
    glG                        d dl mZ d dlmZ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mZ d dlmZ d dlmZmZ d dlmZmZmZmZmZmZmZmZ d d	lmZm Z m!Z! d d
l"m#Z#m$Z$ d dl%m&Z&m'Z'm(Z(m)Z)m*Z* d dl+m,Z, d dl-m.Z. d dl/m0Z0m1Z1m2Z2 d dl3m4Z4m5Z5 d Z6d Z7d Z8d Z9d Z:d Z;d Z<d Z=d Z>d Z?d Z@d ZAd ZBd ZCd ZDd ZEd  ZFe5d!             ZGd" ZHd# ZId$ ZJd% ZKd& ZLd' ZMd( ZNd) ZOd* ZPd+ ZQd, ZRd- ZSd. ZTd/ ZUd0 ZVd1 ZWd2 ZXd3 ZYd4 ZZd5 Z[d6 Z\d7 Z]d8 Z^d9 Z_d: Z`d; Zad< Zbd= Zcd> Zdd? Zed@ ZfdA ZgdB ZhdC ZidD ZjdE ZkdF ZldG ZmdH ZndI ZodJ ZpdK ZqdL ZrdM ZsdN ZtdO ZudP ZvdQ ZwdR ZxdS ZydT ZzdU Z{dVS )W    )AccumBounds)
Derivative	PoleError)EIIntegerRationalpi)S)Symbolsymbols)sign)explog)acoshacothasinhatanhcoshcothsinhtanh)ceilingfloorfrac)cbrtsqrt)asincoscotsintan)limit)O)xyz)raisesXFAILc                     t          j        t           d          t           k    sJ t          j        t           d          t          k    sJ dt           t          z  z                      t          d          dt           t          z  z  k    sJ t          dd                              t           d          t          dd          k    sJ t          j                    t           k    sJ d S )N   n         )r%   nseriesr&   r	        [/var/www/html/ai-engine/env/lib/python3.11/site-packages/sympy/series/tests/test_nseries.pytest_simple_1r5      s    9Q!!!!!9Q!!!!!qsGQ!$$1Q3////Aq>>!!!q!))Xa^^;;;;9;;!r3   c                      t           t          t                     z                      t           d          t           t          t                     z  k    sJ d S )Nr+   r,   )r%   r   r1   r2   r3   r4   
test_mul_0r7      s@    c!ffHa1%%3q66111111r3   c                  (   t           t          dt           z             z                      t           d          t           t          d          z  t           dz  dz  z   t           dz  dz  z
  t           dz  dz  z   t          t           dz            z   k    sJ t           t          dt           z             z                      t           d          t           dz  t           dz  dz  z
  t           dz  dz  z   t          t           dz            z   k    sJ d S )	N   r+   r,   r/      r0      r.   r%   r   r1   r$   r2   r3   r4   
test_mul_1r=      s    c!a%jjL!!!q!))Qs1vvX1Q->Aa-G	1R.AqD''.    c!a%jjL!!	Q "  a4!Q$q&=1a46)AadGG34 4 4 4 4 4r3   c                  r   t           dz                      t           d          t           dz  k    sJ dt           z                      t           d          dt           z  k    sJ dt           dz  z                      t           d          dt           dz  z  k    sJ t           t          dd          z                      t           d          t           t          dd          z  k    sJ t          t                     dz                      t           d          t          t                     dz  k    sJ d S )Nr9   r+   r,   r.   r/   )r%   r1   r	   r   r2   r3   r4   
test_pow_0r?   &   s   qD>>!q>!!QT))))aC==a=  AaC''''adFA##qAv----x1~~&&qA&..1hq!nn3DEEEEGGQJQ''DGGQJ777777r3   c                  D   dt           z   dz                      t           d          t           dz  dt           z  z   dz   k    sJ t          t                     dz   dz                      t                     dt          t                     z  t           z   dz   k    sJ t          t                     t          t                     z   dz                      t                     dt           t	          dd          z  z  t           t	          dd          z  z   t           z   k    sJ d S )Nr.   r9   r+   r,      r/   )r%   r1   r   r   r	   r2   r3   r4   
test_pow_1rB   .   s    UQJQ''1a4!A#:>9999 !WWq[1%%a((Ad1ggIMA,=====!WWtAww"++A..!Ax1~~4E2E
Xa^^
33             r3   c                     ddt           z
  z                      t           d          dt           z   t           dz  z   t           dz  z   t           dz  z   t          t           dz            z   k    sJ t           dt           z
  z                      t           d          t           t           dz  z   t           dz  z   t           dz  z   t           dz  z   t          t           dz            z   k    sJ t           dz  dt           z
  z                      t           d          t           dz  t           dz  z   t           dz  z   t           dz  z   t           d	z  z   t          t           dz            z   k    sJ d S )
Nr.   r+   r,   r9   r/   r0   rA   r:      r%   r1   r$   r2   r3   r4   test_geometric_1rF   7   s6   q1uIqA&&!a%!Q$,A*=1*DqAww*NNNNNq1uIqA&&!ad(QT/AqD*@1a4*G!AqD''*QQQQQqD!a%L!!!q!))QTAqD[1a4-?!Q$-F	1.Aww.      r3   c                     t          dt          z                                 t          d          dt          dz  z   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 d S )
Nr.   r+   r,   r9   r:   r/      r0      r   r%   r1   r$   r2   r3   r4   test_sqrt_1rK   >   s~    A;;qA&&!ac'AqDF*:QT"W*DqAvcz*QTUVWYZVZT[T[*[[[[[[[r3   c                     t          t                                        t          d          dt          z   t          dz  dz  z   t          dz  dz  z   t          dz  dz  z   t          t          dz            z   k    sJ t          t                                        t          d	          dt          z   t          dz  dz  z   t          dz  dz  z   t          dz  dz  z   t          dz  d
z  z   t          dz  dz  z   t          dz  dz  z   t          dz  dz  z   t          dz  dz  z   t          dz  dz  z   t          dz  dz  z   t          t          d	z            z   k    sJ t          dt          z                                t          d          t          dt          z            k    sJ t          ddt          z   z                                t          d          t          dt          z
  dt          dz  z  dz  z
  dt          dz  z  dz  z   z                                  t          t          dz            z   k    sJ t          dt          z                                 t          d          t          d          dt          z   t          dz  dz  z   t          dz  dz  z   t          dz  dz  z   z                                  t          t          dz            z   k    sJ d S )Nr+   r,   r.   r9   r/   rA   r0   r;      x     rD   i  r:   i  	   i 
   i _7    i a   )r   r%   r1   r$   r   expandr2   r3   r4   
test_exp_1rU   B   s   q66>>!q>!!QUQT!V^ad1f%<q!tBw%F1a4%PPPPPq66>>!r>""a!ead1fnq!tAv&=1R&G!Q$s(&R	1S'a49' !tEz'*,-qDK'8:;R%-'H	2h'1b5'" " " " " qs88A##s1Q3xx////q!a%y>>!!!q!))	
