
    \h                     t   S SK Jr  S SKJr  S SKJrJr  S SKJrJ	r	J
r
Jr  S SKJr  S SKJrJr  S SKJrJr  S SKJrJrJr  S S	KJrJr  S S
KJr  S SKJrJr  S SK J!r!J"r"J#r#J$r$J%r%J&r&J'r'J(r(J)r)J*r*J+r+  S SK,J-r-J.r.  S SK/J0r0  S SK1J2r2  S SK3J4r4  S SK5J6r6  S SK J7r7J8r8J9r9J:r:J;r;  S SK<J=r>J?r@JArBJCrC  S SKDJErE  S SKFJGrGJHrH  S SKIJJrJJKrKJLrLJMrM  \C" 5       rNS rOS rPS rQS rRS rSS rTS rUS rVS rWS  rXS! rYS" rZS# r[S$ r\\HS% 5       r]S& r^S' r_S( r`S) raS* rbS:S+ jrcS, rdS- reS. rfS/ rgS0 rhS1 riS2 rjS3 rkS4 rlS5 rmS6 rnS7 roS8 rpg9);    )product)Sum)diffexpand_func)IRationaloopi)S)Symbolsymbols)	conjugate
polar_lift)exp	exp_polarlog)coshsinh)sqrt)cossin)besselibesseljbesselkbesselyhankel1hankel2hn1hn2jnjn_zerosyn)gamma
uppergamma)hyper)Integral)O)series)airyaiairybiairyaiprimeairybiprimemarcumq)random_complex_numberverify_numericallytest_derivative_numerically_randint)
besselsimp)raisesslow)znkxc            	      8   [         [        [        [        [        [
        4 H-  n [        U " [        5       [        5      [        5      (       a  M-   e   [        [        [        [        4 H/  n [        U " [        SS5      [        5      [        5      (       a  M/   e   g )Ni
   )r   r   r   r   r   r   tdrandcplxr5   r    r"   r   r   randintfs    a/var/www/auris/envauris/lib/python3.13/site-packages/sympy/functions/special/tests/test_bessel.pytest_bessel_randrA   !   sl    w'7C!HJ"A&&&& D "c3!GC$a(!,,,,      c                     ^  [         [        [        [        [        [
        [        [        4 H+  m [        [        U 4S j5        [        [        U 4S j5        M-     g )Nc                     > T " S5      $ )N    r>   s   r@   <lambda>'test_bessel_twoinputs.<locals>.<lambda>+   s	    !A$rB   c                     > T " SSS5      $ )NrE         rF   r>   s   r@   rG   rH   ,   s    !Aq!*rB   )
r   r   r   r   r   r   r    r"   r3   	TypeErrorr>   s   @r@   test_bessel_twoinputsrM   )   s4    w'7BKy,'y,- LrB   c                     [        S[        5      R                  [        5      S:X  d   e[        S[        [        5      5      R                  [        5      [        S-  :X  d   e[        SS[	        [        5      -  5      R                  [        5      [	        [        5      :X  d   e[        [
        [        5      [        [
        -  -  R                  [        5      SS[
        -  [        [
        S-   5      -  -  :X  d   eg Nr   rE   rJ   )r   r8   as_leading_termr   r   r5   r#   rF   rB   r@   test_besselj_leading_termrQ   /   s    1a=((+q0001c!f--a0AaC7771aQi 003tAw>>> AqM!Q$//2aAeAEl9J6KKKKrB   c                     [        S[        5      R                  [        5      S[        [        5      -  S[        S5      -  -
  S[        R
                  -  -   [        -  :X  d   e[        S[        [        5      5      R                  [        5      S[        [        -  -  :X  d   e[        SS[        [        5      -  5      R                  [        5      S[        [        [        5      -  -  :X  d   eg )Nr   rJ   rE   )	r   r8   rP   r   r   
EulerGammar
   r   r   rF   rB   r@   test_bessely_leading_termrV   8   s    1a=((+#a&1SV80Can0TVX/XXXX1c!f--a0B1I===1aQi 003r2d1g:FFFrB   c                  L   [        S[        5      R                  [        5      S:X  d   e[        S[        [        5      5      R                  [        5      [        S-  :X  d   e[        SS[	        [        5      -  5      R                  [        5      [	        [        5      :X  d   eg rO   )r   r8   rP   r   r   rF   rB   r@   test_besseli_leading_termrX   >   su    1a=((+q0001c!f--a0AaC7771aQi 003tAw>>>rB   c                  F	   [        S[        5      R                  [        5      [        [        5      * [        R
                  -
  [        S5      -   :X  d   e[        S[        [        5      5      R                  [        5      S[        -  :X  d   e[        SS[        [        5      -  5      R                  [        5      SS[        [        5      -  -  :X  d   e[        [	        S5      S-  [        5      R                  [        5      S[	        S5      S-  -  [        [	        S5      S-  5      -  [        [	        S5      S-  -  -  :X  d   e[        [	        S5      S-  [        5      R                  [        5      [        [	        S5      * S-  [        5      R                  [        5      :X  d   e[        S[        [        5      5      R                  [        5      [        SS5      :X  d   e[        SS[        -  5      R                  [        5      [        [        5      [        S[        -  * 5      -  [        S[        -  5      -  :X  d   e[        SS[        [        5      -  5      R                  [        5      [        [        5      [        S[        -  * 5      -  [        S[        -  5      -  :X  d   e[        SSS9n [        U [        5      R                  [        5      R                  U [	        S5      S	-  05      [        [	        S5      S	-  [        5      R                  [        5      R                  [        5      :X  d   e[        U [        5      R                  [        5      R                  U [	        S
5      S	-  05      [        [	        S
5      S	-  [        5      R                  [        5      R                  [        5      :X  d   e[        U [        5      R                  [        5      R                  U S05      [        S[        5      R                  [        5      R                  [        5      :X  d   e[        U [        5      R                  [        5      R                  U S05      [        S[        5      R                  [        5      R                  [        5      :X  d   eg )Nr   rJ   rE      rK   nzT)nonzero   irS   )r   r8   rP   r   r   rU   r   r   r#   r   r
   r   r   subsr(   )r[   s    r@   test_besselk_leading_termr_   D   sL   1a=((+Aw/EA/NNNN1c!f--a0AaC7771aQi 003q!DG)}DDD1Q461--a0A!QKad1f4MaRSTURVWXRXk4YYYY1Q461--a0GQqTE!GQ4G4W4WXY4ZZZZ1SV,,Q/71Q<???1QqS>))!,RqsV0DT!A#Y0NNNN1Qs1vX..q1T"XcAaC&k5I$qQRs)5SSSS	d	#B2q>))!,112ad1f+>'!A$q&RSBTB[B[\]B^BnBnopBqqqq2q>))!,112afQh-@GAcFSTHVWDXD_D_`aDbDrDrstDuuuu2q>))!,112a&9WQ]=Q=QRS=T=d=def=gggg2q>))!,112b':gb!n>S>STU>V>f>fgh>iiiirB   c                     [        S[        5      R                  [        5      S[        S-  S-  -
  [        S-  S-  -   [        [        S-  5      -   :X  d   e[        S[        S-  5      R                  [        5      S[        S-  S-  -   [        S	-  S-  -
  [        [        S-  5      -   :X  d   e[        S[        S-  [        -   5      R                  [        5      S[        S-  S-  -
  [        S
-  S-  -
  S[        S-  -  S-  -
  [        S-  S-  -   [        [        S-  5      -   :X  d   e[        S[	        [        5      [        -   5      R                  [        SS9S[        S-  -
  S[        S-  -  S-  -
  S[        S
-  -  S-  -   [        [        S
S5      -  S-  -
  [        [        SS5      -  S-  -   S[        [        SS5      -  -  S-  -   [        [        S-  5      -   :X  d   e[        S[        S[        -
  -  5      R                  [        5      S[        S-  S-  -
  [        S
-  S-  -
  S[        S-  -  S-  -
  S[        S-  -  S-  -
  [        [        S-  5      -   :X  d   e[        S[        S[        -   5      5      R                  [        5      S[        S-  S-  -
  [        S
-  S-  -   S[        S-  -  S-  -
  S[        S-  -  S-  -   [        [        S-  5      -   :X  d   e[        S[        [        5      5      R                  [        5      [        S-  S[        S
-  -  S-  -
  S[        S-  -  S-  -   [        [        S-  5      -   :X  d   e[        SS[	        [        5      -  5      R                  [        5      [	        [        5      [        [        S
