
    sgW                     0   d dl mZ d dlmZ d dlmZmZ d dlmZ d dl	m
Z
mZmZmZ d dlmZ d dlmZ d dlmZ d d	lmZmZ d d
lmZmZ d dlmZ d dlmZmZmZm Z m!Z! d dl"m#Z# d dl$m%Z% d dl&m'Z' d dl(m)Z) d dl*m+Z+ d dl,m-Z- d dl*m.Z.m/Z/m0Z0m1Z1m2Z2m3Z3m4Z4m5Z5m6Z6 d dl7m8Z8m9Z9m:Z:  ed      \  Z;Z<Z= edd      \  Z>Z?Z@ ed       ed      cZAZBd ZCd ZDd ZEd ZFd ZGd  ZHe:d!        ZIe:d"        ZJd# ZKd$ ZLd% ZMd& ZNd' ZOe9d(        ZPd) ZQd* ZRd+ ZSd, ZTd- ZUd. ZVy/)0    )Sum)Add)
DerivativeFunction)Mul)IRationaloopi)S)symbols)	factorial)explog)acoshasech)sqrt)acosasinatancossin)airyai)erf)gamma)	integrate)fps)O)	rational_algorithmFormalPowerSeriesFormalPowerSeriesProductFormalPowerSeriesComposeFormalPowerSeriesInversesimpleDErational_independentexp_rehyper_re)raisesXFAILslowzx y zzn m kT)integerfrc                  J
   dt         dz
  dz  t         dz
  z  z  } t        | t         t              dt         dz
  z   dz   t        t        dz         t        t              z  z
  ddfk(  sJ dt         z   t         dz  z   t         dz  z   t         dz
  t         dz
  z  z  } t        | t         t              ddt         dz
  z  z  dz   t         dz   dfk(  sJ t        t
        t        z  t        t         z  z
  t
        t         z  z
  t         dz  z   z  } t        | t         t              t
        t         dz
  z   t        z  t
        t        z
  z  t        t         dz
  z  t        z  t
        t        z
  z  z   ddfk(  sJ t         dt         z
  t         dz  z
  z  } t        | t         t              J t        | t         t        d      t        d	d      t        d
      dz  z   t         dz
  z  t        d
       dz  t        j                  z   z  t        d
       dz  t        j                  z
  t         dz
  z  t        d
      dz  t        j                  z   z  z   ddfk(  sJ dt         dz  dt         z  z   dz   z  } t        | t         t              J t        | t         t        d      t        d	t        z   t         dz
  z  z  dz  t        d	t        z
  t         dz
  z  z  dz  z
  ddfk(  sJ t        dt         z         } t        | t         t              d	t         z   t        z  ddfk(  sJ t        t               } t        | t         t              J t        | t         t        d      t        t        t         z  z  dz  t        t         t         z  z  dz  z
  t        z  ddfk(  sJ t         t        t               z  t        dt         dz  z         dz  z
  } t        | t         t              J t        | t         t        d      t        t        t         dz   z  z  dz  t        t         t         dz   z  z  dz  z
  t        t        dz
  z  z  ddfk(  sJ t        dt         z   dt         z
  z        dz  t        t               z
  } t        | t         t              J t        | t         t        d      d	t         z   dz  t        t        t         z  z  dz  z
  t        t         t         z  z  dz  z   t        j                  z   t        z  ddfk(  sJ t        t        t               t         t              J y )N      r      i   Tfull   
   )xr   kr   zymr	   r   r   Halfr   r   r   r   r,   s    Q/var/www/html/venv/lib/python3.12/site-packages/sympy/series/tests/test_formal.pytest_rational_algorithmr@      s   	a!eaZ1q5!"AaA&
qb1f+	Yq1u-	!<	=q!DE E E 
QA1	!a%AE!23AaA&	Q!a[1	a!eQ'( ( ( 	
QqS1Q3Y1_q!t#$AaA&QK<>a!e
$!qb1f+a-AE)B	CQJK K K 	
QUQT\AaA&...aAD1B?T!WQY&1"q&
17(2+
 G8A:1"q&
)
q'"*qvv

  "#A	'' ' ' 	
QTAaCZ!^AaA&...aAD1
R!VrAv
!	#q"q&QBF);';q&@	@!QGH H H 	AE
AaA&!*q!Q      	QAaA&...aAD1Q!W9/Qqbz\Q.
.!	3Q:; ; ; 	
$q'	CAqDMA%%AaA&...aAD1Q!a[=A
QB1"q&>!1Q 6
6
QU)
	   	QUq1u"T!W,AaA&...aAD11":+/QqA2wY!O
+q1"|q.@
@
&&
!	   c!fa+333    c            
      t   t         }  | g t              g k(  sJ  | t        t              t        t              gt              t        t              t        t              gk(  sJ  | t        dz  t        t              t        t        t              z  t        dz  gt              t        dz  t        dz  z   t        t        t              z  t        t              z   gk(  sJ  | t        j
                  t        t        t              z  t        t              t        t              t        z  t        t              t        t              t        gt              t        dz   t        t        t              z  t        t              z   t        t              t        z  t        t              z   t        t              gk(  sJ y )Nr0   r1   r/   )r%   r8   r   r   r   Oner   )ris    r?   test_rational_independentrE   P   s!   	Bb!9??s1vs1v"s1vs1v&6666q!tSVQs1vXq!t,a0	
A1aAhQ'() ) )quuaAhAAq#a&#a&!DaH	
Q#a&3q6!3q6!8c!f#4c!f=> > >rA   c            	      d   t        t        t              t        t              D ]8  } | t        t               t	        t        t              t              z   dfk(  sJ  n t        t        t              t        t              D ]<  } | t        t              t	        t        t              t        t              z   dfk(  sJ  n t        t        dt        z         t        t              D ]P  } | t        dz   t	        t        t              t        d      z  t	        t        t              t              z   dfk(  sJ  n t        t        t              t        t              D ]^  } | t        t	        t        t              t              z  t        dz  dz
  t	        t        t              t        t              z  z   dfk(  sJ  n t        t        t              t        t              z  t        t              D ][  } | dt        t              z  dt	        t        t                    z  z
  t	        t        t              t        t              z   dfk(  sJ  n t        dt        z   dt        z
  z  t        z  t        t              D ]N  } | dt        z  t        t              z  t        dz  dz