AEBq!tGAI!Q$q(	)1133a1gg=> > > >q1u::a1%%	QQAa!Q$q&(1a472	3;;==!Q$GH H H H H Hr3   c                     t          dt          t                    z                                 t          d          t          d          dt          t                    z   t          dz  z   t          t                    t          z  dz  z   z                                  t          t          t                    dz            z   k    sJ d S )Nr.   r/   r,   r9   rA   )r   r   r%   r1   rT   r$   r2   r3   r4   test_exp_sqrt_1rW   N   s    q477{##A#++	QT!WWqs"T!WWQYq[0	199;;aQ
mmKL L L L L Lr3   c                     t          t          t          t                    z                                t          d          dt          t          t                    z  z   t          dz  t          t                    dz  z  dz  z   t          dz  t          t                    dz  z  dz  z   t	          t          dz  t          t                    dz  z            z   k    sJ d S Nr0   r,   r.   r9   r/   rA   )r   r%   r   r1   r$   r2   r3   r4   test_power_x_x1rZ   S   s    #a&&MM""1"**	Ac!ffHq!tCFFAI~a''!Q$s1vvqy.*::Qq!tCFFAI~=N=NNO O O O O Or3   c                     t           t           z                      t           d          dt           t          t                     z  z   t           dz  t          t                     dz  z  dz  z   t           dz  t          t                     dz  z  dz  z   t          t           dz  t          t                     dz  z            z   k    sJ d S rY   )r%   r1   r   r$   r2   r3   r4   test_power_x_x2r\   X   s    qD>>!q>!!	Ac!ffHq!tCFFAI~a''!Q$s1vvqy.*::Qq!tCFFAI~=N=NNO O O O O Or3   c                  2   t          ddt          z  z                                 t          d          t          t          t                    z
  t          dz  dz  z
  t          dz  dz  z   t          dz  dz  z
  t          t          dz            z   k    sJ d S )Nr.   r+   r,   r9   r/   r0   )r   r%   r1   r$   r2   r3   r4   test_log_singular1r^   ]   s    q1Q3w<<Q''1s1vv:1Q+>Aa+G	1Q,1a4,      r3   c                  F   ddt           z  t           t          d          t          d          z  z  z   z  } |                     t           d          t           t          d          t          d          z  dz   z   t           z   t          t           dz            z   k    sJ d S )Nr.   r/   r9   r+   r,   r<   es    r4   test_log_power1rb   b   s    	QqS1QA''(A99Q!9SVVCFF]Q%6!7 7! ;a1gg EEEEEEEr3   c                      t          d          } ddt          t                    z
  z  }|                    t          d|           dd| z
  z  k    sJ d S )Nlr.   r+   r-   logxr   r   r%   r1   rd   ra   s     r4   test_log_seriesri   g   sQ    sA	1s1vv:A99Q!!9$$1q5	111111r3   c                      t          dt          z            } |                     t          d          t          t                     t          d          z   k    sJ d S )Nr+   r,   )r   r%   r1   r`   s    r4   	test_log2rl   m   sK    BqD		A99Q!9Q#b'' 1111111r3   c                      t          d          } dt          dt          z            z  }|                    t          d|           d|  t          d          z   z  k    sJ d S )Nrd   r.   rk   r0   re   rg   rh   s     r4   	test_log3rn   r   sZ    sA	#bd))A99Q!!9$$A2B<(8888888r3   c                  :   t          t                    } |                     t          dd          dk    sJ |                     t          dd          t          dt                    k    sJ |                     t          dd          t          t          t                    k    sJ |                     t          dd          t          t          t          dz  t                    z   k    sJ |                     t          dd          t          t          t          dz  t                    z   k    sJ |                     t          dd          t          t          dz  dz  z
  t          t          dz  t                    z   k    sJ t	          t                    dz
  t          z  } |                     t          dd          dt          dz  z   t          dz  dz  z   t          t          dz            z   k    sJ t          j        t          dd          t          k    sJ d S )Nr   r.   r9   r/   r0   rA   )r!   r%   r1   r$   r   r`   s    r4   test_series1rp   x   s   AA99Q1""""99Q11a((((99Q11a((((99Q1Qq!tQZZ////99Q1Qq!tQZZ////99Q1QT!Va1ajj!88888	Q!QA99Q1QqS1a46!1AadGG!;;;;;9Q1""""""r3   c                      t          j        t           dd          t          dt                     k    sJ t          j        t           dd          t          t           t                     k    sJ d S )Nr   r.   rE   r2   r3   r4   test_series1_failingrr      sV    9Q11a((((9Q11a((((((r3   c                      dt           z                      t           dd          dt           z  k    sJ t           dt           z  z                       t           dd          t           dt           z  z   k    sJ d S )Nr.   r   r/   )r%   r1   r2   r3   r4   test_seriesbug1rt      sd    aC==Aq!!QqS((((!GQ1%%QqS000000r3   c                     t           dz   dz                      t           dd          ddt           z  z
  dt           dz  z  z   dt           dz  z  z
  t          t           dz  t                     z   k    sJ t           dz   dz                      t           dd          dt           z
  t           dz  z   t           dz  z
  t          t           dz  t                     z   k    sJ t           dz   dz                      t           dd          dk    sJ t           dz   dz                      t           dd          dt           z   k    sJ t           dz   dz                      t           dd          t           dz  dt           z  z   dz   k    sJ t           dz   dz                      t           dd          ddt           z  z   dt           dz  z  z   t          t           dz            z   k    sJ ddt           z   z                      t           dd          dt           z
  t           dz  z   t           dz  z
  t          t           dz  t                     z   k    sJ t           dddt           z  z   z  z                       t           dd          ddt           z  z
  d	t           dz  z  z   d
t           dz  z  z
  t          t           dz  t                     z   k    sJ dt           z  dz   dz                      t           dd          ddt           z  z   dt           dz  z  z   t           dz  z   k    sJ dddt           z  z   z                      t           dd          t           t           dz  z