S5      -  S-  -
  [        [        SS5      -  S-  -   [        [        SS5      -  S-  -
  [        [        SS5      -  S-  -   [        [        S S5      -  S!-  -
  [        [        S-  5      -   :X  d   e[        S"[        [        5      5      R                  [        SS9[        S[        [        5      5      R                  [        SS9:X  d   eg )#Nr   rE   rJ      @      皙?皙@皙@rK      rZ      r6       	     r]     /   )         `   0   I           	   @     Q rS   )r   r8   r(   r'   r   r   r   r   rF   rB   r@   test_besselj_seriesr|   U   s   1a="a!Q$q&j1a47&:Qq!tW&DDDD1a#h&&q)QC]QVAX-E!Q$-OOOO1adQh&&q)QAaZ!Q$q&-@
QT'"*.!tBw.!"1a4.) ) ) )1d1gk"))!q)1Q1Wr!Q$wrz5I
ad(4-6Xa^+A-6.018Aq>0A"0D6E
QA
s
"6#%&q!tW6- - - - 1aQi ''*a!Q$q&j1a46.ABq!tGBJ.N
QT'"*/Aw/   1c!a%j!((+q1a46zAqDF/B
QT'#+01a4
0#%&q!tW0- - - -1c!f$$Q'1Q31a4?R1WT\+IAaQRdG+SSSS1aQi ''*d1g8Aq>8I!8K.K
Xa^
B
/!"HQN!23!6/79:HQN9J49O/P
Xb!_
U
"/#%&q!tW/- - - - 2s1v%%a1%-CF1C1J1J1PQ1J1RRRRrB   c                     S[         R                  -  [        -  S[        S5      -  [        -  -
  S[        [        5      -  [        -  -   n [        S[        5      R                  [        SS9U [        S-  [        [        5      * S[        -  -  SS[         R                  -  -
  S[        -  -  -   [        S5      S[        -  -  -   -  -   [        [        S-  [        [        5      -  5      -   :X  d   e[        S[        5      R                  [        SS9S[        [        -  -  [        [        [        5      [        -  [        S5      [        -  -
  SS[         R                  -  -
  S[        -  -  -
  -  -   [        S-  [        [        5      * S[        -  -  [        S	5      S-  S[         R                  -  -
  S
[        -  -  -   [        S5      S[        -  -  -   -  -   [        [        S-  [        [        5      -  5      -   :X  d   e[        S[        5      R                  [        SS9S[        [        S-  -  -  S[        -  -
  [        S-  [        [        5      S[        -  -  [        S5      S[        -  -  -
  [        S5      S-  S[         R                  -  -
  S[        -  -  -
  -  -   [        [        S-  [        [        5      -  5      -   :X  d   e[        S[        5      R                  [        SS9S[        [        S-  -  -  S[        [        -  -  -
  [        S[        -  -  -
  [        S-  [        [        5      S[        -  -  [        S5      S[        -  -  -
  [        S5      S-  S[         R                  -  -
  S[        -  -  -
  -  -   [        [        S-  [        [        5      -  5      -   :X  d   e[        S[        S-  5      R                  [        SS9S[         R                  -  [        -  S[        S5      -  [        -  -
  S[        [        5      -  [        -  -   [        S-  S[        [        5      -  [        -  SS[         R                  -  -
  S[        -  -  -   [        S5      S[        -  -  -   -  -   [        [        S-  [        [        5      -  5      -   :X  d   e[        S[        S-  [        -   5      R                  [        SS9U SS[         R                  -  -
  [        S-  * S[        -  -  [        S-  S[        -  -  -
  -  -
  S[        -  [        -  -   [        S-  [        [        5      * S[        -  -  S[        -  -
  [        S5      S[        -  -  -   -  -   [        S-  [        [        5      * [        -  SS[        -  -  -   [        S5      [        -  -   -  -   [        [        S-  [        [        5      -  5      -   :X  d   e[        S[        S[        -
  -  5      R                  [        SS9U S[        -  [        -  -   [        S-  [        [        5      * S[        -  -  SS[         R                  -  -
  S[        -  -  -   [        S5      S[        -  -  -   S[        -  -   -  -   [        [        S-  [        [        5      -  5      -   :X  d   e[        S[        S[        -   5      5      R                  [        SS9U [        [        -  -
  [        S-  [        [        5      * S[        -  -  SS[         R                  -  -
  S[        -  -  -   [        S5      S[        -  -  -   S	S[        -  -  -   -  -   [        [        S-  [        [        5      -  5      -   :X  d   e[        S[        [        5      5      R                  [        SS9S[        [        S-  * S-  [        S-  -   -  -  SS[         R                  -  -
  [        S-  * S-  [        S-  -   -  [        -  -
  [        [        [        5      [        -  [        S5      [        -  -
  -  -   [        S-  S[        [        5      -  S[        -  -  SS[        -  -  -
  [        S	5      S-  S[         R                  -  -
  S
[        -  -  -   S[        S5      -  S[        -  -  -   -  -   [        [        S-  [        [        5      -  5      -   :X  d   e[        SS[        [        5      -  5      R                  [        SS9S[        [        [        5      -  -  [        [        5      [        [        5      [        -  SS[         R                  -  -
  [        -  -
  -  -   [        [        S5      S-  -  [        [        5      * S[        -  -  [        S	5      S-  S[         R                  -  -
  S[        -  -  -   -  -   [        [        S	5      S-  -  [        [        5      S[        -  -  [        S5      S-  S[         R                  -  -
  S[        -  -  -
  -  -   [        [        S-  [        [        5      -  5      -   :X  d   e[        S[        [        5      5      R                  [        SS9[        S[        [        5      5      R                  [        SS9:X  d   eg )NrJ   r   ra   ri   rE   rS   rK      rZ   rh   i   rz   rc   rs   rd   rf   grv   rT   r]   r:   )
r   rU   r
   r   r8   r   r(   r'   r   r   consts    r@   test_bessely_seriesr   h   s	   allN2#a&+aAhrk9E1a=Q'51a4#a&!B$q~"
%:&(+A":6 47 ,79:1a4A;,H H H H1a=Q'2r!t9q#a&)c!fRi:O	
Qq||^	ad#;$ 8% ,%'(!tc!fWad^	
1a!ALL.	 2b5).*,/FAbDM.: (;,;=>q!tCF{^,L L L L 1a=Q'2r!Q$w<!B$+>As1vqQSt}A"H1a!ALL.01R48H9 B: ,:<=ad3q6kN,K K K K1a=Q'31a4=1bd8+C	1R4,a4QB#a&"R%.8	
2q1Q\\>	!BrE*+ ,,,./1SVn,= = = = 1a#h&&qA&.!ALL.2C
CF(2+3CF
23&()3c!fRq~"
%1&(+A"16 )7379:1a4A;3H H H H 1adQh&&qA&.Qq||^#q!teQrTlQT1R4[&@AAAaCFJ
Q$Q22%A"5
6	7
Q$Q
Q"X%Ar	1
2	356q!tCF{^	DD D D 1aQi ''Q'/5
A#b&4a4#a&!B$1q~+="*EE
a&!B$-B$  4 "#AqDQK.41 1 1 1 1c!a%j!((a(0E
B$5AAw"~Qq||^);ad(CC
a&!B$-RU)$ %5%'(Ac!f~56 6 6 6 1c!f$$Q!$,BAb1Q34G0H	
Qq||^	q!teBh1n-b01134c!fRi#a&)6K3L1M	1bQiB!QrT(*ad1fq~.E2-NN	#a&"R% 	1 ad3q6kN1+ + + + 1aQi ''Q'/2r$q'z?QQa!ALL.0"4454689AaDFc!fWaPRd^	
1a!ALL.	 1R4(F) 9*4*,-!QKQB	
2q1Q\\>	!BrE*:+ -,4, /01SVn4= = = = 2s1v%%a1%-CF1C1J1J1PQ1J1RRRRrB   c            	         [        S[        5      R                  [        5      S[        S-  S-  -   [        S-  S-  -   [        [        S-  5      -   :X  d   e[        S[        S-  5      R                  [        5      S[        S-  S-  -   [        S	-  S-  -   [        [        S-  5      -   :X  d   e[        S[        S-  [        -   5      R                  [        5      S[        S-  S-  -   [        S
-  S-  -   S[        S-  -  S-  -   [        S-  S-  -   [        [        S-  5      -   :X  d   e[        S[	        [        5      [        -   5      R                  [        SS9S[        S-  -   S[        S-  -  S-  -   S[        S
-  -  S-  -   [        [        S
5      S-  -  S-  -   [        [        S5      S-  -  S-  -   S[        [        S5      S-  -  -  S-  -   [        [        S-  5      -   :X  d   e[        S[        S[        -
  -  5      R                  [        5      S[        S-  S-  -   [        S
-  S-  -   S[        S-  -  S-  -   S[        S-  -  S-  -   [        [        S-  5      -   :X  d   e[        S[        S[        -   5      5      R                  [        5      S[        S-  S-  -   [        S
-  S-  -
  S[        S-  -  S-  -   S[        S-  -  S-  -