  t	        t        t              t              z  z   dfk(  sJ  n t        t        t              t        t              D ]D  } | t         t        t              z  t	        t        t              t        t              z   dfk(  sJ  y  y )Nr/   r0   )
r$   r   r8   r,   r   r   r   r   nr   )DEs    r?   test_simpleDErI   Z   s   s1vq!$ qtej1q1115555 s1vq!$ adZ!a33Q7777 s1q5z1a( q1uj1q!44z!A$7JJANNNN tAw1% a
1Q4++q!taxAaD!Q9O.OO  	  s1vc!f}a+ a!fqAaD!111JqtQ4JJANNNN QQ!+Q2 ac!A$h!Q$(JqtQ,?!??CCCC vay!Q' qb1g
1Q4A 66::::rA   c                  "   t        t               t        t        t              t              z   } t        | t        t
              t	        t
               t	        t
        dz         z   k(  sJ t        t              t        t        t              t        t              z   } t        | t        t
              t	        t
              t	        t
        dz         z   k(  sJ t        t              t        t        t              t              z   t        t        t              t        t              z   } t        | t        t
              t	        t
              t	        t
        dz         z   t	        t
        dz         z   k(  sJ t        t        t              t              t        t        t              t        t              z   } t        | t        t
              t	        t
              t	        t
        dz         z   k(  sJ t        t        t              t        d      t        t        t              t        d      z   t        t        t                    z   } t        | t        t
              t	        t
              t	        t
        dz         z   t	        t
        dz         z   k(  sJ y )Nr/   r0   r1   r2   )r,   r8   r   r&   r-   r9   ds    r?   test_exp_rerM   t   s   	
1
1Q4##A!Q?qteaAh....	!z!A$1%%A!Q?adQq1uXo---	!z!A$""Z!a%;;A!Q?adQq1uXo!a%88881Q4j1q!44A!Q?adQq1uXo---1Q4AAaD!Q!77*QqT:JJA!Q?adQq1uXo!a%8888rA   c                     t        t              t        t        t              t        t              z   } t        | t        t
              t	        t
              t
        dz   t
        dz   z  t	        t
        dz         z  z   k(  sJ t         t        t              z  t        t        t              t        t              z   } t        | t        t
              t
        dz   t
        dz   z  t	        t
        dz         z  t	        t
              z
  k(  sJ dt        t              z  dt        t        t              t              z  z
  t        t        t              t        t              z   } t        | t        t
              dt
        z  dz
  t	        t
        dz         z  t
        dz   t
        dz   z  t	        t
        dz         z  z   dt	        t
              z  z   k(  sJ dt        z  t        t              z  t        dz  dz
  t        t        t              t              z  z   } t        | t        t
              t
        t	        t
              z  dt        z  t	        t
        dz         z  z   t
         dz
  t	        t
        dz         z  z   k(  sJ t        dz  dz   t        t        t              t              z  t        t        dz  dz
  z  t        t        t              t        t              z  z   } t        | t        t
              t
        t
        dz
  z  t
        z   t	        t
              z  dt
        z  t
        dz   t
        dz   z  z
  dz   t	        t
        dz         z  z   k(  sJ t        dz  dz
  t        t        t              t        d      z  dt        z  t        t        t              t        t              z  z   t        t        t              t              z   } t        | t        t
              t
        t
        dz
  z  t
        dz
  z  dt
        z  t
        dz
  z  z   t
        z   t	        t
              z  t
         t
        dz   z  t
        dz   z  t	        t
        dz         z  z   k(  sJ y )	Nr/   r0   r1   r7   r2   	   (   )r,   r8   r   r'   r-   r9   rG   rK   s    r?   test_hyper_rerR      s2   	!z!A$1%%AAq!!!ac{1QU8'; ;;;;	
1Q4*QqT1a((AAq!QQ!a% 81Q4 ????	!A$:adA&&&AaD!Q)??AAq!	A1QU8q1uq1uoaAh661Q4?@ @ @ 	
!AaDAqD1Hj1q111AAq!	!A$1Qq1uX!a1q5 112 2 2 
BJqtQ''!QUQY-
1Q4A8N*NNAAq!	
AEQ!!q1uq2v&6 6 ;Qq2vYFFG G G Q$(JqtQ*	*QqSAaD!Q1G-G	G	AaD!	
AAq!
QU)QU
ac1q5k
)A
-qt	3"a!e*a!e
aAh	&
'( ( (rA   c                  v   t        d      dk(  sJ t        dt              dk(  sJ t        dt        d      dk(  sJ t        dt        d      dk(  sJ t        dt        z  dt        dz  z  z         dt        z  dt        dz  z  z   k(  sJ t        t        dt        z         dd      t        dt        z         k(  sJ t        t        dz  t        z   dz         } t        | t              sJ | j
                  t        dz  t        z   dz   k(  sJ | d   dk(  sJ | d   t        dz  k(  sJ | j                  d	      t        dz  t        z   dz   t        t        d	z        z   k(  sJ | j                         t        dz  t        z   dz   k(  sJ t        t        dt        z               } t        | t              sJ | j
                  t        dt        z         k(  sJ | j                  t        t              | k(  sJ | d d
 dt        t        dz   dz  t        dz  dz  t        d	z   d	z  gk(  sJ | j                  t              t        k(  sJ | j                  d      t        t        dz  dz  z
  t        dz  dz  z   t        d	z  d	z  z
  t        d
z  d
z  z   k(  sJ | j                  j                  d   }| j                  t        d| z   t        |z  z  |z  |dt         f      k(  sJ | j                  d       | d d
 }}t#        |      D ]  \  }}|d
k(  r n||   |k(  rJ  t%        t              j                  t              } t        | t              sJ | j                         t        t        dz  dz  z
  t        d
