp \7        VV,          VVV,          ,          W4      w  pp VV\9        WV4      ,          ,          p\;        V\=        W4      4      pVR8X  g   Kd   M	  \?        V4      pVR8X  d   \        R4       R# \        RV4       ^ RI H!p \?        V! V4      4      pVPE                  \F        4      '       g   V\G        V4      ,          pVP                  W"V,           4      p\        V\H        4      '       g   VP                  W"V,           4      p^RI%H&p \3        VP                  W$4      V3V! VP                  W$4      WR4      R34      # R#   \         d    ^ p EL6i ; i  \         d    ^ p ELi ; i)a  
Compute the inverse laplace transform
$\int_{c+i\infty}^{c-i\infty} f(x) e^{tx}\, dx$,
for real c larger than the real part of all singularities of ``f``.

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

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

Examples
========

>>> from sympy.abc import x, t
>>> from sympy.integrals.meijerint import meijerint_inversion
>>> meijerint_inversion(1/x, x, t)
Heaviside(t)
r   Tr  zLaplace-invertingzBut expression is not analytic.Nz.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:r.  )InverseLaplaceTransform)'r   r   r~  r8  r   r   is_Mulr   rj   r   r*   popr   r   r   r   r   r   r   r,   r   r   r   r    r>   r4   rD  r  r  rR   r  r3  r   r/  rk   r?   rj  r7  r  )rd   r{   r   r?  t_shiftrj   rh  exponentialsr!   rd  r   r   r   rl   r
  r   r  r  r  ru  r/  r  s   &&&                    re   meijerint_inversionr  !  s   $ 
B	
BcA	rA
"01 FFExxxAFF|	As		std((*C#s##c{;;;II%D)#((1+q9DA 6 ''*NN3'c{;;;II%DHH111-cggq9 Av$++Ac#((mO<s#s#\"M?J"U)Q5=HI
19%%BCN#((1/4011	1B	~RD4c2qAffGAq!%c!eR1Wa;DAq1^A!,,,Ct9!?@Du}  d#5=./7=2 S!12C779%%y|#((1%i(CdD))yyI.;chhqot45bgganaTRTXY[ [/ I + A / s$   Q 9Q( Q%$Q%(Q98Q9r   )F)__conditional_annotations__r   
__future__r   r  sympyr   
sympy.corer   r   sympy.core.addr   sympy.core.basicr   sympy.core.cacher   sympy.core.containersr	   sympy.core.exprtoolsr
   sympy.core.functionr   r   r   r   r   sympy.core.mulr   sympy.core.intfuncr   sympy.core.numbersr   r   sympy.core.relationalr   r   r   sympy.core.sortingr   r   sympy.core.symbolr   r   r   r   sympy.core.sympifyr   (sympy.functions.combinatorial.factorialsr   $sympy.functions.elementary.complexesr   r    r!   r"   r#   r$   r%   r&   r'   r(   r)   &sympy.functions.elementary.exponentialr*   r+   r,   #sympy.functions.elementary.integersr-   %sympy.functions.elementary.hyperbolicr.   r/   r0   r1   (sympy.functions.elementary.miscellaneousr3   $sympy.functions.elementary.piecewiser4   r5   (sympy.functions.elementary.trigonometricr6   r7   r8   r9   sympy.functions.special.besselr:   r;   r<   r=   'sympy.functions.special.delta_functionsr>   r?   *sympy.functions.special.elliptic_integralsr@   rA   'sympy.functions.special.error_functionsrB   rC   rD   rE   rF   rG   rH   rI   rJ   rK   rL   'sympy.functions.special.gamma_functionsrM   sympy.functions.special.hyperrN   rO   -sympy.functions.special.singularity_functionsrP   	integralsrQ   sympy.logic.boolalgrR   rS   rT   rU   rV   sympy.polysrW   rX   sympy.utilities.iterablesrY   sympy.utilities.miscrZ   r~  r[   r  r\   ra   r   sympy.utilities.timeutilsr   timeitr   r:  r   r   r   r   r  r  r  r  r  r$  r&  r)  __annotations__r0  r+  r8  rW  rl  rr  rx  r  r  r  r  r  r  r  r  r8  rB  rD  rQ  rW  rV  rp  r  rr  r  r3  r2  r  )r  s   @re   <module>r     s(  8 #     " $ ' -8 8  # + : : 8 : : & >    G F 79 9 9 J  M M I M6 6 6 6 9 8 M  J J & 9 0 2 
#JJPb /	)		N	* 	HDBI
!H$N.I>0 QD4 +-
& ,	#Oyv"	*m`&<F.Rj`	2:	Qwt  	I  	IX#3:"&JWJt G& G&T@<0% D> D>Nn[rm   