  [        [        S-  5      -   :X  d   e[        S[        [        5      5      R                  [        5      [        S-  [        S
-  S-  -
  S[        S-  -  S-  -
  [        [        S-  5      -   :X  d   e[        SS[	        [        5      -  5      R                  [        5      [	        [        5      [        [        S
5      S-  -  S-  -   [        [        S5      S-  -  S-  -   [        [        S5      S-  -  S-  -   [        [        S5      S-  -  S-  -   [        [        S5      S-  -  S -  -   [        [        S-  5      -   :X  d   e[        S![        [        5      5      R                  [        SS9[        S[        [        5      5      R                  [        SS9:X  d   e[        S[        5      R                  [        [        SS9[	        S5      [	        S[        -  5      S[        -  [        S
5      S-  -  S"-  -
  S
S[        -  [        S5      S-  -  -  S#-  -
  S$S[        -  [        S5      S-  -  -  S%-  -
  [        S[        -  [        S5      S-  -  [        [        45      -   -  [        [        5      -  S[	        [        5      -  -  :X  d   e[        S[        5      R                  [        [        * SS9[	        S5      [	        S&[        -  5      S&[        -  [        S
5      S-  -  S"-  -
  S
S&[        -  [        S5      S-  -  -  S#-  -
  S$S&[        -  [        S5      S-  -  -  S%-  -
  [        S&[        -  [        S5      S-  -  [        [        * 45      -   -  [        [        * 5      -  S[	        [        5      -  -  :X  d   eg )'Nr   rE   rJ   ra   rb   rc   rd   re   rf   rK   rq   rZ   rh   ri      rk      r]   rm   1   rn   rp   rl   rr   rs   ru   rv   rw   rx   ry   rz   r{   rS   r~      rg      rT   )r   r8   r(   r'   r   r   r   r   r	   r   r
   rF   rB   r@   test_besseli_seriesr      s   1a="a!Q$q&j1a47&:Qq!tW&DDDD1a#h&&q)QC]QVAX-E!Q$-OOOO1adQh&&q)QAaZ!Q$q&-@2ad72:-M	1R.AqD'.   1d1gk"))!q)1Q1Wr!Q$wrz5IAqD6AaDFA6&()AaDFB679;A!QK9K6LNOPQSTPTg6V V V V1aQi ''*a!Q$q&j1a46.ABq!tGBJ.N
1a4
/q!tW/   1c!a%j!((+q1a46zAqDF/BR1WS[/P
1a4
0q!tW0   1c!f$$Q'1Q3Ab=2ad74<+G!AqD'+QQQQ1aQi ''*d1gAaDFA.EAaDQRFTV.V	AaDFC/ad1f+d*/+-.2q\%-?/@BCAqD'/J J J J2s1v%%a1%-CF1C1J1J1PQ1J1RRRR 1Q<q"*d1gtAaCyAaC1Q4PQ6?STCT7T	1Q3!A$q&/#8 "AaC1Q46? 24 788:;QqSAaDFOaQSW:U8V /WWZ[\W]/^_`aefhai_i/k k k k1Q<q"*d1gtBqDzRTQqTRSVDTUVDV7VYZ\^_`\`defgdhijdj[kYkloYo7o
BqDAaDFD 8!#$bdad1f%5B3x#@8A /BBEqb'/JKLTRTX:/W W W WrB   c            	         [        S5      [        R                  -
  [        [        5      -
  n [	        S[        5      R                  [        SS9U [        S-  [        [        5      * S-  [        R                  S-  -
  [        S5      S-  -   [        S5      S-  -   -  -   [        [        S-  [        [        5      -  5      -   :X  d   e[	        S[        5      R                  [        SS9S[        -  [        [        [        5      S-  [        S5      S-  -
  [        S5      S-  -
  [        R                  S-  -   -  -   [        S-  [        [        5      S-  [        S5      S	-  -
  [        S5      S-  -
  [        R                  S-  -   -  -   [        [        S-  [        [        5      -  5      -   :X  d   e[	        S[        5      R                  [        SS9S[        S-  -  [        S5      S-  -
  [        S-  [        [        5      * S
-  [        R                  S
-  -
  [        S5      S
-  -   [        S5      S-  -   -  -   [        [        S-  [        [        5      -  5      -   :X  d   e[	        S[        5      R                  [        SS9S[        S-  -  [        S5      S-  -
  [        [        5      -   :X  d   e[	        S[        S-  5      R                  [        SS9[        S5      [        R                  -
  S[        [        5      -  -
  [        S-  S[        [        5      -  [        R                  S-  -
  [        S5      S-  -   [        S5      S-  -   -  -   [        [        S-  [        [        5      -  5      -   :X  d   e[	        S[        S-  [        -   5      R                  [        SS9U SS[        R                  -  -
  [        S-  S-  [        S-  S
-  -   -  -   [        -
  [        S-  [        [        5      * S-  [        S5      S-  -   [        S5      S-  -   -  -   [        S-  [        [        5      * S-  [        S5      S-  -
  [        S5      S-  -   -  -   [        [        S-  [        [        5      -  5      -   :X  d   e[	        S[        S[        -
  -  5      R                  [        SS9U [        -
  [        S-  [        [        5      * S-  [        S5      S-  -
  [        R                  S-  -
  [        S5      S-  -   -  -   [        [        S-  [        [        5      -  5      -   :X  d   e[	        S[        S[        -   5      5      R                  [        SS9U [        S-  -   [        S-  [        [        5      * S-  [        R                  S-  -
  [        S5      S-  -   [        S5      S-  -   -  -   [        [        S-  [        [        5      -  5      -   :X  d   e[	        SS[        [        5      -  5      R                  [        SS9SS[        [        5      -  -  [        [        5      [        [        5      S-  [        S5      S-  -
  [        R                  -   -  -   [        [        S5      S-  -  [        [        5      S-  [        S5      S
-  -
  [        R                  S-  -   -  -   [        [        S5      S-  -  [        [        5      S-  [        S5      S-  -
  [        R                  S-  -   -  -   [        [        S-  [        [        5      -  5      -   :X  d   e[	        S[        [        5      5      R                  [        SS9[	        S[        [        5      5      R                  [        SS9:X  d   e[	        S[        S-  5      R                  [        SS9S[        S-  -  [        S5      S-  -
  [        [        S-  5      -   :X  d   e[	        S[        S-  5      R                  [        SS9S[        S-  -  [        S5      S-  -
  [        S-  [        [        5      * S-  [        R                  S
-  -
  [        S5      S
-  -   [        S5      S-  -   -  -   [        [        S-  [        [        5      -  5      -   :X  d   e[        S-  [	        S[        5      -  R                  [        SS9S[        [        S-  5      -   :X  d   e[	        S[        5      R                  [        [        SS9[        S5      [        [        5      -  [        S[        -  5      S[        -  [        S5      S-  -  S
-  -   SS[        -  [        S5      S-  -  -  S-  -
  SS[        -  [        S5      S-  -  -  S-  -   [        S[        -  [        S5      S-  -  [        [        45      -   -  [        [        * 5      -  S-  :X  d   e[	        S[        5      R                  [        [        * SS9[        S5      [        [        5      -  [        * [        S[        -  5      -  [        S[        -  [        S5      S-  -  -  S
-  -   S[        -  S[        -  [        S5      S-  -  -  S-  -   S[        -  S[        -  [        S5      S-  -  -  S-  -   [        S[        -  [        S5      S-  -  [        [        * 45      -   -  [        [        * 5      -  S-  :X  d   eg )NrJ   r   ra   ri   rE   rK   rh   rZ   rb   r~       rd   rf   gѿr]   rv   r   $   rS   rc   r   rg   r   rx   rT   )r   r   rU   r8   r   r(   r'   r   r   r	   r
   r   r   r   s    r@   test_besselk_seriesr      s	   FQ\\!CF*E1a=Q'5	1s1vgai!,,q.(3q6!83ad1f<=,>@A!Q$s1v+,O O O O1a=Q'1Q3CF1Hs1vax4G	!Q5a5  2! ,!#$a4QQqT"W)<s1vby)H	R* $,QT#a&[>,* * * * 1a=Q'1QT6AaDF?QTCF719	QDQD"$%aDGD, >- ,-/0Ac!f~,> > > >1a=Q'1QT6AaDF?QqT+AAAA1a#h&&qA&.#a&1<<2GCF
3VVCF]Q\\!^;AqQ46 3q!tCF{^3, , , , 1adQh&&qA&.%	