z  dz  z   t        t        dz        z   k(  sJ t'        t(        d        t'        t*        d        y )Nr/   r0   +dir-F)hyperrationalr   r2   r6   r1      r5   rG   x   c                  .    t        t        t        z        S N)r   r;   r8    rA   r?   <lambda>ztest_fps.<locals>.<lambda>   s    AaC rA   c                  $    t        t        d      S )Nr   rU   )r   r8   r_   rA   r?   r`   ztest_fps.<locals>.<lambda>   s    s1!} rA   )r   r8   r   
isinstancer    functiontruncater   
polynomialsubsr;   as_leading_termak	variablesinfiniter   r
   	enumerater   r(   NotImplementedError
ValueError)r,   r9   ftsits         r?   test_fpsrr      s\   q6Q;;q!9>>q!"""q!"""qsQq!tV|!a1f,,,s1q5z73q1u:EEEAqD1HqLAa*+++::AA%%%Q4199Q41a4<<::a=AqD1HqL1QT72222<<>QTAX\)))CAJAa*+++::QU###66!Q<1Ra5QAqD57AqDFQTE!G4444Q1$$$<<?a!Q$q&j1a461AqDF:QT!VCCCC	qA::qbzk!Q$.1Aq":>>>>JJJ"1B" 16tqyy
 	A

1Aa*+++::<1q!tAv:1S01QT7::::
 01
:,-rA   c                  "   t        dt        z        dt        z  k(  sJ t        t        dz  t        z   dz   t        dz  z  d      t        dz  t        z   dz   t        dz  z  k(  sJ dt        dz
  dz  t        dz
  z  z  } t        | t              j                         t        dd      t        t        dd      z  z
  dt        dz  z  d	z  z
  d
t        dz  z  dz  z
  dt        dz  z  dz  z
  dt        dz  z  dz  z
  t	        t        dz        z   k(  sJ dt        z   t        dz  z   t        dz  z   t        dz
  t        dz
  z  z  } t        | t              j                         t
        j                  t        t        dd      z  z   dt        dz  z  d	z  z   d
t        dz  z  dz  z   dt        dz  z  dz  z   dt        dz  z  dz  z   t	        t        dz        z   k(  sJ t        dt        z
  t        dz  z
  z  } t        | t        d      j                         t        t        dz  z   dt        dz  z  z   dt        dz  z  z   dt        dz  z  z   t	        t        dz        z   k(  sJ dt        dz  dt        z  z   dz   z  } t        | t        d      j                         t
        j                  t        dz  z
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  dz  t        df      z   k(  sJ t        | t        dd      j                         t        d      t        dd      z
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  dz  t        df      z   k(  sJ t        t              } t        | t        d      j                         t        t        dz  dz  z
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  dz  z  dz  z   t        dz  z   t	        t        dz
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  dz  t        df      z   k(  sJ t        t        t              z  t        dt        dz  z         dz  z
  } t        | t        d      j                         t        dz  dz  t        dz  d z  z
  t	        t        dz        z   k(  sJ t        dt        z   dt        z
  z        dz  t        t              z
  } t        | t        d      j                  d!"      dt        dz  z  dz  dt        d#z  z  d#z  z   t	        t        d!z        z   k(  sJ y )$Nr/   r0   r1   r5   rU   r6   r2         1             iA  @   rZ   q      Tr3      Q   D    )r4   rV         iw  iq  )   i	=     r7   r[      )	r   r8   rd   r	   r   r   r=   r   r   r>   s    r?   test_fps__rationalr      s   qs8!1q11$"-!Q$(Q,!Q$1FFFF	a!eaZ1q5!"Aq!9	"a1Xa^+	+bAgai	7"QT'"*	Ds1a4xPR{	R	QT"
Aw
      
QA1	!a%AE!23Aq!9	
!HQN"	"R1WQY	.AqD	;c!Q$hrk	I	QT"
Aw
      	
QUQT\Aq!$((*	AqD1QT6AadF"Qq!tV+a1g56 6 6 	
QTAaCZ!^Aq!$((*	1q!tAv1Q&Aa/!AqD'9: : : 	AE
Aq!9	AqDF
QT!Vad1f$q!tAv-!Q$78 8 8q!$$&#a*;*D*D*FFFFq!Q<  "	Q(1a.	 AEA:b=	0AEA:b=	@
a%!C
q51*T/
*,-aC
023QUQJA2G
HI I I q!QB((*	Q(1a.	 QBFQ;r>	1aR!VaKN	B"q&1S
BFQ;t+
,./c
245q1uqj1a&4I
JK K K 	QAq!$((*a!Q$q&j1a46.AAadG.KKKKq!$A&//1Aqt$--/0 0 0q!QT"++-	a8Aq>	!Aq1uqjLO	3b!a%!mC6G	G	
AEA:c	
AzM%/
023A#
6891q51*q!f8M
NO O O q!QTr*335	a8Aq>	!ArAvkM"$4	4rA26A+~c7I	I	
QBFQ;s	
!a!^E1
245aC
8:;QUQJA:O
PQ Q Q 	
$q'	CAqDMA%%Aq!$((*ad1fq!tBw.>1a4.HHHHQUq1u"T!W,Aq!$((2(.!AqD&(Qq!tVAX2E!R%2PPPPrA   c            
      v   t        t              } t        | t              j                         t        t        dz  dz  z
  t        dz  dz  z   t	        t        dz        z   k(  sJ t        t              } t        | t              j                         dt        dz  dz  z
  t        dz  dz  z   t	        t        dz        z   k(  sJ t        t              } t        | t              j                         dt        z   t        dz  dz  z   t        dz  dz  z   t        dz  dz  z   t        dz  dz  z   t	        t        dz        z   k(  sJ t        t              } t        | t              j                         t        t        dz  dz  z
  t        dz  dz  z   t	        t        dz        z   k(  sJ t        t        t                    } t        | t              j                         t        t        dz        t        t        t        dz        z  z
  t        dz  t        t        dz        z  dz  z   t        dz  t        t        dz        z  dz  z
  dt        dz  z  t        t        dz        z  dz  z   t        dz  t        t        dz        z  dz  z
  t	        t        dz        z   k(  sJ t        t        t                    } t        | t              j                         t        t        z   t        t        dz  z  dz  z
  t        t        dz  z  d	z  z
  t	        t        dz        z   k(  sJ t        dt        z        } t        | t              j                         t        dz  t        z
  t        dz  dz  z   t        dz  dz  z
  t	        t        dz        z   k(  sJ t        t        t              z  t        dt        dz  z         dz  z
  } t        | t        d
      j                         t        dz  dz  t        dz  dz  z
  t	        t        dz        z   k(  sJ t        dt        z         } t        | t        d
      j                         t        t        dz  dz  z
  t        dz  dz  z   t        dz  dz  z