  t           dz  z   t          t           dz  t                     z
  k    sJ dddt           dz  z  z   z                      t           dd          t           dz  t           dz  z
  t          t           dz  t                     z   k    sJ d S )Nr.   r   r0   r9   r/   rk   r+   rM   r;   rA   rE   r2   r3   r4   test_series2xrx      sE   UbM""1a++q1Q3w1a4/?!AqD&/H1QPQTST::/UUUUUUbM""1a++q1uq!t|ad/BQq!tQZZ/OOOOOUQJ1a((A----UQJ1a((AE1111UQJ1a((AqD1Q3JN::::UQJ1a((A!Ga1f,<qAww,FFFFFq1uIq!Q''1q51a4<!Q$+>1a4+KKKKK1qs7O$$Q1--QqS2ad71BR1W1LqQRTUQUWXzz1YYYYYqS1WqL!!!Q**a!A#g!Q$.>R.HHHHHq1Q3wK  Aq))QAX1_qAqzz-IIIIIq1QT6zN##Aq!,,1q!ta1ajj0HHHHHHHr3   c                  r   t          d          } | dz  | t          d           t          d          dz  z  z  z   dz  d| t          d           t          d          dz  z  z  z  d| dz  z  z   z  }|                                }|                    | dd                              | d          dk    sJ d S )Nwrk   r/   r9   r   r0   )r   r   rT   r1   subsrz   ra   s     r4   	test_bug2r}      s    sA	
R1	QA 
!#	%&'SVVGCFFRL,@(A&AAa"gI&M	OA	

A99Q1""1a((A------r3   c                  6   dt           z   dt           z  z  } |                     t           d          t          d          t           t          d          z  dz  z
  dt          d          z  t           dz  z  dz  z   t          t           dz            z   k    sJ d S )Nr.   r/   r,   r9   rR   r;   )r%   r1   r   r$   r`   s    r4   test_expr      s~    	
Q!A#A99Q!9A3q66! 3bQi1nR6G G!AqD'' QQQQQQQr3   c                     t          d          } | dt          t                    t          d          t          t                    z   z  z
  z  }t          d          }|                    | dd|          t	          |t          d          z  t          t                    t          d          z   z            k    sJ d S )Nrz   r.   r9   logwr   )rf   )r   r   r%   r1   r   )rz   ra   r   s      r4   	test_exp2r      s    sA	AAAQ(()A&>>D99	1ad   "4A;AQ#@AAB B B B B Br3   c                     dt           z  dt           dz  z  z   dt           z  dt           dz  z  z   z  } |                     t           d          dt           z
  t           dz  z   t          t           dz            z   k    sJ d S )Nr9   r/   r.   r,   rE   r`   s    r4   	test_bug3r      sn    	
1qAv!a1f%A99Q!9A1qAww 6666666r3   c                      d} dt           z  dt           | z  z  z   dt           z  dt           | z  z  z   z  }|                    t           dd          dt           z
  t           dz  z   t          t           dz            z   k    sJ t          j        } dt           z  dt           | z  z  z   dt           z  dt           | z  z  z   z  }|                    t           dd          dt           z
  t          t                     z   t           t          d          dz  z  z   t          t           dz            z   k    sJ dt          t                     z   }|                    t           dd          dt          t                     z   k    sJ d S )Nr9   r/   r.   r   r0   )r%   r1   r$   r   Halfr   pra   s     r4   test_generalexponentr      s/   	A	
1qAv!a1f%A99Q1QA!Q$!77777	A	
1qAv!a1f%A99Q1Qa1qttAv;!>1a4!HHHHH	DGGA99Q1T!WW,,,,,,r3   c                  $   ddt          t                    z   z  } |                     t          dd          dt          z   t          t                    z
  t          t                    dz  z
  t          t          dz  t                    z   k    sJ d S )Nr.   r   r9   r/   rJ   r`   s    r4   test_genexp_xr      sw    	1tAww;A99Q1	AQ$q''1*$qAqzz12 2 2 2 2 2r3   c                     t          dd          } dt          z  dt          | z  z  z   dt          z  dt          | z  z  z   z  }|                    t          dd          dt          z   t          dz  z   t          t                    z
  t          t	          d          dz  z  z
  t          t	          d          dz  z  z
  t          t          dz            z   k    sJ d S )Nr/   r9   r.   r   r+   )r	   r%   r1   r   r   r$   r   s     r4   test_genexp_x2r      s    AA	
1qAv!a1f%A99Q1QAQ!7!add1f+!EAaDDQRF!SVWXY[\X\V]V]!]]]]]]]r3   c                      t          d          } d| z  | z  d| z   z  }|                    | dd          dt          | |           z   k    sJ |                    | dd                              | d          dk    sJ d S )Nrz   r9   r.   r   )r   r1   r$   r{   r|   s     r4   test_seriesbug2r      s    sA
A#qAEA99Q1Qq!WW,,,,99Q1""1a((A------r3   c                      t          d          } t          d| z            | z  }|                    | dd          dd| dz  z  dz  z
  t          | dz            z   k    sJ d S )Nrz   r9   r   r/   r0   )r   r!   r1   r$   r|   s     r4   test_seriesbug2br      sf    sAAaC
A99Q1Qq!tVAX!Q$!7777777r3   c                     t          dd          } t          t          d| z            | z            }|                    | d          t          d          d| dz  z  dz  z
  d| dz  z  d	z  z
  t	          | dz            z   k    sJ d S )
Nrz   Trealr9   r+   r,   r/   r0   -   r   r   r!   seriesr$   r|   s     r4   test_seriesbug2dr      s    sAC!HHQJA88A8s1vv!Q$q01QT6"9<qAwwFFFFFFFr3   c                     t          dd          } t          d| z            | z  d| z   z  }|                    | dd          dt          |           z   k    sJ |                    | dd          dd| z  t	          d          z  z   | dz  t          dd          t	          d          dz  z   z  z   t          | dz            z   k    sJ |                    | dd                              | d          dk    sJ d S )	Nrz   Tr   r9   r.   r   r/   )r   r!   r   r$   r   r	   r{   r|   s     r4   test_seriesbug2cr      s   sA	QqS!q1uA88Aq!AaDD((((88Aq!AaCAJ	1hr1ooA	)*!+-.q!tWW!5 5 5 5 588Aq!!!!Q''1,,,,,,r3   c                  <   t          dd          } t          t          d| z            | z            d| z   z                      | dd          t          d          | t          d          z  z   t	          | dz  |           z   k    sJ t          t          t          d| z            | z            d| z   z                                | dd          dd| z  t          d          z  z   t	          | dz            z   k    sJ t          t          d          t	          |           z                                 | dd          dt	          |           z   k    sJ dt	          |           z   d| z   z                      | dd          dt	          |           z   k    sJ d S )Nr%   Tr   r9   r.   r   )r   r   r!   r   r$   r   r1   r%   s    r4   test_expbug4r      s   sAAaC
 U#VAq!__A3q660AAadAJJ0NO O O OC!HHQJQ! !!'1aA!CFF