Qq||^	ad1fq!tAvo.3/123356TCF719Aq<Q46< 63TCF719qtBw#6Q#AB3CEFq!tCPQF{^3T T T T 1aQi ''Q'/519q!tc!fWQY	!QFaF "%a&(F+ @, 4,./1SVn4= = = =1c!a%j!((a(0EAaCK	1s1vgai!,,q.(1Q472SVAX=>5?AB1a4A;5P P P P1aQi ''Q'/1aQi=QQAaDF"Q\\1243561a[#a&(QqTRSVBS	QC 64ad1f+s1vby1Q47':Q\\"_'LM4NPQRSUVRVWZ[\W]R]P^4_ _ _ _ 2s1v%%a1%-CF1C1J1J1PQ1J1RRRR1ad""1"*a1fqtAvo!Q$.GGGG1ad""1"*a1fqtAvo1s1vgaiRSR^R^_`R`F`cfghcijkckFknopqnrsunuFu@v.vyz{|~{  AD  EF  AG  |G  zH  /H  H  H  HqDA&&qA&.!a1g+=== 1Q<q"*d1gd2h.>QqS	QqSTUVWTXYZTZO\]L]@]qsad1foc!A"$&!qtAv$6t$;A<>?1!QRSUWQX>YAZ /[[^`a_a[b/ccd/e e e e1Q<q2#+tAwtBx/?!DAJQRTVWXTX\]^_\`ab\bScQcdeQeAeaCA1a  $B%')!tRTQqT!V,<'<T'ABBDEr!tqQRtTUvFVYZ]_\_X`DaBb 0ccfhigicj0kkl0m m m mrB   c                  B   [        [        S5      S-  [        5      R                  [        SS9S[        S5      S-  -  [	        [        S5      S-  5      -  [        [        S5      S-  -  -  S[	        [        S5      S-  5      -  [        [        S5      S-  -  -  SS[        S5      S-  -  -  -  -
  [	        [        S5      * S-  5      [        [        S5      S-  -  -  SS[        S5      S-  -  -  -  -   [        [        S-  5      -   :X  d   e[        [        S5      S-  [        5      R                  [        SS9[        [        S-  5      [        [        5      -  [        [        [        -  S-  5      -
  [        [        S5      S-  -  [        [        S-  5      -  S-  -   [        [        S-  5      -   :X  d   e[        [        S5      S-  [        [        5      5      R                  [        SS9[        [        S-  5      [        [        S5      S-  -  -  [        [        S-  5      [        [        S5      S-  -  -  -
  [        [        S-  5      [        [        S5      S-  -  -  S-  -   [        [        S-  5      [        [        S5      S-  -  -  S-  -
  [        [        S-  5      [        [        S5      S-  -  -  S	-  -   [        [        S-  5      -   :X  d   e[        [        S5      S-  [        S-  5      R                  [        SS9[        [        S-  5      [        -  [        [        S-  5      [        -  -
  [        [        S-  5      -   :X  d   e[        [        S5      * S-  [        5      R                  [        5      [        [        S5      S-  [        5      R                  [        5      :X  d   e[        [        S5      * S-  [        5      R                  [        5      [        [        S5      S-  [        5      R                  [        5      :X  d   eg )
NrZ   rK   rJ   ri   rE   ra   rc   r]   r   )r   r   r8   r(   r#   r'   r   r
   rF   rB   r@   test_besselk_frac_order_seriesr      sl   1Q461$$Q!$,AaDFE!A$q&M0I!aPQdSTf+0U	%!Q-AaDF#Qq1Q46{]3141Q4%'N1qtAv;&!ad1f+6179:1a41A A A A 1Q461$$Q!$,RT
470BRT!V11Q46{41:-a/1023AqD'1: : : :1Q4647#**1*2d2a4jQqT!V6LRT
1qtAv;7!%bdA!QK!7!97:AJq1Q46{"1$7%'+BqDz!ad1f+'=b'@7ACDQT77K K K K 1Q461a4 ''Q'/41:a<
r!t*Q,41a44! ! ! !AaD57A%%a(GAaDFA,>,E,Ea,HHHHAaD57A%%a(GAaDFA,>,E,Ea,HHHHrB   c                     [        [        [        5      R                  [        5      [        [        S-
  [        5      S-  [        [        S-   [        5      S-  -
  :X  d   e[	        [        [        5      R                  [        5      [	        [        S-
  [        5      S-  [	        [        S-   [        5      S-  -
  :X  d   e[        [        [        5      R                  [        5      [        [        S-
  [        5      S-  [        [        S-   [        5      S-  -   :X  d   e[        [        [        5      R                  [        5      [        [        S-
  [        5      * S-  [        [        S-   [        5      S-  -
  :X  d   e[        [        [        5      R                  [        5      [        [        S-
  [        5      S-  [        [        S-   [        5      S-  -
  :X  d   e[        [        [        5      R                  [        5      [        [        S-
  [        5      S-  [        [        S-   [        5      S-  -
  :X  d   eg )NrE   rJ   )	r   r6   r5   r   r   r   r   r   r   rF   rB   r@   	test_diffr      s   1a=a GAE1$5a$7'!a%:KA:M$MMMM1a=a GAE1$5a$7'!a%:KA:M$MMMM1a=a GAE1$5a$7'!a%:KA:M$MMMM1a=a WQUA%6$6q$871q5!;LQ;N$NNNN1a=a GAE1$5a$7'!a%:KA:M$MMMM1a=a GAE1$5a$7'!a%:KA:M$MMMMrB   c                     [        [        [        5      R                  [        5      [        S[        -  [        -  5      [	        [        [        R                  -
  [        5      -  :X  d   e[        [        [        5      R                  [        5      [        S[        -  [        -  5      [        [        [        R                  -
  [        5      -  :X  d   e[        [        [        5      R                  [         5      [        [        * [        -  [        -  S-  5      [        [        [        [        5      [        -  5      -  :X  d   e[        [        [        5      R                  [        5      [        [        [        -  [        -  S-  5      [        [        [        [        * 5      [        -  5      -  :X  d   e[        5       n [!        [        U [        5      [        U [        5      R                  [        5      [        5      (       d   e[!        [        U [        5      [        U [        5      R                  [        5      [        5      (       d   e[!        [        U [        5      [        U [        5      R                  [         5      [        5      (       d   e[!        [        U [        5      [        U [        5      R                  [        5      [        5      (       d   e[!        [        U [        5      [        U [        5      R                  [         5      [        5      (       d   e[!        [        U [        5      [        U [        5      R                  [        5      [        5      (       d   e[!        [#        U [        5      [#        U [        5      R                  [         5      [        5      (       d   e[!        [#        U [        5      [#        U [        5      R                  [        5      [        5      (       d   e[!        [#        U [        5      [#        U [        5      R                  [        5      [        5      (       d   e[        U [        5      [        U [        5      R                  [        5      :w  d   e[%        U [        5      [%        U [        5      R                  [        5      :w  d   e['        U [        5      ['        U [        5      R                  [        5      :w  d   e[	        U [        5      [	        U [        5      R                  [        5      :w  d   e[%        U [        5      [%        U [        5      R                  [        5      :w  d   e['        U [        5      ['        U [        5      R                  [        5      :w  d   eU n[         [        4 Hi  n[%        U[        5      [%        U[        5      R                  U5      :X  d   e['        U[        5      ['        U[        5      R                  U5      :X  a  Mi   e   [	        U[        5      R                  [         5      [        S5      [        [        5      -  [        S[        -  5      -  [        U[        R                  -   [        5      -  S-  :X  d   e[	        U[        5      R                  [        5      SU -  [        S5      -  [        [        5      -  [        S[        -  5      -  [        U* [        R                  -
  [        5      -  S-  :X  d   e[)        SSS9n[+        SS5      nXC4 H  n[         [        4 H  n[        U[        5      [        U[        5      R                  U5      :w  d   e[	        U[        5      [	        U[        5      R                  U5      :w  d   e[%        U[        5      [%        U[        5      R                  U5      :w  d   e['        U[        5      ['        U[        5      R                  U5      :w  a  M   e   M     [-        [        [        [$        [&        4[        [        [         [        45       HB  u  pV[!        U" U[        5      U" U[        5      R                  U5      [        5      (       a  MB   e   g )	NrJ   rE   rT   r6   Tintegerr:   )r   r6   r5   rewriter    r   r