  t        dz  dz  z   t	        t        dz        z   k(  sJ t        t        dz        } t        | t              j                         dt        dd      z  t        t        dd            z  dt        z  z  dt        dd      z  t        dz  z  dt        t        dd            z  z  z
  t	        t        dz        z   k(  sJ t        t              t        t              z  } t        | t              j                         t        t        dz  z   t        dz  dz  z   t        dz  dz  z
  t	        t        dz        z   k(  sJ t        t              t        t              z  t        z  } t        | t              j                         dt        z   t        dz  dz  z   t        dz  dz  z
  t        dz  dz  z
  t	        t        dz        z   k(  sJ t        t              t        t              z  } t        | t              j                         t        dt        dz  z  dz  z
  dt        dz  z  dz  z   t	        t        dz        z   k(  sJ y )Nr1   rZ   r6   r\   r/   r0   r2      ru   FrY   r      Z      )r   r8   r   rd   r   r   r   r   r   r   r   r   r   r   r	   r   r>   s    r?   test_fps__hyperr      sw   AAq!91q!tAv:1S#81QT7#BBBBAAq!91q!tAv:1R#7!AqD'#AAAAAAq!9	A1QAa!Q$r')AqDH4qAw>? ? ? 	QAq!91q!tAv:1Q#61a4#@@@@DGAq!9	RTQs2a4y[	 1a4BqD	>!#3	3ad3r!t9nQ6F	F	
1a4BqD		"	
 !tC1I~a/
023AqD'
:; ; ; 	E!HAq!91q51QT6!8#3a1fQh#>1a4#HHHHQqS	Aq!92a4!8ad1f#4q!tAv#=!Q$#GGGG	$q'	CAqDMA%%Aq!e$--/1a46AqDG3Ca1g3MMMMAE
Aq!e$--/	AqDF
QT!Vad1f$q!tAv-!Q$78 8 8 	q!tAq!9	
HQN	5!Q0	0!B$	7	
HQN	1a4	5!Q#8!8	9
:<=adG
DE E E 	As1vAq!91q!t8ad1f#4q!tBw#>1a4#HHHHAs1vaAq!91q51a46>AqDG#;ad2g#E!Q$#OOOOAQAq!91qAvax<!AqD&)#;a1g#EEEErA   c                     t         dz  t        t               z  } t        | t               j                         dt         dz  z  ddt         dz  z  z  z
  t	        dd      z   t         dz  dz  z
  t         dz  dz  z   t        t         dz        z   k(  sJ t         dz  t        t               z  } t        | t         d	
      j                         t         dz  t         dz  dz  z
  t        t         dz        z   k(  sJ t        t        t                     t         z  } t        | t               j                         t         t         dz  dz  z
  t         dz  dz  z   t         dz  dz  z
  t         dz  dz  z   t        t         dz        z   k(  sJ t         dz  t        t        t                     z  } t        | t               j                         t         dz  t         dz  dz  z
  t         dz  dz  z   t         dz  dz  z
  t        t         dz        z   k(  sJ y )Nr/   r2   rZ   r0   r\     选 Fr   r1   r6   r     逝  )	r8   r   r   rd   r	   r   r   r   r   r>   s    r?   test_fps_shiftr   ,  s   	2c!fAq!9	!Q$AadFhq#..Ad:QT&[H1QPQT7RS S S 	
1T!WAq!e$--/	1q!tAv!Q$      	DGQAq!9	AqDF
QT"Wq!tCx'!Q$u*4qAw>? ? ? 	
1Sa\Aq!9	1q!tAv1R!Q$s(*Qq!tW45 5 5rA   c                  t   t         t        t               z  t        dt         dz  z         dz  z
  } t        | t               j	                         t         dz  dz  t         dz  dz  z
  t        t         dz        z   k(  sJ t        t               t        t               z   t        t               z
  t        dt         z         z   } t        | t               j	                         t         dt         dz  z  dz  z
  t         dz  dz  z
  t         dz  dz  z   t        t         dz        z   k(  sJ dt         z  t        t               z   } t        | t               j	                         dt         z  t         z   t         dz  dz  z
  t         dz  dz  z   t        t         dz        z   k(  sJ t        t               t        t               z
  dt         dz
  z  z   } t        | t               j	                         d	t         dz  dz  z
  d
t         dz  z  dz  z
  dt         dz  z  dz  z
  dt         dz  z  dz  z
  t        t         dz        z   k(  sJ y )Nr/   r0   r2   r   rZ   r1   r6   r\   rO   r   r   r   w   )	r8   r   r   r   rd   r   r   r   r   r>   s    r?   test_fps__Add_exprr   >  s   	$q'	CAqDMA%%Aq!91a46AqDG#3a1g#====AQ#a& 3q1u:-Aq!91qAvax<!Q$q&#81a46#AAadG#KKKK	!c!fAq!91Q37QT!V#3ad3h#>1a4#HHHHAQ!QU)#Aq!9
QT!Va1fQhAqD+c!Q$hsl:Qq!tWDE E ErA   c                     t        t              } t        | t        t              | k(  sJ t        | t        t               j	                         t        dt        dz  z  t        t        f      k(  sJ t        t              } t        | t        t              j	                         dt        dt        dz  z  t        t        f      z   k(  sJ t        | t        t               j	                         dt        dt        dz  z  t        t        f      z   k(  sJ t        t              } t        | t        t        d      j	                         ddt        dz  z  z  ddt        dz  z  z  z   dt        z  z
  t        dz  z   t        dt        dz  z  t        t        f      z   k(  sJ t        | t        t         d      j	                         ddt        dz  z  z  ddt        dz  z  z  z   dt        z  z
  t        dz  z
  t        dt        dz  z  t        t        f      z   k(  sJ t        dt        z         } t        | t        t              ddt        dz  z  z  dd	t        d	z  z  z  z
  ddt        dz  z  z  z   ddt        dz  z  z  z
  dt        z  z   t        dt        z        z
  t        dt        dz  z  t        t        f      z   k7  sJ t        | t        t               ddt        dz  z  z  dd	t        d	z  z  z  z
  ddt        dz  z  z  z   ddt        dz  z  z  z
  dt        z  z   t        t        z  z   t        dt        z        z
  t        dt        dz  z  t        t        f      z   k7  sJ y )
Nr/   rZ   r5   Tr3   r6   r1   r0   r2   )r   r8   r   r
   rd   r   r   r   r   r   r   r>   s    r?   test_fps__asymptoticr   M  s   AAq!R=Aq!bS>""$!AqD&1b'(::::AAq!R=!!#q1Qq!tVaW+='====q!bS>""$QqAv2w-?(????QAq!Rd#,,.