NQq!tWW4LM M M M s1vv!}%%aA..!add(::::1XQ((Aq11Q1X======r3   c                      t          dt          t                    z                                 t          dd          t          d          t          t          t                    z   k    sJ d S )Nr9   r   )r   r$   r%   r1   r2   r3   r4   test_logbug4r     sN    q1Q44x==  Aq))SVVa1gg-=======r3   c                     t          t          dt          z             t          z                                t          d          t          d          t          d           t          z  dz  z   dt          d          z  t          dz  z  dz  z   t	          t          dz            z   k    sJ t          t	          t                                                  t          dd          dt	          t                    z   k    sJ d S )Nr.   r/   r,   r9   rR   r;   r   )r   r   r%   r1   r$   r2   r3   r4   test_expbug5r     s    s1q5zz!|$$Q!$,,A#a&&10Dr#a&&yQRTUQU~VXGX0X[\]^`a]a[b[b0bbbbbqtt99Q1%%QqTT111111r3   c                     t          t          t                                                  t          d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 d S )Nr   r:   r/   r+   rQ   rD   i;  r!   r%   r1   r$   r2   r3   r4   test_sinsinbugr     ss    s1vv;;q!Q''1q!tAv:1R+?!AqD&*+LqQRTUQUww+VVVVVVVr3   c                     t           t          t                     dz
  z  } |                     t           dd          dt           dz  z
  t           dz  dz  z
  t           dz  dz  z   t          t           dz            z   k    sJ d S )Nr.   r   r+   r9   r0   rO   rM   )r%   r   r1   r$   as    r4   test_issue_3258r     sm    	3q66A:A99Q1QqS1a48!3ad2g!=!Q$!GGGGGGGr3   c                      t          dd          } t          | dz            t          dd          z  }|                    | dd          | | dz  d	z  z
  | d
z  dz  z
  t	          | dz            z   k    sJ d S )Nr%   T)nonnegativer/   r.   r      rD      rS   i  )r   r!   r	   r1   r$   )r%   fs     r4   test_issue_3204r     s    s%%%AAqD		8Aq>>!A99Q2!ad2g+2d
":Qq"uXX"EEEEEEEr3   c                  >   t          dt          t                    z
            } |                     t          dd          dt          t                    dz  z
  t          dz  z
  t          t                    dz  dz  z
  t          t          dz            z   k    sJ d S )Nr.   r   r9   r:   r/   rH   )r   r&   r1   r$   )r   s    r4   test_issue_3224r     sy    Qa[A99Q1T!WWQY1!4tAwwz"}!DqAww!NNNNNNNr3   c                     t          d          \  } }t          d          t          d          z  }| d|z   z  }dt          z  t          | d|z   z             t          | | |z  z             z   z  }| t          |           z  t          z  |t          z  z   |dz  dt          z  z  z
  t          |           z   }|                    | d          |k    sJ d S )Nzw,ir+   r/   rk   r.   r9   r,   )r   r   r%   r$   r1   )rz   irr   ra   e_sers         r4   test_issue_3463r      s    5>>DAqAs1vvA	BFA	!c!a!e*ooA1H-.ABs1vvIaK!A#1ac
*QqTT1E99Q!9%%%%%%r3   c                  z   t          dt          z                                t          d          dt          z  dt          dz  z  dz  z
  t          t          dz            z   k    sJ t          t          t          z                                 t          d          t          t                    t          t                    z   k    sJ t          t          t          z                                 t          d          t          t                    t          t                    t          z  z   t          t          dz            z   k    sJ t          t          t          z                                 t          d          t          t                    t          t                    t          z  z   t          t                    t          dz  z  dz  z
  t          t                    t          dz  z  d	z  z
  t          t                    t          dz  z  d
z  z   t          t          dz            z   k    sJ d S )Nr:   r0   r,      r/   r.   r9   r+   rA   r;   )r!   r%   r1   r$   r&   r   r2   r3   r4   test_sinr   )  s   qs88A##qsSAXaZ'7!AqD'''AAAAAq1u::a1%%Q!A$$6666q1u::a1%%Q#a&&():Qq!tWW)DDDDDq1u::a1%%Q#a&&():SVVAqD[])JAq!tA*Aq!tB*')*1a4*1 1 1 1 1 1 1r3   c                      t          dt          z            t          z  } |                     t          d          d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 S )
Nr:   rA   r,   r   r9   r/   i   r0      r   r`   s    r4   test_issue_3515r   1  sl    AaC
A99Q!9C1HQJad2 =!Q$ GGGGGGGr3   c                  <   t          t                    dz  t          t          t                              t          t                    dz  z  t          t                    t	          dd          z  t          t                    dz  z  z
  z  } |                     t          d          t	          dd          d	t          dz  z  d
z  z
  dt          dz  z  dz  z
  dt          dz  z  dz  z
  dt          dz  z  dz  z   t          t          dz            z   k    sJ d S )Nr   r9   r.   r/   rP   r,   rk   rM   rD   i   +   r0   i(  ic  rA   i % iy  r:   i )r!   r%   r   r   r	   r1   r$   r`   s    r4   test_issue_3505r   6  s   Ad3q66ll3q6619,AAs1vvqy() 	*A99Q!9R 0 01QT6#: =
1a4!QT	')!*,/1HX,=!>@A!Q$!H H H H H H Hr3   c                     t          d          } t          dz  t          t          | t          z             z  t          t          |           z  z
  z  }|                    t          d          t	          |           t          |           t          z  dz  z
  t	          |           t          dz  z  dz  z
  t          dz  t          |           z  dz  z   t          dz  t	          |           z  d	z  z   t          d
z  t          |           z  dz  z
  t          t          dz            z   k    sJ t          dz  t          t	          | t          z             z  t          t	          |           z  z
  z  }|                    t          d          t          |            t	          |           t          z  dz  z
  t          |           t          dz  z  dz  z   t	          |           t          dz  z  dz  z   t          dz  t          |           z  d	z  z
  t          d
z  t	          |           z  dz  z