   r   Halfr   r"   r   r   r   r   r<   tnr   r   r   r   r=   r   )nuorderr?   Nrifuncrefuncs          r@   test_rewriter      s   1a=  $QqSVRAFF
A5F(FFFF1a=  $QqSVRAFF
A5F(FFFF1a=  )QBqDGAIwq*Q-/223 3 31a=  )AaCF1HgaQB!1223 3 3 
Bgb!ngb!n44W=qAAAAgb!ngb!n44W=qAAAAgb!ngb!n44W=qAAAAgb!ngb!n44W=qAAAAgb!ngb!n44W=qAAAAgb!ngb!n44W=qAAAAgb!ngb!n44W=qAAAAgb!ngb!n44W=qAAAAgb!ngb!n44W=qAAAA b!92q	))"----r1:R++B////r1:R++B////b!92q	))"----r1:R++B////r1:R++B//// Ew5!}E1 5 5a 88885!}E1 5 5a 8888   eQ<(DGDH,<T!A#Y,FwuWXW]W]~_`Ga,abc,cccceQ<(R"HT!W,<T"X,Ed1Q3i,OPWY^X^abagagXgijPk,klm,mmmm 	sD!A	b	B7#AeQ<2eQ<#7#7#::::eQ<2eQ<#7#7#::::ua=CqM$9$9!$<<<<ua=CqM$9$9!$<<<<	 $   Rc 2!#R' :<$r1+tB{226:A>>>><rB   c            
         [        [        [        R                  [        5      R                  [        5      5      [        S5      [        [        5      -  [        [        5      [        [        5      -  -  :X  d   e[        [        [        R                  [        5      R                  [        5      5      [        S5      * [        [        5      -  [        [        5      [        [        5      -  -  :X  d   e[        [        [        R                  [        5      5      [        S5      [        [        5      -  [        [        5      [        [        5      -  -  :X  d   e[        [        [        SS5      [        5      5      [        S5      [        [        5      -  [        [        5      [        [        5      -  -  :X  d   e[        [        [        SS5      [        5      5      [        S5      * [        S-  [        [        5      -  S[        -  [        [        5      -  -   S[        [        5      -  -
  -  [        [        5      [        [        SS5      -  -  -  :X  d   e[        [        [        SS5      [        5      5      [        S5      * [        S-  [        [        5      -  S[        -  [        [        5      -  -
  S[        [        5      -  -
  -  [        [        5      [        [        SS5      -  -  -  :X  d   e[        [        [        R                  [        5      5      [        S5      [        [        5      -  * [        [        5      [        [        5      -  -  :X  d   e[        [        [        SS5      [        5      5      [        S5      [        [        5      -  [        [        5      [        [        5      -  -  :X  d   e[        [        [        SS5      [        5      5      [        S5      [        S-  [        [        5      -  S[        -  [        [        5      -  -
  S[        [        5      -  -
  -  [        [        5      [        [        SS5      -  -  -  :X  d   e[        [        [        SS5      [        5      5      [        S5      * [        S-  [        [        5      -  S[        -  [        [        5      -  -   S[        [        5      -  -
  -  [        [        5      [        [        SS5      -  -  -  :X  d   e[        [        [        R                  [        5      5      [        S5      [!        [        5      -  [        [        5      [        [        5      -  -  :X  d   e[        [        [        SS5      [        5      5      [        S5      [#        [        5      -  [        [        5      [        [        5      -  -  :X  d   e[        [        [        SS5      [        5      5      [        S5      [        S-  [!        [        5      -  S[        -  [#        [        5      -  -
  S[!        [        5      -  -   -  [        [        5      [        [        SS5      -  -  -  :X  d   e[        [        [        SS5      [        5      5      [        S5      [        S-  [#        [        5      -  S[        -  [!        [        5      -  -
  S[#        [        5      -  -   -  [        [        5      [        [        SS5      -  -  -  :X  d   e[        [%        [        R                  [        5      5      [        [%        [        SS5      [        5      5      s=:X  aA  [        [        5      ['        [        * 5      -  [        S5      [        [        5      -  -  :X  d   e   e[        [%        [        SS5      [        5      5      [        [%        [        SS5      [        5      5      s=:X  a_  [        S5      [        [        5      -  [        S-  S[        -  -   S-   -  ['        [        * 5      -  S[        [        SS5      -  -  -  :X  d   e   e[)        SSSS9n [        [        U S-   [        5      5      [        U [        5      S	U -  S-
  S	U -  [        U [        5      -  [        -  [        U S
-
  [        5      -   -  [        -  -   :X  d   e[        [        U S-   [        5      5      [        U [        5      * SU -  S-   SU -  [        U [        5      -  [        -  [        U S
-
  [        5      -
  -  [        -  -   :X  d   e[        [%        U S-   [        5      5      [%        U [        5      SU -  S-   SU -  [%        U [        5      -  [        -  [%        U S
-
  [        5      -   -  [        -  -   :X  d   e[        [        U S-   [        5      5      [        U [        5      * SU -  S-   SU -  [        U [        5      -  [        -  [        U S
-
  [        5      -
  -  [        -  -   :X  d   e[        [        U [        R                  -   [        5      R                  [        5      5      [        S5      [        [        5      -  ['        [*        * [        -  U [        R                  -   -  S-  5      -  [-        [*        [        -  S-  5      -  [        U [        [-        [*        [        -  S-  5      -  5      -  [        [        5      -  :X  d   e[        [        U [        R                  -   [        5      R                  [        5      5      [        S5      [        [        5      -  [        U [        5      -  [        [        5      -  :X  d   e[)        SSS9n[)        SSS9n[)        SSS9n[        [        [        [$        4 HR  nU" X25      R.                  SL d   eU" U[0        5      R.                  b   eU" [0        [        5      R.                  c  MR   e   [        [        4 H  nU" X15      R.                  SL a  M   e   [        [$        4 H  nU" X15      R.                  c  M   e   [        [        [        [$        4 Ha  n[        U" [2        [0        5      5      U" [2        [0        SS9:X  d   e[        U" [2        * [0        5      5      U" [2        * [0        SS9:X  a  Ma   e   g )NrJ   rT   rZ   rK   r6   T)r   positiverS   rE   ra   rrealpr   ir   Fevaluate)r   r   r   r   r5   r   r    r   r   r
   r   r"   r   r2   r   r   r   r   r   r   r   r   r   is_extended_realr8   r	   )r6   r   r   r   besselxs        r@   test_expandr     s	   wqvvq)11"56QARa()* * *wqvvq)11"56	aQb$q')*+ + +
 gaffa()T!WSV^T"Xd1g=M-NNNNghr1oq12d1gc!fnd2htTUwFV6WWWWghq!na01	a!Q$s1v+!CF
*Qs1vX56RHQPQNAR8RST T Tghr1oq12	a!Q$s1v+!CF
*Qs1vX56RHQPQNAR8RST T T gaffa()
q'#a&.48DG+,- - -ghr1oq12d1gc!fnd2htTUwFV6WWWWghq!na01QAc!fqs3q6z)Ac!fH45tBx8Aq>@Q7QRS S Sghr1oq12	a!Q$s1v+!CF
*Qs1vX56RHQPQNAR8RST T T gaffa()T!WT!W_d2htAw>N-OOOOghr1oq12QQb$q')*+ + +ghq!na01QAd1g!DG+aQi78$r(1hqRSnCT:TUV V Vghr1oq12QAd1g!DG+aQi78$r(1hqRSnCT:TUV V V gaffa()78B?A./V378CG3CT!WTRSW_3UV V V V Vghq!na0178B?A./HQR!Q$1*q.)#qb'11QA5F3FGH H H H H 	sD40Awq1ua()1ABqDA$6q$871q5!;L$LMaOOP P Pwq1ua()	A!A#'AaC1$5a$7'!a%:K$KLQNNO O Owq1ua()11q1Q3wq!}#4Q#6Q9J#JKAMMN N Nwq1ua()	A!A#'AaC1$5a$7'!a%:K$KLQNNO O O wq166z1-55b9:	aaaRUAJ/12	2	1R46	
a9QrT!V#4!45
66:2h
?@ @ @ wq166z1-55b9:QQ1a b)* * * 	sAsT"AsD!AWgw7q}--555q!}--555q!}--555 8
 W%q}--555 &W%q}--555 & Wgw772q>*gb!e.LLLL7B3?+wsA/NNNN 8rB   c            	         S n [        SSSSS9n[        [        [        [        4 HS  n[        S[        SS5      -  S-   5      S-  n[        [        U" U[        5      5      U" U[        5      5      (       a  MS   e   U " [        [        U[        5      5      [        US-