AadFa1a4j 1Q3&A-!AqD&1b'0BBC C Cq!bSt$--/
AadFa1a4j 1Q3&A-!AqD&1b'0BBC C C 	AE
Aq!R=	Qq!tVq!AqD&z	!AqAvJ	.AadF	;ac	AC!H	L	
1QT6Ar7	
   q!bS>	Qq!tVq!AqD&z	!AqAvJ	.AadF	;ac	AAbD	H	RT
qAv2w'
() ) )rA   c                  	   t        t        t                    t        z  } t        | t              j	                         dt        t              z  t        t              dz  z
  t        t        dd      z  dz  z   t        t        dd      z  dz  z
  t        t        dd      z  d	z  z   t        t        d
d      z  dz  z
  t        t        dd      z  dz  z   t        t        dz        z   k(  sJ t        t        t                    t        z  } t        | t              j	                         t        t        dd      z  t        t        dd      z  dz  z
  t        t        dd      z  dz  z   t        t        d
d      z  dz  z
  t        t        dd      z  d	z  z   t        t        dz        z   k(  sJ t        t        t                    t        dz  z  } t        | t              j	                         t        t        dd      z  t        t        dd      z  dz  z
  t        t        j                  z  dz  z   t        t        dd      z  dz  z
  t        t        dd      z  d
z  z   t        t        dd      z  dz  z
  t        t        d
d      z  dz  z   t        t        dd      z  dz  z
  t        t        dz        z   k(  sJ t        t        t                    } t        | t              j	                         j                         dt        dz  z   t        dz  dz  z   t        dz  dz  z   t        dz  dz  z   t        dz  dz  z   t        t              z   t        t        dd      z  dz  z   t        t        dd      z  dz  z   t        t        dd      z  dz  z   t        t        d
d      z  d	z  z   t        t        dd      z  dz  z   t        t        dz        z   k(  sJ t        t        t                    t        z  } t        | t              j	                         j                         t        t        dz  dz  z   t        dz  dz  z   t        dz  dz  z   t        dz  dz  z   t        t        dd      z  z   t        t        dd      z  dz  z   t        t        dd      z  dz  z   t        t        d
d      z  dz  z   t        t        dd      z  d	z  z   t        t        dz        z   k(  sJ y )Nr/   rZ   r1   r0   r\   r6   r   r   r   rP   i ar   l    LQf r5      r   r   r   r2   r   i _7 )r   r   r8   r   rd   r	   r   r   r   r=   r   expandr>   s    r?   test_fps__fractionalr   e  s   DGqAq!9	
47T!WQY	HQN!23!6	6	
HQN	4	
 "#Xa^"3F":
;	
HQN	8	#
$&'"a&8&C
DFG1g
NO O O
 	DGqAq!9	
HQN	QA.q0	01hq!n3DS3H	H	
HQN	4	
 "#Xb!_"4V";
<>?1g
FG G G 	T!W1Aq!9	
HRO	a"a02	2QYq[	@	
HQN	1	
 (1a.02
356A5Fr5I
J	
HQN	2	
 !8B? 22 5
689!Q$
@A A A
 	DGAq!9&&(	
QqS1a47	QT#X	%1U
	2QT'\	ADG	K	
HQN	1	
 (1a.04
578(1a.7H7M
N	
HQN	6	!
"$%xA$6x$?
@BCAqD'
JK K K
 	DGQAq!9&&(	
QT!Vad2g	1S	(1a4:	58Aq>8I	I	
HQN	1	
 (1a.04
578(1a.7H7M
N	
HRO	F	"
#%&q!tW
-. . .rA   c                     t        ddt        z  z         } t        | t              t        t               t        z   t        dz  dz  z
  t        dz  dz  z   t        dz  dz  z
  t        dz  dz  z   t        t        dz        z   k7  sJ t        | t        d      t        t               t        z   t        dz  dz  z
  t        dz  dz  z   t        dz  dz  z
  t        dz  dz  z   t        t        dz        z   k7  sJ y )	Nr/   r0   r1   r2   r6   rZ   Fr   )r   r8   r   r   r>   s    r?   !test_fps__logarithmic_singularityr     s    A!GAq!9	Q!ad1fq!tAv%1Q.Aa7!AqD'AB B Bq!e$	Q!ad1fq!tAv%1Q.Aa7!AqD'AB B BrA   c                      t        t              } t        | t              t        d      t        t              z
  t        dz  dz  z
  dt        dz  z  dz  z
  t	        t        dz        z   k(  sJ y )Nr0   r2   r1   rz   rZ   )r   r8   r   r   r   r>   s    r?   &test_fps__logarithmic_singularity_failr     sV    aAq!9AQ!Q$q&01QT6"9<qAwFFFFrA   c                  .   t         t        z  t        t         dz        z  } t        | t               j	                  d      t         t        dz   z  t         t        dz   z  dz  z
  t        t         t        dz   z  t               z   k(  sJ t         t        z  t        dt         z         z  } t        | t               }|j                  j                  d   }|j                  t        d| z   t         |t        z   z  z  |z  |dt        f      k(  sJ t         dz
  t        z  t        dt         z         z  } t        | t         d      j	                         t         dz
  t        z  t        d      z  t         dz
  t        dz   z  dz  z   t         dz
  t        dz   z  dz  z
  t         dz
  t        dz   z  d	z  z   t         dz
  t        d
z   z  dz  z
  t         dz
  t        dz   z  dz  z   t        t         dz
  t        dz   z  t         df      z   k(  sJ t         t        dz
  z  t        t               z  } t        | t               j	                         t         t        dz
  z  t         t        z  dz  z
  t         t        dz   z  dz  z   t        t         t        d
z   z  t               z   k(  sJ t         t        dz
  z  t        t               z  t         t        z  t        t               z  z   } t        | t               j	                         t         t        dz
  z  t         t        dz   z  z   t         t        dz   z  dz  z   t         t        z  z   t         t        d
z   z  dz  z   t         t        dz   z  dz  z   t        t         t        dz   z  t               z   k(  sJ t         t        z  t        t               z  } t        | t         t              j	                         t         t        dz
  z   dz  t         t        dz
  z  dz  z   t         t        z  t        dz  dt         z  z
  z  z   t        dt         z  t         z  t         dz  z  t         t        f      z   k(  sJ t         t        dz  z  t        t               z  } t        | t               j	                         t         t        dz  z  t         t        dz  dz   z  dz  z