  t          t          dz            z   k    sJ d S )Nr   rv   rA   r,   r9   r/   r;   r0   rN   r+   rO   )r   r%   r!   r1   r   r$   r   ra   s     r4   test_issue_3501r   =  s   sA	B3q1u::#a&&()A99Q!9AQ! 3c!ffQTk!m C	1SVVB!Ac!ffS!)+,a4A;s?!;=>q!tWW!E E E E E	B3q1u::#a&&()A99Q!9Q#a&&(1* 4s1vvad{1} DAq!tB!Ac!ffS!)+,a4A;s?!;=>q!tWW!E E E E E E Er3   c                     t          dt          z            t          dt          z            z  } |                     t          d          t          dd          t	          t          dz            z   k    sJ |                     t          d          t          dd          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 S )	Nr+   r9   r,   rA   #   r0   iI  0   )r!   r%   r1   r	   r$   r`   s    r4   test_issue_3502r   G  s    AaCQqSA99Q!9A1a4 8888899Q!9AAqD"SAXb[01QT77:; ; ; ; ; ;r3   c                  J   t          dt          z             dt          z   z  } |                     t          d          t          d          dz  t          t          d          z  dz  z   t          t          d          z  dz  z
  t	          t          dz            z   k    sJ d S )Nr9   r,   r0   )r!   r%   r1   r   r$   r`   s    r4   test_issue_3503r   N  s    AE

AEA99Q!9Aq1SVV8A: 5#a&&
 BQq!tWW LLLLLLLr3   c                      t           t          dt           z            z   dz  t           t           t          dt           z            z   z  t           t          dt           z            z   t          dt           z            z  z
  z  } |                     t           d          t          dd          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 S )Nr/   rv   r9   rD   r,   rk   r0   r+   `   [   i   im+  rA   i  )r%   r!   r1   r	   r$   r`   s    r4   test_issue_3506r   S  s    	
S1XXaS1XX.!c!A#hh,AaC1HHIA99Q!9Q!AqD&)#bAgck1E!Q$Jv4EE!Q$OP P P P P Pr3   c            	         t          dd          } t          t          |                                         | d          t          |           | dz  dz  z
  | dz  d	z  z
  t	          | dz            z   k    sJ t          |            | t          |            t          t          d| z                      z   z  z   t          t          d| z                      z   }|                    | d          t          d          t          d          | z  z   d| dz  z  d
z  z
  d| d
z  z  d
z  z
  d| dz  z  dz  z
  t	          | dz            z   k    sJ d S )Nr%   Tr   r+   r,   r9   rA   r0      r/   r   r   )r%   ra   s     r4   test_issue_3508r   Y  sL   sAs1vv;;a1%%Q!Q$q&1a48)Ca1gg)MMMMM	Q!c!ffWs3qs88}},--C!HH=A88A8AQAadF1H$qAvax/!AqD&);a1ggEF F F F F Fr3   c                     t           dz  t           dz  t           dz  t          t          t                               z  z
  z  } |                     t           d          t	          dd          t           dz  dz  z   dt           dz  z  d	z  z   d
t           dz  z  dz  z   dt           dz  z  dz  z   t          t           dz            z   k    sJ d S )Nr   r9   rP   r,   r.   r0   r      i  i/  rA   i 	 iq  r:   i )r%   r   r   r1   r	   r$   r`   s    r4   test_issue_3507r   a  s    	BA1T#a&&\\))*A99Q!9AAb 2ad74</#ad(6/AE!Q$JyDXX[\]^`a]a[b[bbc c c c c cr3   c                     t          t          t                                                  t          d          t          d          t          dz  t          d          z  dz  z
  t          dz  t          d          z  dz  z
  t          dz  t          d          z  dz  z   t	          t          dz            z   k    sJ d S )Nr+   r,   r.   r9   r0   r:   r;   )r!   r   r%   r1   r$   r2   r3   r4   test_issue_3639r   g  s    s1vv;;qA&&AAc!ffQAc!ffQ.Ac!ffR?!AqD''IJ J J J J Jr3   c                  ~   t          t                                        t          d          t          t          dz  dz  z   t          dz  dz  z   t          t          dz            z   k    sJ t	          t                                        t          d          dt          dz  dz  z   t          dz  d	z  z   t          t          dz            z   k    sJ t          t                                        t          d          t          t          dz  dz  z
  dt          dz  z  d
z  z   t          t          dz            z   k    sJ t          t                                        t          d          dt          z  t          dz  dz  z
  t          dz  z   dt          dz  z  dz  z   t          t          dz            z   k    sJ t          t                                        t          d          t          t          dz  dz  z
  dt          dz  z  dz  z   t          t          dz            z   k    sJ t          t                                        t          d          t          t          z  dz  t          t          z  z
  dt          z  t          dz  z  dz  z
  t          t          dz  z  dz  z
  t          t          dz            z   k    sJ t          t                                        t          d          t          t          dz  dz  z   t          dz  dz  z   t          t          dz            z   k    sJ t          t                                        t          d          t           t          z  dz  t          z   t          dz  dz  z   t          dz  dz  z   t          t          dz            z   k    sJ d S )NrA   r,   r/   r+   rN   r.   r9   r0   r;   r   r   i  (   )r   r%   r1   r$   r   r   r   r   r   r
   r   r   r   r2   r3   r4   test_hyperbolicr   l  s   77??1?""a!Q$q&j1a48&;a1gg&EEEEE77??1?""a!Q$q&j1a47&:Qq!tWW&DDDDD77??1?""a!Q$q&j1QT6"9&<qAww&FFFFF77??1?""	!ad2g!a1fSj(1QT7723 3 3 388A##q1a46zAadF2I'=!Q$'GGGGG88A##
1Q1qs1a4x{"Qq!tVAX-!Q$78 8 8 888A##q1a46zAqDF':Qq!tWW'DDDDD88A##r"uQw{QT!V';ad1f'DqAww'NNNNNNNr3   c                     t          dd          } t          dd          }| dz  | t          d|z  | z
            z  | t          d|z            z  z
  z  }|                    | d          t          d|z             | t          d|z            z  d	z  z   | d	z  t          d|z            z  d
z  z
  | dz  t          d|z            z  dz  z   t          | dz            z   k    sJ d S )Nrz   Tr   r%   rv   r.   r0   r,   r9   rA   r/   r;   )r   r   r1   r$   )rz   r%   ra   s      r4   test_series2r   y  s    sAsA	B3qsQw<<!C!HH*,-A99Q!9QqS	Ac!A#hhJqL 81a4AaC=? JQPQTRUVWXYVYRZRZ][]M] ]`abcefbf`g`g gggggggr3   c                  L   t          dd          } | dz  | dz  t          |           z  | dz  t          |           z  z
  z  }|                    | d          t	          d          d	z  | d	z  dz  z   d
| dz  z  dz  z   d| dz  z  dz  z   t          | dz            z   k    sJ d S )Nrz   Tr   ir/   r:   r,   r.   r9   rS   r0      i  rA   i^  )r   r"   r!   r1   r   r$   r|   s     r4   test_series3r     s    sA	BAc!ffq!tCFF{*+A99Q!9