  [        5      SUS-
  -  [        US-
  [        5      -  [        -  -
  5      (       d   eU " [        [        U* [        5      5      [        U* S-   [        5      SU* S-   -  [        U* S-   [        5      -  [        -  -   5      (       d   eU " [        [        U[        5      5      [        US-
  [        5      * SUS-
  -  [        US-
  [        5      -  [        -  -   5      (       d   eU " [        [        U* [        5      5      [        U* S-   [        5      * SU* S-   -  [        U* S-   [        5      -  [        -  -   5      (       d   eU " [        [	        U[        5      5      [	        US-
  [        5      SUS-
  -  [	        US-
  [        5      -  [        -  -   5      (       d   eU " [        [	        U* [        5      5      [	        U* S-   [        5      SU* S-   -  [	        U* S-   [        5      -  [        -  -
  5      (       d   eU " [        [        U[        5      5      [        US-
  [        5      * SUS-
  -  [        US-
  [        5      -  [        -  -   5      (       d   eU " [        [        U* [        5      5      [        U* S-   [        5      * SU* S-   -  [        U* S-   [        5      -  [        -  -   5      (       d   eg )Nc                 .    [        X5      =(       a    X:H  $ N)r   )eqanss     r@   checktest_slow_expand.<locals>.checke  s    "{(ry(rB   rE   r   rJ   )abdcr   r:   )r<   r   r   r   r   r   r=   r   r2   r5   r   r8   )r   rnr   r   s       r@   test_slow_expandr   c  sd   ) 
Aa1	%BWgw7qb!!A%&**WR^,gb!n==== 8 WR^,a#aaja1C&CA&EEG G G GWbS!_-"q!$q2#'{7B37A3F'Fq'HHJ J J J WR^,"q&!$$q"q&z'"q&!2D'DQ'FFH H H HWbS!_-2#'1%%B37GRC!GQ4G(G(IIK K K K WR^,a#aaja1C&CA&EEG G G GWbS!_-"q!$q2#'{7B37A3F'Fq'HHJ J J J WR^,"q&!$$q"q&z'"q&!2D'DQ'FFH H H HWbS!_-2#'1%%B37GRC!GQ4G(G(IIK K K KrB   c                 *    [        [        X5      5      $ r   )r   r    r6   r5   s     r@   mjnr         r!x  rB   c                 *    [        [        X5      5      $ r   )r   r"   r   s     r@   mynr     r   rB   c                     [        S5      n [        SS5      S:X  d   e[        SS5      S:X  d   e[        SS5      [        R                  :X  d   e[        U S5      [        U SSS9:X  d   e[        S[        5      S:X  d   e[        S[        * 5      S:X  d   e[        SU 5      [        U 5      U -  :X  d   e[        SU 5      [        U 5      U S-  -  [        U 5      U -  -
  :X  d   e[        SU 5      SU S-  -  SU -  -
  [        U 5      -  SU S-  -  [        U 5      -  -
  :X  d   e[        SU 5      S	U S
-  -  SU S-  -  -
  [        U 5      -  SU -  S	U S-  -  -
  [        U 5      -  -   :X  d   e[        S
U 5      SU -  SU S-  -  -   SU S-  -  -
  [        U 5      -  SU S
-  -  SU S-  -  -   [        U 5      -  -   :X  d   e[        SU 5      SU S-  -  SU S
-  -  -
  S	U S-  -  -   [        U 5      -  SU -  SU S-  -  -
  SU S-  -  -   [        U 5      -  -   :X  d   e[        SU 5      SU -  SU S-  -  -   SU S-  -  -
  SU S-  -  -   [        U 5      -  SU S-  -  SU S
-  -  -   SU S-  -  -
  [        U 5      -  -   :X  d   e[        [        [        U 5      5      [        [        U 5      :X  d   e[        SS5      R                  5       [        SS5      :X  d   e[        [        SS5      R                  S5      /SS[        -  -
  /5      (       d   eg )Nr5   r   rE   rT   Fr   rJ   rK   rg   ra   rc   i   rZ   -   ir:   i  i  i(  r]   iu     iei            @      @@333333?gMs?gM?)r   r    r   ComplexInfinityr	   r   r   r   r   r6   evalfr   r   r5   s    r@   test_jnr     sF   Aa8q==a8q==b!9)))))a8r!Q////a9>>a":??q!9Aq   q!9Aq!tc!fQh....q!91a4!A#s1v-1a43q60AAAAAq!9AqD1QT6)3q61QqS2ad7]CF4JJJJJq!91s1a4x"QT'13q69	adR1W	c!f$% % % %q!9QTC1H,r!Q$w6A>	AAqD3q!t8	#SV+, , , ,q!9Aad
*T!Q$Y6QTA3q6I	1tAqDy	 2ad7	*CF23 3 3 3 r!Qx Bq!H,,, dH##%D();;;;r!X$$R()}Q./1 1 1 1rB   c                  X   [        S5      n [        SU 5      [        U 5      * U -  :X  d   e[        SU 5      [        U 5      * U S-  -  [        U 5      U -  -
  :X  d   e[        SU 5      SU S-  -  SU -  -
  [        U 5      -  SU S-  -  [        U 5      -  -   * :X  d   e[	        [        [        U 5      5      [        [        U 5      :X  d   e[        SS5      R                  5       [        SS5      :X  d   e[        [        SS5      R                  S5      /S	S
[        -  -   /5      (       d   eg )Nr5   r   rE   rJ   rK   r   r   r:   g=;H?gLs?)
r   r   r   r   r   r"   r6   r   r   r   r   s    r@   test_ynr     s   Aq!9Q	!!!q!9Q1s1vax////q!91QT6AaC<Q/1QT63q6/ABBBBr!Qx Bq!H,,, dH##%D();;;;r!X$$R()]1_,-/ / / /rB   c                      [        S5      [        S[        5      R                  5       ;   d   e[        S[        5      S[        S-  -  S[        S-  -  -
  :X  d   eg )Nrg   rK   ra   rc   rJ   )r   r   r
   atomsrF   rB   r@   test_sympify_ynr     sL    R5C2J$$&&&&q":BEAb!eG++++rB   c                 R    [        X5       H  u  p4[        X4-
  5      U:  a  M    g   g)NFT)zipabs)r   r   toluvs        r@   r   r     s(    A	AE
S   rB   c                  :   [        [        SS5      / SQ5      (       d   e[        [        SS5      / SQ5      (       d   e[        [        SS5      / SQ5      (       d   e[        [        SS5      / S	Q5      (       d   e[        [        SS5      / S
Q5      (       d   eg )Nr   ra   )gz !	@gަ?!@g/^|"@gަ?!)@rE   )gP5z5@@g2@g@%@gr!,@rJ   )gX32@g@#H0"@g#EdX(@g^z/@rK   )g:r@gW%}$@g,IEce+@gmr0@)gx] @geqh'@g7N.@g4M2@)r   r!   rF   rB   r@   test_jn_zerosr     s    hq!nGHHHHhq!nHIIIIhq!nHIIIIhq!nIJJJJhq!nIJJJJrB   c                  F   [        SSS9[        S5      [        SSSS9p!n [        [        4 GHl  nU" SS5      [        R                  L d   eU" S	S5      [        R
                  L d   eU" S
S5      [        R
                  L d   eU" SS5      [        R                  L d   eU" SS[        -  -   S5      [        R
                  L d   eU" S
[        -   S5      [        R                  L d   eU" S[        -  S5      [        R                  L d   eU" U S5      [        R                  :w  a  U" U S5      [        R
                  :w  d   eU" US5      [        R                  :w  a  U" US5      [        R
                  :w  d   eU" US5      [        R
                  L a  GMm   e   [        SS5      [        R                  L d   e[        SS5      [        R                  L d   e[        [        4 HF  nU" S[        -   S5      [        R                  L d   eU" [        S5      [        R                  L a  MF   e   [        [        4 HW  nU" U[        R                  5      [        R
                  L d   eU" U[        R                  5      [        R
                  L a  MW   e   [        [        4 He  nU" U[        [        R                  -  5      [        R
                  L d   eU" U[        [        R                  -  5      [        R
                  L a  Me   e   [        [        4 H}  nU" S[        5      U" S[        5      :X  d   eU" S
[        5      U" S[        5      :X  d   eU" U * [        5      U" U [        5      :X  d   eU" U* [        5      U" U[        5      :w  a  M}   e   [        [        4 H  nU" S[        5      U" S[        5      :X  d   eU" S