  t         t        dz  d
z   z  dz  z   t        t         t        dz  dz   z  t               z   k(  sJ t         t        t         z   z  t        t               z  } t        | t               j	                         t         t         t        z   dz   z  t         t         t        z   dz   z  dz  z
  t         t         t        z   dz   z  dz  z   t        t         t         t        z   dz   z  t               z   k(  sJ y )Nr0   ru   rZ   r/   r   r5   r1   r}   r~   r2   r   r6   r   r   <   r\   )r8   rG   r   r   rd   r   r   rh   ri   rj   r   r
   r   r   r   r   r<   )r,   fpr9   s      r?   test_fps_symbolicr     s   	1SAYAq!9a AAJQUA$=!a!e*a@P$PPPP	1SQZA	QB
A;;raRj[QU#Q&Ar
34 4 4 
Q
3q1u:Aq!Q<  "
a%!CF	a!eq1u-a/	/1q5AE2B22E	EQRSVWRWHXY[H[	[
a%1q5	#	
!"Q!a% 0 5
6891q5AE:JQPQF8S
TU U U 	
AE
3q6Aq!9	
QUad1f	q1q5z"}	,qQUQ/?	?A A A 	
AE
3q6AqDQK'Aq!9	
QUa!a%j	 1q1u:a<	/!Q$	6	
QUB
QUB
')*1q1u:q)9
:; ; ; 	
1T!WAq!R=!!#
a!e*QQUA	%1bdQqSj(9	9	
AaCA2;q!taW	%
&' ' ' 	
AaCQAq!9	AaC1qsQw<>!A!aLO3aAaC!Ga6HHI I I 	
AE
3q6Aq!9	AEAIQUQY))AA	N3,>>1q1uqy>STAUUV V VrA   c                     t         t        t               z  t        dt         z        z  } t        | t               j	                         dt         dz  z  dt         dz  z  z   t         dz  dz  z
  t         dz  z
  t        t         dz        z   k(  sJ y )Nr0   r1   r2   r6   rZ   )r8   r   r   r   rd   r   r>   s    r?   test_fps__slowr     sp    	#a&QqSAq!91QT6AadF?QT!V#;ad#BQq!tW#LLLLrA   c                  >   t        t        t                    t        t        t                    c} | z   }|j                  t        t              t        t              z   k(  sJ |j                         dt        z   t        dz  dz  z
  t        dz  dz  z
  t        dz  dz  z   t        dz  dz  z   t        t        dz        z   k(  sJ dz   }|j                  t        t              dz   k(  sJ |j                         dt        z   t        dz  dz  z
  t        dz  dz  z   t        t        dz        z   k(  sJ d| z   }|j                  t        t              dz   k(  sJ |j                         dt        dz  dz  z
  t        dz  dz  z   t        t        dz        z   k(  sJ t        z   t        t              k(  sJ | j                          d	t        dz  dz  z   t        dz  dz  z
  t        t        dz        z   k(  sJ z
  t        j                  u sJ | z
  }|j                  t        t              t        t              z
  k(  sJ |j                         d	t        z   t        dz  dz  z   t        dz  dz  z
  t        dz  dz  z
  t        dz  dz  z   t        t        dz        z   k(  sJ dz
  }|j                  t        t              dz
  k(  sJ |j                         d	t        z   t        dz  dz  z
  t        dz  dz  z   t        t        dz        z   k(  sJ d| z
  }|j                  t        t               dz   k(  sJ |j                         t        dz  dz  t        dz  dz  z
  t        t        dz        z   k(  sJ t        t        fd
       t        t        fd       dz  }|j                  dt        t              z  k(  sJ |j                         dt        z  t        dz  dz  z
  t        dz  dz  z   t        t        dz        z   k(  sJ d| z  }|j                  dt        t              z  k(  sJ |j                         ddt        dz  z  dz  z
  t        dz  dz  z   t        t        dz        z   k(  sJ | z  t        |       k(  sJ t        z  t        t              k(  sJ j                         }|j                  t        t              k(  sJ |j                         dt        dz  dz  z
  t        dz  dz  z   t        t        dz        z   k(  sJ | j                         }|j                  t        t               k(  sJ |j                         t         t        dz  dz  z   t        dz  dz  z
  t        t        dz        z   k(  sJ | j                         j                         }|j                  t        t               k(  sJ |j                         d	t        dz  dz  z   t        dz  dz  z
  t        t        dz        z   k(  sJ t        t        t        t                          }|j                         }|j                         j!                         ddt        t              z  z  t        j"                  z   t        dz  z   t        dz  dz  z   t        dz  dz  z   t        dz  dz  z   t        dz  dz  z   t        t              dz  z   t        t%        dd      z  dz  z   t        t%        dd      z  dz  z   t        t%        dd      z  dz  z   t        t%        dd      z  dz  z   t        t%        dd      z  dz  z   t        t        dz        z   k(  sJ j'                  t        ddf      t        d       dz   k(  sJ t'        t        ddf      t        d       dz   k(  sJ t'        t              }|j                  t        t               k(  sJ |j                         d	t        dz  dz  z   t        dz  dz  z
  t        t        dz        z   k(  sJ | j'                  t              }|j                  t        t              k(  sJ |j                         t        t        dz  dz  z
  t        dz  dz  z   t        t        dz        z   k(  sJ y )Nr/   r0   r1   rZ   r2   r   r6   r\   r5   c                  >     t        t        t              d      z   S )Nr5   rU   r   r   r8   f1s   r?   r`   z&test_fps__operations.<locals>.<lambda>  s    rCAB$77 rA   c                  >     t        t        t              d      z   S )Nr/   x0r   r   s   r?   r`   z&test_fps__operations.<locals>.<lambda>  s    rCA1$55 rA   rQ   ru   r      i`'  i  i *0   i  r   i ; rP   i n r   i 9r   )r   r   r8   r   rc   rd   r   r   r   Zeror(   rm   r   diffr   r   r   r=   r	   r   )f2fsumfsubfmfdf3fir   s          @r?   test_fps__operationsr     sw   Q[#c!f+FB7D==CFSVO+++==?	A1QAa!Q$r')AqDH4qAw>? ? ? 6D==CFQJ&&&==?a!ead1fnq!tCx7!AqD'AAAAr6D==CFQJ&&&==?a!Q$q&j1a472Qq!tW<<<<Fs2qz!!!KKM>R!Q$q&[1a472Qq!tW<<<<G7D==CFSVO+++==?