1q!tAv 51a4 Cc!Q$hun TWXYZ\]Y]W^W^ ^^^^^^^r3   c                     t          d          } t          | dz  t          dz  | dz  z  z   dt          z  | dz  z  z   z  | dz  z  }|                    | d                                                                          t          ddt          z  z   t          dz  z   z  ddt          dz  z   dt          z  dz  z   z  dz  dt          z  ddt          z  z   t          dz  z   z  fv sJ d S )Nrz   r0   r9   r,   r.   rv   )r   r%   r1   removeOrT   r|   s     r4   	test_bug4r     s    sA	1a4!Q$q!t)ac!Q$h&'1,A99Q!9$$&&--//Aq1Q3wA~4F	1qs7QqSU?AqsA!Ga"g$5648 8 8 8 8 8 8r3   c                     t          d          } t          d          }t          |            t          d| t          t                    z  z             z   dz  | dz  z  t          |            t          dt          | z  z             z   t          |            t          d| t          t                    z  z             z   z  | z  t          t          |            t          d| t          t                    z  z             z   z  | z  z
  z  }|                    | d|          t          | z  |z  d| z  z   t	          d|           z   k    sJ |                    | d|          t          | z  |z  d| z  z   t          |z  z
  d|z  t          t                    z  z   t          t          t                    z  |dz  z  z   t	          |           z   k    sJ d S )Nrz   rd   r.   rv   r   re   r9   )r   r   r%   r1   r$   )rz   rd   ra   s      r4   	test_bug5r     s   sAsA
a&&3q1SVV8|$$	$+AG3s1vvgA!G7AwQ3q66\!2!2264456689CFF7A#a&&L< 996 	A 99Q!!9$$!A!a1gg(=====99Q!!9$$!A!ac(9AaCAJ(F
CFF(1a4-)A$$)      r3   c                     t          t                    dt          t                    z
  z                      t          d          dt          z  t	          t                    z   k    sJ t          t                    dz  dt          t                    z
  z                      t          d          dt	          t                    z   k    sJ d S )Nr.   r,   r9   )r!   r%   r   r1   r$   r2   r3   r4   test_issue_4115r     s    FFAAJ((a(00AaC!A$$J>>>>FFAIq3q66z"++A+33q1Q44x??????r3   c                      t          t          d            t          t          d            t          t          d            d S )Nc                  d    t          dt          z                                t          dd          S Nr.   r   r+   r!   r%   r   r2   r3   r4   <lambda>ztest_pole.<locals>.<lambda>  s     c!A#hhooaA66 r3   c                  j    t          ddt          z  z                                 t          dd          S r   r   r2   r3   r4   r   ztest_pole.<locals>.<lambda>  s&    c!ac'll11!Q:: r3   c                  t    t           t          dt           z            z                      t           dd          S r   )r%   r!   r   r2   r3   r4   r   ztest_pole.<locals>.<lambda>  s&    qQqSz11!Q:: r3   )r(   r   r2   r3   r4   	test_poler     sD    
966777
9::;;;
9::;;;;;r3   c                     t          t          t                                                  t          dd          t	          dt                    k    sJ t          t          t                                                  t          dd          dt	          t                    z   k    sJ t          t          t                                                  t          dd          dt          z   t	          t          dz            z   k    sJ t          t          t                                                  t          dd          dt          z   t          dz  dz  z   t	          t          dz            z   k    sJ t          t          t                                                  t          dd          dt          z   t          dz  dz  z   t	          t          dz            z   k    sJ t          t          t                                                  t          dd          dt          z   t          dz  dz  z   t          dz  dz  z
  t	          t          dz            z   k    sJ d S )Nr   r.   r9   r/   r0   r+   r:   )r   r!   r%   r   r$   r2   r3   r4   test_expsinbugr     s   s1vv;;aA&&!Aq''1111s1vv;;aA&&!add(2222s1vv;;aA&&!a%!AqD''/9999s1vv;;aA&&!a%!Q$q&.1QT77*BBBBBs1vv;;aA&&!a%!Q$q&.1QT77*BBBBBs1vv;;aA&&!a%!Q$q&.1a46*AAadGG*KKKKKKKr3   c                      t          d          } t          |                               |           dk    sJ t          |                                |           dk    sJ t          t          |                                         |           dk    sJ t          t          |                                          |           dk    sJ t          | dz                                |           dk    sJ t          | dz                                 |           dk    sJ t          t	          |                                         |           dk    sJ t          t	          |                                          |           dk    sJ t          dt          |           z                                 |           dk    sJ t          dt          |            z                                 |           dk    sJ t          |                               | d          dk    sJ t          |                                | d          dk    sJ t          dd	
          } t          | dz                                 |           dk    sJ d S )Nr%   r   rk   r/   r+   r0   r9   rw   T)negativeg      ?r.   )r   r   r   r!   r   r   s    r4   
test_floorr     sN   sA88??1""""!99A"$$$$Q==""a''''aR>>  ##r))))A;;a  A%%%%!Q$<<q!!R''''Q==""a''''aR>>  ##q((((SVV##A&&!++++S!WW$$Q''1,,,,88??1a  A%%%%!99Aq!!R''''sT"""AS>>  ##q((((((r3   c                  f   t          t                                        t          d          t          k    sJ t          t                                        t          d          dt          z   k    sJ t          dt          z  dz                                 t          d          dt          z  k    sJ t          dt          z  dz                                 t          d          ddt          z  z   k    sJ t          t          dz                                t          d          t          dz  k    sJ t          t          dz                                t          d          t          dz  k    sJ t          t          t                    dz                                 t          d          t          t          dz  dz  z
  t          dz  dz  z   t	          t          dz            z   k    sJ t          t          t                    dz                                 t          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          t                    t
          j        z                                 t                    t