[        5      U" S[        5      * :X  d   eU" U * [        5      SU -  U" U [        5      -  :X  d   eU" U* [        5      SU-  U" U[        5      -  :w  a  M   e   [        [        4 H8  nU" U[        * 5      [        * U-  [        U* -  -  U" U[        5      -  :X  a  M8   e   [        S[        * 5      [        S[        5      :X  d   e[        S[        * 5      [        S[        5      * :X  d   e[        S[        * 5      [        S[        5      :X  d   e[        S[        * 5      [        S[        5      * :X  d   e[        S[        [        -  5      [        S[        5      :X  d   e[        S[        [        -  5      [        [        S[        5      -  :X  d   e[        S[        [        -  5      [        * [        S[        5      -  :X  d   eg )Nr6   Tr   mr7   F)r   zeror   g @gffffff$rE   rK   rS   r   ra   rT   rJ   )r   r   r   r   OneZeror   r   NaNr   NegativeInfinityr   Infinityr5   )r6   r   r7   r?   s       r@   test_bessel_evalr     s   S$'fS$UZ6[!AwAw!%%ayAFF"""Qx166!!!{a/////QqS!}&&&a|q00000AqzQUU"""Aw!%%AaGqvv$555Aw!%%AaGqvv$555Aw!&&      1a=A.....1a=AJJ&&&wQ{a/////Aw!%%   wAJJ166)))A&&'166111   wAajjL!QVV+++Aa((()QVV333   wQx1Q7"""Qx1Q7"""!Qx1Q7"""!Qx1Q7"""	   wQx1Q7"""QxAaG8###!QxB71Q7?***!QxB71Q7?***	   wQBxQB71r7?1Q72222   1qb>WQ]***1qb>gam^+++1qb>WQ]***1qb>gam^+++1ac?gam+++1ac?a1o---1ac?qbA....rB   c                      [         [        [        [        [        [
        [        [        4 H4  n U " S[        R                  5      U " S[        R                  SS9:X  a  M4   e   g )NrE   Fr   )
r   r   r   r   r   r   r"   r    r   r   r>   s    r@   test_bessel_nanr     sC    w'7BKAEE{a15559999 LrB   c                     [        S[        5      R                  [        S5      S:X  d   e[        S[        5      R                  [        S5      S:X  d   e[        S[        5      R                  [        [        5      S:X  d   e[        [	        S5      S-  [        5      R                  [        S5      S:X  d   e[        [	        S5      S-  [        5      R                  [        S5      S:X  d   e[        [	        S5      S-  [        5      R                  [        [        5      S:X  d   e[        [        S[        -  5      R                  [        S5      S:X  d   e[        S[        5      R                  [        S5      S:X  d   e[        S[        5      R                  [        S5      S:X  d   e[        S[        5      R                  [        S5      S:X  d   e[        S[        5      R                  [        S5      S:X  d   e[        S[        5      R                  [        S5      S:X  d   e[        S[        5      R                  [        S5      S:X  d   e[        S[        5      R                  [        S5      S:X  d   e[        S[        S-  S[        -  -   5      R                  [        S5      S:X  d   e[        S[        5      R                  [        S5      S:X  d   e[        SS5      R                  [        S	5      S:X  d   e[        S
S[        S-  -  S-   5      R                  [        S5      S:X  d   e[        SS[        -  5      R                  [        S5      S:X  d   e[        S	S[        -  S-   5      R                  [        S5      S:X  d   e[        S[        S-  S-   5      R                  [        S
5      S:X  d   eg )NrJ   rE   Tr   FrK   rz   ra   rZ   rc   r]   rx   r~   )r   r8   is_meromorphicr	   r   r   r   r   r   r   r   r   r    r"   rF   rB   r@   test_meromorphicr     s   1a=''1-5551a=''1-5551a=''2.%7771Q461,,Q2d:::1Q461,,Q2e;;;1Q461,,Q3u<<<1ac?))!Q/58881a=''1-5551a=''1-5551a=''1-5551a=''1-5551a=''1-5551a=''1-6661a=''1-5551adQqSj!00A6$>>>1a=''1-6662q>((A.$666q!AqD&1*,,Q2d:::q!A#;%%a+t333a1q>((A.$666aA**1a0D888rB   c            
      	   [        S5      n [        SSS9n[        SSS9n[        SSS9n[        S	SS
9n[        [        [        [        [
        [        4 H  nU" U S5      R                  5       U" [        U 5      S5      :w  d   eU" X5      R                  5       U" [        U 5      U5      :w  d   eU" X5      R                  5       U" [        U 5      U5      :w  a  M   e   [        SS9n[        [        [        [        4 GH:  nU" U S[        -   5      R                  5       U" [        U 5      S[        -
  5      :X  d   eU" U S5      R                  5       U" [        U 5      S5      :X  d   eU" U S5      R                  5       U" [        U 5      S5      :X  d   eU" X5      R                  5       U" [        U 5      [        U5      5      :X  d   eU" X5      R                  5       U" [        U 5      U5      :X  d   e[        U" X5      R                  5       U" [        U 5      [        U5      5      5      (       a  GM;   e   [        U S[        -   5      R                  5       [        [        U 5      S[        -
  5      :X  d   e[        U S5      R                  5       [        [        U 5      S5      :X  d   e[        U S5      R                  5       [        [        U 5      S5      :X  d   e[        X5      R                  5       [        [        U 5      U5      :X  d   e[        X5      R                  5       [        [        U 5      [        U5      5      :X  d   e[        [        X5      R                  5       [        [        U 5      [        U5      5      5      (       d   e[        U S[        -   5      R                  5       [        [        U 5      S[        -
  5      :X  d   e[        U S5      R                  5       [        [        U 5      S5      :X  d   e[        U S5      R                  5       [        [        U 5      S5      :X  d   e[        X5      R                  5       [        [        U 5      U5      :X  d   e[        X5      R                  5       [        [        U 5      [        U5      5      :X  d   e[        [        X5      R                  5       [        [        U 5      [        U5      5      5      (       d   eg )Nr6   r5   F)extended_realr8   Tyr   tnegativerT   g      ?)r   rE   r   )r   r   r   r   r   r   r   r   r<   r   r   )r6   r5   r8   r   r   r?   rzs          r@   test_conjugater     s   sAs%(As$'AsT"AsT"Aw'7CBx!!#q1r'::::w  "a	!a&8888w  "a	!a&8888 D
 
CBw1AE{$$&!IaL!a%*@@@@Aw  "a	!a&8888Aw  "a	!a&8888w  "a	!il&CCCCw  "a	!a&8888!A($$&)A,	"(FGGGG 2 1a!e&&(GIaL!a%,HHHH1a=""$	!a(@@@@1a=""$	!a(@@@@1=""$	!a(@@@@1=""$	!il(KKKKgan&&(')A,	"*NOOOO1a!e&&(GIaL!a%,HHHH1a=""$	!a(@@@@1a=""$	!a(@@@@1=""$	!a(@@@@1=""$	!il(KKKKgan&&(')A,	"*NOOOOrB   c                  0   [        [        [        5      [        5      [        [        [        5      :X  d   e[	        [        [        5      [        5      [	        [        [        5      :X  d   e[        SSS9n [        U [        S[        -  [        -  5      [        -  5      [        U [        5      :X  d   e[        U [        [        5      5      [        U [        5      :X  d   e[	        U [        S[        -  [        -  5      [        -  5      [	        U [        5      :X  d   e[	        U [        [        5      5      [	        U [        5      :X  d   eS n[        S5      n[        U[        S[        -  [        -  5      [        -  5      [        S[        -  [        -  U-  5      [        U[        5      -  :X  d   e[	        U[        S[        -  [        -  5      [        -  5      [        S[        -  [        -  U-  5      [	        U[        5      -  :X  d   eU" [         S5      (       d   eU" [         [        5      (       d   eU" [         [        5      (       d   eU" [        S5      (       d   eU" [        [        5      (       d   eU" [        [        5      (       d   eg )Nr6   Tr   rJ   c           	      t   SSK Jn  U" SS5      nU " X[        [        [        -  5      -  5      U " X[        [        * [        -  5      -  5      -
  nSnU " X-   U* U[        -  -   5      U " X-   U* U[        -  -
  5      -
  n[        UR                  5       UR                  5       -
  5      R                  5       S:  $ )Nr   )uniformrE   rZ   gV瞯<g|=)sympy.core.randomr   r   r   r
   r   r6   )r   sr   r   exprepsexpr2s          r@   r   test_branching.<locals>.tnN  s    -AqMA1R4()Di26F4F,GGQWqb3q5j)D1"s1u*,EE4668eggi'(**,u44rB   r   )
r   r   r7   r8   r   r   r   r