QAa!Q$q& 1a47*QT#X5!Q$?@ @ @ 6D==CFQJ&&&==?b1fq!tAvo1S81QT7BBBBr6D==SVGaK'''==?ad1fq!tBw.1a48888
:78
:56	aB;;!CF(""";;=AaC!Q$q&L1a472Qq!tW<<<<	
RB;;!CF(""";;=A!Q$qL1a461AadG;;;;GB###Fs2qz!!!	B;;#a&   ;;=A1QJAb01QT7::::	B;;3q6'!!!;;=QBAaK!Q$s(2Qq!tW<<<<		B;;3q6'!!!;;=BAaK!Q$r'1AadG;;;;	Sa\	B	B;;=!	
Ad1gI	!B$	&Ac	1AqDJ	>Af	L	
Ah
a
#%&A%6r%9
:<=x1~<Md<R
S	
HQN	5	 
!#$hq!n#4W#<
=?@(2q/?QR[?[
\ 
1a4
   <<Aq	"s1vgk111R!Q#Aw{222	2q	B;;3q6'!!!;;=BAaK!Q$r'1AadG;;;;	aB;;#a&   ;;=A1QJAc1AadG;;;;rA   c                     t        t        t                    t        t        t                    t        t	        t                    c} }t        t        fd       t        t        fd       t        t        fd       t        t        fd       j                  | t              t        t              sJ t        j                  t              sJ t        j                  t              sJ j                  t        t              k(  sJ j                  t        t              k(  sJ j                  t        t              t        t              z  k(  sJ j!                  d      t        t        dz  z   t        dz  dz  z   k(  sJ j#                  d      t        t        dz  z   t        dz  dz  z   t%        t        dz        z   k(  sJ j'                  d      t        t        dz  z   t        dz  dz  z   k(  sJ t        t(        fd       t        t(        fd	       t        t(        fd
       t        t(        fd       j                  |t              j!                  d      t        dt        dz  z  dz  z
  k(  sJ j                  |t              j#                  d      t        dt        dz  z  dz  z
  t%        t        dz        z   k(  sJ y )Nc                  J     j                  t        t              t              S r^   )productr   r8   r   s   r?   r`   z#test_fps__product.<locals>.<lambda>      rzz#a&!4 rA   c                  b     j                  t        t        t              d      t        d      S Nr5   rU   r2   r   r   r   r8   r   s   r?   r`   z#test_fps__product.<locals>.<lambda>      rzz#c!f"*=q!D rA   c                  b     j                  t        t        t              d      t        d      S Nr/   r   r2   r   r   s   r?   r`   z#test_fps__product.<locals>.<lambda>      rzz#c!f*;QB rA   c                  ^     j                  t        t        t                    t        d      S Nr2   )r   r   r   r;   r8   r   s   r?   r`   z#test_fps__product.<locals>.<lambda>      rzz#c!f+q!< rA   r2   r0   r1   c                  &     j                  d      S Nr6   
_eval_termfprods   r?   r`   z#test_fps__product.<locals>.<lambda>!      (8(8(; rA   c                       j                   S r^   rj   r   s   r?   r`   z#test_fps__product.<locals>.<lambda>"       rA   c                  .     j                  t              S r^   _eval_derivativer8   r   s   r?   r`   z#test_fps__product.<locals>.<lambda>#      (>(>q(A rA   c                  .     j                  t              S r^   r   r8   r   s   r?   r`   z#test_fps__product.<locals>.<lambda>$      (: rA   )r   r   r8   r   r   r(   rm   r   rb   r!   ffpsr    gfpsr,   grc   _eval_termsrd   r   re   rl   )r   r   r   r   s     @@r?   test_fps__productr     s%   SVc#a&k3s1v;JBB
:45
:DE
:BC
:<=JJr1Ee5666ejj"3444ejj"344477c!f77c!f>>SVc!f_,,,Q1q!t8ad1f#4444>>!AqD1a46 1AadG ;;;;A!ad(QT!V"3333
 ;<
 67
 AB
 :;::b!((+q1QT6!8|;;;::b!%%a(A!Q$qL1QT7,BBBBrA   c                  
    t        t        t                    t        t        t                    t        t	        t                    c t        t         fd       t        t         fd       t        t         fd       t        t         fd       t        t         fd       t        t        fd        j                  t              t        t              sJ t        j                  t              sJ t        j                  t              sJ j                  t        t              k(  sJ j                  t        t              k(  sJ j                  t        t        t                    k(  sJ j!                  d      dt        z   t        d	z  d	z  z   t        d
z  dz  z
  t        dz  dz  z
  k(  sJ j#                         dt        z   t        d	z  d	z  z   t        d
z  dz  z
  t        dz  dz  z
  t%        t        dz        z   k(  sJ j#                  d      dt        z   t        d	z  d	z  z   t        d
z  dz  z
  t%        t        dz        z   k(  sJ t        t&        fd       t        t&        fd       t        t&        fd       t        t&        fd        j                  t              j#                  d
      dt        z   t        d	z  d	z  z   t%        t        d
z        z   k(  sJ  j                  t              j#                  d      dt        z   t        d	z  d	z  z   t        d
z  dz  z
  t        dz  dz  z
  t        dz  dz  z
  t        dz  dz  z   t%        t        dz        z   k(  sJ  j                  t              j#                  d      dt        z   t        d	z  d	z  z   t        d
z  dz  z
  t        dz  dz  z
  t%        t        dz        z   k(  sJ j                  t              j#                  d
      t        t        dz  dz  z
  t%        t        d
z        z   k(  sJ j                  t              j#                  d      t        t        dz  dz  z
  t        dz  dz  z   dt        dz  z  dz  z
  t%        t        dz        z   k(  sJ j                  t              j#                  d      t        t        dz  dz  z