          j        t          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          d          t	          t          dz            k    sJ t          dt          z                                t                    t          d
d          t	          t          dz            z   k    sJ d S )Nr.   cdirrk   r9   r+   r/   rA   rN   r:   r   )r   r%   r   r!   r$   r   r   r   r2   r3   r4   	test_fracr     s   77>>!!>$$))))77>>!">%%Q....!a==**ac1111!a==++q1Q3w66661::QQ''1a4////1::QR((AqD0000A
""11"--QT!Vad3h1F1a41PPPPPA
""12"..!a%!Q$q&.1a482KaPQSTPTgg2UUUUUA  ''**afqj1a46.AAqDH.LqQRTUQUww.VVVVV1::QQ''1QT772222!99A+a"3"3a1gg"=======r3   c                     t          t                                        t                    dk    sJ t          t                                         t                    dk    sJ t          t          t                                                  t                    dk    sJ t          t          t                                                   t                    dk    sJ t          dt	          t                    z
                                t                    dk    sJ t          dt	          t                     z
                                t                    dk    sJ t          t                                        t          d          dk    sJ t          t                                         t          d          dk    sJ d S )Nr.   r   r9   r/   rv   )r   r%   r   r!   r   r2   r3   r4   test_ceilingr     sb   1::Q1$$$$A2;;a  A%%%%3q66??!!!$$))))3r77""1%%****1s1vv:%%a((A----1sA2ww;&&q))Q....1::Q""a''''A2;;a##r))))))r3   c                  h   t          d          } t          t                                        t          d          t          k    sJ t          t                                         t          d          t          k    sJ t          t          dz                                 t          d          t          dz   k    sJ t          t	          t                                                  t          d          t          t          dd          t          dz  z  z
  t          t          dz            z   k    sJ t          t	          t                                                   t          d          t          t          dd          t          dz  z  z
  t          t          dz            z   k    sJ t          t          | z
                                t          d          |  t          d| z
            z  t          dz
  t          d| z
            z  z   t          d| z
            z   k    sJ d S )Nr   r0   r,   r.   rA   r/   )r   absr%   r1   r!   r	   r$   r   r   s    r4   test_absr     s   sAq66>>!q>!!Q&&&&r77??1?""a''''q1u::a1%%Q....s1vv;;qA&&!hq!nnQT.A*AAadGG*KKKKKsA2ww<<Q''1x1~~ad/B+BQq!tWW+LLLLLq1u::a##r$q1u++~QQU8K'KdSTWXSXkk'YYYYYYYr3   c                     t          t                                        t          dd          t          k    sJ t          t                                        t          dd          t           k    sJ t          t          dz                                 t          dd          dk    sJ t          t          dz                                 t          dd          dk    sJ t          t          dz                                 t          dd          dk    sJ t	          t          dz                                 t          dd          dk    sJ t          t          t          z                                 t          dd          t          t          t          z                                 t          dd          k    sJ d S )	Nr   +)dir-r9   r.   皙@r/   r   r%   r   r   r   r!   r&   r2   r3   r4   test_dirr     sm   q66==A3=''1,,,,q66==A3=''A2----Q<<q!--2222Q<<q!--2222S>>  A3 //144441s7""1aS"11Q6666q1u::Qs++s1q5zz/@/@A3/@/O/OOOOOOOr3   c                     t          t                                        t          dd          t          k    sJ t          t                                        t          dd          t           k    sJ t          t          dz                                 t          dd          dk    sJ t          t          dz                                 t          dd          dk    sJ t          t          dz                                 t          dd          dk    sJ t	          t          dz                                 t          dd          dk    sJ t          t          t          z                                 t          dd          t          t          t          z                                 t          dd          k    sJ d S )Nr   r.   r   rk   r9   r   r/   r   r2   r3   r4   	test_cdirr    sm   q66==AA=&&!++++q66==AB=''A2----Q<<q!!,,1111Q<<q!"--2222S>>  AA ..!33331s7""1ab"11Q6666q1u::Q++s1q5zz/@/@AA/@/N/NNNNNNNr3   c            	      r   t          d          } t          | t          z            t          z  }|                    t          dd                                          t          dz
  | dt          d| dz  z  dz             z  z  t          d| z            dz  z
  z  t          d| z            dz  z   k    sJ d S )Nr   r0   r9   r,   ir.   rH   )r   r   r%   r   r   r   r   s     r4   test_issue_3504r    s    sAQqS		!A88AqA8&&((	
QAd3q!t8a<((()D1IIbL89D1IIaKGH H H H H Hr3   c                     t          d          \  } }dd| t          z  z   z  }|                    t          dd          d| t          z  z
  | dz  t          dz  z  z   | dz  t          dz  z  z
  | dz  t          dz  z  z   t          t          dz            z   k    sJ dd| |z   t          z  z   z  }|                    t          dd          dt          |  |z
  z  z   t          dz  | |z   dz  z  z   t          t          dz            z   k    sJ d S )Nza,br.   r   r+   r9   r/   r0   )r   r%   r   r$   )r   br   s      r4   test_issue_4441r    s   5>>DAq	1qs7A88Aq!AaC!Q$q!t) 3ad1a4i ?	1QT	!adGG!    	1Aqy=A88Aq!ArAvJ
Q$Az/!adGG!$ $ $ $ $ $ $r3   c                     t          t                                        t          t          dz  d                                          t           dz  t          dz  z   dt          t          dz  z
  z  z
  k    sJ t          t                                        t          t          d                                          t           dz  t          dz  z   dt          t          z
  z  z   k    sJ t          t          t                    t          dt          z            z  t          t          dz            t          d          k    sJ d S )Nr9   r/   r,   rA   r.   r0   rk   )r"   r%   r   r
   r   r    r#   r   r2   r3   r4   test_issue_4329r	    s    q66==BqDA=&&..00	A!aRTl"# # # #q66==B!=$$,,..	