  t        dz  dz  z   t%        t        dz        z   k(  sJ y )Nc                  J     j                  t        t              t              S r^   )composer   r8   r   s   r?   r`   z#test_fps__compose.<locals>.<lambda>-  r   rA   c                  b     j                  t        t        t              d      t        d      S r   r   r   r   r8   r   s   r?   r`   z#test_fps__compose.<locals>.<lambda>.  r   rA   c                  b     j                  t        t        t              d      t        d      S r   r   r   s   r?   r`   z#test_fps__compose.<locals>.<lambda>/  r   rA   c                  ^     j                  t        t        t                    t        d      S r   )r   r   r   r;   r8   r   s   r?   r`   z#test_fps__compose.<locals>.<lambda>0  r   rA   c                  0     j                  t              S r^   r   r8   )r   r   s   r?   r`   z#test_fps__compose.<locals>.<lambda>2      rzz"a0 rA   c                  0     j                  t              S r^   r   )r   r   s   r?   r`   z#test_fps__compose.<locals>.<lambda>3  r   rA   rZ   r/   r0   r2   ru   r6   r   c                  &     j                  d      S r   r   fcomps   r?   r`   z#test_fps__compose.<locals>.<lambda>@  r   rA   c                       j                   S r^   r   r   s   r?   r`   z#test_fps__compose.<locals>.<lambda>A  r   rA   c                  .     j                  t              S r^   r   r   s   r?   r`   z#test_fps__compose.<locals>.<lambda>B  r   rA   c                  .     j                  t              S r^   r   r   s   r?   r`   z#test_fps__compose.<locals>.<lambda>C  r   rA   r   r   r   r1   r7   i;  )r   r   r8   r   r   r(   rm   r   rb   r"   r   r    r   r,   r   rc   r   rd   r   rl   )r   r   r   r   s   @@@@r?   test_fps__composer   *  s   SVc#a&k3s1v;JBB
:45
:DE
:BC
:<=
:01
:01JJr1Ee5666ejj"3444ejj"344477c!f77c!f>>SQ[(((Q1q51a46>AqDF#:QT"W#DDDD>>q1uq!tAv~1Q6Ab@1QT7JJJJ>>!A1QAa 7!AqD' AAAA
 ;<
 67
 AB
 :;::b!%%a(AEAqDFNQq!tW,DDDD::b!%%a(	A1QAa!Q$r')AqDH4q!tBw>1a4HI I I::b!%%a(	A1QAa!Q$r')AadG34 4 4 ::b!%%a(A1QJ1a4,@@@@::b!%%a(A1QJAb,@1QT6#:,MPQRSUVRVPW,WWWW::b!%%a(A1QJAb,@1QT7,JJJJrA   c                  D   t        t        t                    t        t        t                    t        t	        t                    c} }t        t        fd       | j                  t              t        t              sJ t        j                  t              sJ t        t        fd       j                  t        t              k(  sJ j                  t        t               k(  sJ j                  d      dt        z
  t        dz  dz  z   t        dz  dz  z
  t        dz  d	z  z   k(  sJ j                         dt        z
  t        dz  dz  z   t        dz  dz  z
  t        dz  d	z  z   t        dz  d
z  z
  t!        t        dz        z   k(  sJ j                  d      dt        z
  t        dz  dz  z   t        dz  dz  z
  t        dz  d	z  z   t!        t        dz        z   k(  sJ t        t"        fd       t        t        fd       t        t"        fd       t        t"        fd       t        t"        fd       | j                  t              j                  d      dt        z
  t        dz  dz  z   t        dz  dz  z
  t        dz  d	z  z   t        dz  d
z  z
  t        dz  dz  z   t        dz  dz  z
  t!        t        dz        z   k(  sJ |j                  t              j                         dt        dz  dz  z   dt        dz  z  d	z  z   t!        t        dz        z   k(  sJ |j                  t              j                  d      dt        dz  dz  z   dt        dz  z  d	z  z   dt        dz  z  dz  z   t!        t        dz        z   k(  sJ y )Nc                  .     j                  t              S r^   )inverser8   r   s   r?   r`   z#test_fps__inverse.<locals>.<lambda>S  s    rzz!} rA   c                       j                   S r^   )r   finvs   r?   r`   z#test_fps__inverse.<locals>.<lambda>X  s    tyy rA   r6   r/   r0   r1   rZ   r2   r   r\   c                  &     j                  d      S r   r   r   s   r?   r`   z#test_fps__inverse.<locals>.<lambda>`  s    (: rA   c                       j                   S r^   )r   r   s   r?   r`   z#test_fps__inverse.<locals>.<lambda>a  s    tvv rA   c                       j                   S r^   r   r   s   r?   r`   z#test_fps__inverse.<locals>.<lambda>b  s     rA   c                  .     j                  t              S r^   r   r   s   r?   r`   z#test_fps__inverse.<locals>.<lambda>c  s    (=(=a(@ rA   c                  .     j                  t              S r^   r   r   s   r?   r`   z#test_fps__inverse.<locals>.<lambda>d  s    q(9 rA   ru   r   r   r   =   )r   r   r8   r   r   r(   rm   r   rb   r#   r   r    r,   rc   r   rd   r   rl   )r   r   r   r   s     @@r?   test_fps__inverser   P  s   SVc#a&k3s1v;JBB
:,-::a=Dd4555dii!2333
:()66SV==CG###A!a%!Q$q&.1a46"9AqDG"CCCC==?a!ead1fnq!tAv51R?!Q$s(JQqRStWTTTT==q1uq!tAv~1Q6Ab@1QT7JJJJ
 :;
:~&
 56
 @A
 9:::a=!!!$	A1QAa!Q$r')AqDH4q!tCx?!Q$t)KaPQSTPTgUV V V ::a=!!#q1a46zAadF2I'=!Q$'GGGG::a=!!!$AqDF
Qq!tVBY(>AqD(LqQRTUQUw(VVVVrA   N)Wsympy.concrete.summationsr   sympy.core.addr   sympy.core.functionr   r   sympy.core.mulr   sympy.core.numbersr   r	   r
   r   sympy.core.singletonr   sympy.core.symbolr   (sympy.functions.combinatorial.factorialsr   &sympy.functions.elementary.exponentialr   r   %sympy.functions.elementary.hyperbolicr   r   (sympy.functions.elementary.miscellaneousr   (sympy.functions.elementary.trigonometricr   r   r   r   r   sympy.functions.special.besselr   'sympy.functions.special.error_functionsr   'sympy.functions.special.gamma_functionsr   sympy.integrals.integralsr   sympy.series.formalr   sympy.series.orderr   r   r    r!   r"   r#   r$   r%   r&   r'   sympy.testing.pytestr(   r)   r*   r8   r;   r:   rG   r<   r9   r,   r-   r@   rE   rI   rM   rR   rr   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r_   rA   r?   <module>r     s6   )  6  4 4 " % > = @ 9 Q Q 1 7 9 / #  I I I 5 4
'
1a
'4
(1a}hsm1/4d>49"(4&.R 0Q 0Qf ,F ,F^5$E)0.>B G G
#VLM
O<bC8#KLWrA   