1r!taRj ! ! ! !QQqS!1bd++s2ww666666r3   c                  d   t          t          t          dz  z                                 d          t          t                    k    sJ t          t          t          dz  z                                 d          t          t          t          dz            z   k    sJ dt          z   dz                      t          d          t          dz  dt          z  z   dz   k    sJ ddt          z  z                                   ddt          z  z   k    sJ t	          t          t                                                    t                                                    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 d S )	Nr9   r.   r,   rA   r/   r0   r;   r+   )r   r%   r   r$   r   r   doitr2   r3   r4   test_issue_5183r    s`   q1a4x==!$$!,,,,q1a4x==!$$AadGG3333UQJqA&&!Q$1*q.8888!GQqS((((c!ffmmooq))..00	A1QAa!Q$r')AadGG34 4 4 4 4 4r3   c                     t          d          } dt          dz  | dz  z   dz  z                      t          t          | z  d          t           d| dz  z  t           | z  t          z   z  z  dd| dz  z  t           | z  t          z   dz  z  z  z
  t	          dt          t          | z  f          z   k    sJ dt          dz  | dz  z   dz  z                      t          t          | z  d          dd| dz  z  z  t          d| dz  z  t           | z  t          z   z  z  z
  dd| dz  z  t           | z  t          z   dz  z  z  z
  t	          t           | z  t          z   t          t          | z  f          z   k    sJ d S )	Nr   r.   r9   r   )x0r-   r0   r/   rH   )r   r%   r1   r   r$   r   s    r4   test_issue_5654r    s   sAq!tAqDy1n%%aAaC1%55	
AadFQBqD1HAadFQBqD1Hq=$8!99Aa!QqSNNJK K K Kq!tAqDy1n%%aAaC1%55Bq!tG	
AadFQBqD1H	:!"AadFQBqD1Hq=$8!9::<=qbdQhAaC<Q<Q:R R R R R R Rr3   c                  0   t          t          t          z                                 t          dd          } t          t          t          z   t          z                                 t          dd          }|                     t          t          t          z             |}}||z
                                                                                                  dk    sJ t          t          t          z                                 t          dd          } t          t          t          z   t          z                                 t          dd          }|                    t          dt          di          |                     t          d          k    sJ d S )Nr   r.   r9   r/   )	r   r%   r'   r   r&   r{   rT   r   simplify)sxsxys1s2s       r4   test_issue_5925r  !  s   	a!e		Aq!	$	$B
q1uqy//
 
 Aq
)
)CWWQABG%%''0022a7777	a!e		Aq!	$	$B
q1uqy//
 
 Aq
)
)C88Qq!AJ2771a==000000r3   c                     t          t          dz                                t          dd          dt          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 S )
Nr/   r      r.   rA   r9   rP   rM   r;   )r   r%   r1   r$   r2   r3   r4   
test_exp_2r  ,  sy    q!t99Q2&&!ad(QT!V*;ad1f*Dq"uRx*ORSTUWYTYRZRZ*ZZZZZZZr3   N)|sympy.calculus.utilr   sympy.core.functionr   r   sympy.core.numbersr   r   r   r	   r
   sympy.core.singletonr   sympy.core.symbolr   r   $sympy.functions.elementary.complexesr   &sympy.functions.elementary.exponentialr   r   %sympy.functions.elementary.hyperbolicr   r   r   r   r   r   r   r   #sympy.functions.elementary.integersr   r   r   (sympy.functions.elementary.miscellaneousr   r   (sympy.functions.elementary.trigonometricr   r   r    r!   r"   sympy.series.limitsr#   sympy.series.orderr$   	sympy.abcr%   r&   r'   sympy.testing.pytestr(   r)   r5   r7   r=   r?   rB   rF   rK   rU   rW   rZ   r\   r^   rb   ri   rl   rn   rp   rr   rt   rx   r}   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r  r  r  r	  r  r  r  r  r2   r3   r4   <module>r)     s;   + + + + + + 7 7 7 7 7 7 7 7 < < < < < < < < < < < < < < " " " " " " / / / / / / / / 5 5 5 5 5 5 = = = = = = = = f f f f f f f f f f f f f f f f f f f f F F F F F F F F F F A A A A A A A A O O O O O O O O O O O O O O % % % % % %                       . . . . . . . .  2 2 24 4 48 8 8       \ \ \	H 	H 	HL L L
O O O
O O O
  
F F F
2 2 22 2 2
9 9 9# # # ) ) )
1 1 1
I I I . . .R R R
B B B7 7 7
	- 	- 	-2 2 2^ ^ ^. . .8 8 8G G G- - -> > >> > >2 2 2W W WH H H
F F FO O O
& & &1 1 1H H H
H H HE E E; ; ;M M M
P P PF F Fc c cJ J J

O 
O 
Oh h h_ _ _8 8 8  @ @ @
< < <L L L) ) )(> > >* * *Z Z ZP P PO O OH H H$ $ $7 7 74 4 4R R R1 1 1[ [ [ [ [r3   