   r   r   )r6   r   r   s      r@   test_branchingr   D  s   :a=!$1555:a=!$1555sD!A1i"Q')*gam;;;1jm$15551i"Q')*gam;;;1jm$15555 
B2y2a(*+s1R46"9~gb!n/LLLL2y2a(*+s1R46"9~gb!n/LLLLgq>>>gr???gq>>>gq>>>gr???gq>>>rB   c            	         [        S5      n [        SSS9n[        SSS9n[        [        U 5      5      [        [        U 5      5      :X  d   e[        U5      R                  (       d   e[        U[        U-  -   5      R                  5       [        U[        U-  -
  5      S-  [        U[        U-  -   5      S-  -   [        [        U[        U-  -
  5      [        U[        U-  -   5      -
  -  S-  4:X  d   eg )Nr5   r8   Tr   r   rJ   )r   r   r)   r   r   as_real_imag)r5   r8   r   s      r@   test_airy_baser   a  s    sAsAsAVAY6)A,#7777!9%%%%!AaC%=%%'1qs7OAq1Q3w 11va!A#gAaC01!3,5 5 5 5rB   c                  	   [        SSS9n [        SSS9n[        SSS9n[        [        U 5      [        5      (       d   e[        S	5      S
[        SS
5      -  S
[	        [        SS
5      5      -  -  :X  d   e[        [
        5      S	:X  d   e[        [
        * 5      S	:X  d   e[        [        U 5      U 5      [        U 5      :X  d   e[        [        U 5      U S	S
5      S
[        SS5      -  [	        [        SS
5      5      -  S[        -  -  S
[        SS5      -  U -  [	        [        SS
5      5      -  S[        -  -  -
  [        U S
-  5      -   :X  d   e[        U 5      R                  [        5      S
[        SS
5      -  * U -  [        S[        SS
5      4U S
-  S-  5      -  S
[	        [        SS
5      5      -  -  S
[        SS
5      -  [        S[        SS
5      4U S
-  S-  5      -  S
[	        [        SS
5      5      -  -  -   :X  d   e[        [        U 5      R                  [        5      [        5      (       d   e[        U5      R                  [        5      [        U* 5      [        [        SS
5      SU* [        S
S5      -  -  S
-  5      [        [        SS
5      SU* [        S
S5      -  -  S
-  5      -   -  S
-  :X  d   e[        U 5      R                  [        5      U * [        [        SS
5      SU [        S
S5      -  -  S
-  5      -  S
U [        S
S5      -  [        SS
5      -  -  -  U [        S
S5      -  [        SS
5      -  [        [        SS
5      SU [        S
S5      -  -  S
-  5      -  S
-  -   :X  d   e[        U5      R                  [        5      [        U5      [        [        SS
5      SU[        S
S5      -  -  S
-  5      [        [        SS
5      SU[        S
S5      -  -  S
-  5      -
  -  S
-  :X  d   e[!        [        SS
U S-  -  [        SS
5      -  -  5      5      [        S
5      * SU S-  [        SS
5      -  U [        SS
5      -  -  -   -  [#        SS
[        SS
5      -  -  U [        SS
5      -  -  5      -  S-  SU S-  [        SS
5      -  U [        SS
5      -  -  -   [        SS
[        SS
5      -  -  U [        SS
5      -  -  5      -  S-  -   :X  d   eg )Nr5   Fr   r   Tr   r   r   r   rK   rE   rJ   rZ   rc   rF   ra   rx   rT   )r   
isinstancer)   r   r#   r	   r   r+   r(   r
   r'   r   r%   r   r   r   r   r*   r5   r   r   s      r@   test_airyair  n  s   sAsT"AsT"AfQi((((!98Aq>)1U8Aq>-B+BCCCC":??2#;!q	1Q///&)Q1%	8Aq>%A//26HQN9J19LUS[\]_`SaMb9bdefhdh9iilmnoqrnrlssu u u !9U#	
HQN	1U2A'8!Q$q&AA1U8TUWX>EZCZ[	
HQN	5hq!n%61Q?	?5RSUVCXAX	Y	Z[ [ [ fQi''0&9999!9W%aR'(2q/1qb8Aq>-A+A!+CD(1a.!aR(1a.,@*@*BCD 	EEF	GH H H !9W%	
78Aq>1QA%6#6q#8991a!Q>ORZ[\^_R`=`;`a
Xa^
hq!n	,WXb!_a8TUWX>HYFYZ[F[-\	\]^	^	_` ` ` !9W%Q"a!Ax1~,=*=a*?@!Q1hq!n+<)<Q)>?@ 	AAB	CD D D va1a4(1a. 889:	a"1x1~-a!Q.???@!XVWYZ^J[H[\]_ghikl_m\mHmAnnopp
q!thq!n$QA%66
6qHQPQNAR?RSTV^_`bcVdSd?d8e	efg	g	hi i irB   c                  	   [        SSS9n [        SSS9n[        SSS9n[        [        U 5      [        5      (       d   e[        S	5      S
[        SS5      -  S
[	        [        SS
5      5      -  -  :X  d   e[        [
        5      [
        L d   e[        [
        * 5      S	:X  d   e[        [        U 5      U 5      [        U 5      :X  d   e[        [        U 5      U S	S
5      S
[        SS
5      -  [	        [        SS
5      5      -  S[        -  -  S
[        SS
5      -  U -  [	        [        SS
5      5      -  S[        -  -  -   [        U S
-  5      -   :X  d   e[        U 5      R                  [        5      S
[        SS5      -  U -  [        S[        SS
5      4U S
-  S-  5      -  [	        [        SS
5      5      -  S
[        SS5      -  [        S[        SS
5      4U S
-  S-  5      -  S
[	        [        SS
5      5      -  -  -   :X  d   e[        [        U 5      R                  [        5      [        5      (       d   e[        U5      R                  [        5      [        U* 5      [        [        SS
5      SU* [        S
S5      -  -  S
-  5      [        [        SS
5      SU* [        S
S5      -  -  S
-  5      -   -  S
-  :X  d   e[        U 5      R                  [         5      [        S
5      U [!        [        SS
5      SU [        S
S5      -  -  S
-  5      -  U [        S
S5      -  [        SS
5      -  -  U [        S
S5      -  [        SS
5      -  [!        [        SS
5      SU [        S
S5      -  -  S
-  5      -  -   -  S
-  :X  d   e[        U5      R                  [         5      [        S
5      [        U5      -  [!        [        SS
5      SU[        S
S5      -  -  S
-  5      [!        [        SS
5      SU[        S
S5      -  -  S
-  5      -   -  S
-  :X  d   e[#        [        SS
U S-  -  [        SS
5      -  -  5      5      [        S
5      SU S-  [        SS
5      -  U [        SS
5      -  -  -
  -  [        SS
[        SS
5      -  -  U [        SS
5      -  -  5      -  S-  SU S-  [        SS
5      -  U [        SS
5      -  -  -   [        SS
[        SS
5      -  -  U [        SS
5      -  -  5      -  S-  -   :X  d   eg )Nr5   Fr   r   Tr   r   r   r   rK   rZ   rc   rJ   rE   rF   ra   rx   rT   )r   r   r*   r   r#   r	   r   r,   r(   r
   r'   r   r%   r   r)   r   r   r   r  s      r@   test_airybir    s   sAsT"AsT"AfQi((((!98Aq>)1U8Aq>-B+BCCCC":2#;!q	1Q///&)Q1%	8Aq>%A//26HQN9J19LUS[\]_`SaMb9bdefhdh9iilmnoqrnrlssu u u !9U#	8Aq>!E"x1~&7Aa@@xPQST~AVV	8Aq>%Xa^$5q!tAv>>%QRTUBW@WX	YZ Z Z fQi''0&9999!9W%aR'(2q/1qb8Aq>-A+A!+CD(1a.!aR(1a.,@*@*BCD 	EEF	GH H H !9W%Q78Aq>1QA->+>q+@AA1hqRSnCTW_`acdWeBeeXa^#hq!n4WXb!_aPQS[\]_`SaPaNabcNc5dde 	ffg	hi i i !9W%QQ"a!Ax1~4E2Ea2GH !Q1hq!n3D1DQ1FGH 	IIJ	KL L L va1a4(1a. 889:QadXa^+Ax1~,===>va8TUWX>HYFYZ[]efgij]kZkFk?llmnn	
adXa^#Ax1~$55	5va8Aq>@Q>QRSU]^_abUcRc>c7ddeff	gh h hrB   c                  	   [        SSS9n [        SSS9n[        SSS9n[        [        U 5      [        5      (       d   e[        S	5      S
[        SS
5      -  * S
[	        [        SS
5      5      -  -  :X  d   e[        [
        5      S	:X  d   e[        [        U 5      U 5      U [        U 5      -  :X  d   e[        [        U 5      U S	S
5      S
[        SS
5      -  * S
[	        [        SS
5      5      -  -  S
[        SS
5      -  U S-  -  S[	        [        SS
5      5      -  -  -   [        U S
-  5      -   :X  d   e[        U 5      R                  [        5      S
[        SS
5      -  U S-  -  [        S[        SS
5      4U S
-  S-  5      -  S[	        [        SS
5      5      -  -  S
[        SS
5      -  [        S[        SS
5      4U S
-  S-  5      -  S
[	        [        SS
5      5      -  -  -
  :X  d   e[        [        U 5      R                  [        5      [        5      (       d   e[        U5      R                  [        5      [        U* 5      [        [        SS
5      SU* [        S
S5      -  -  S
-  5      [        [        SS
5      SU* [        S
S5      -  -  S
-  5      -   -  S
-  :X  d   e[        U 5      R                  [        5      U S-  [        [        SS
5      SU [        S
S5      -  -  S
-  5      -  S
U [        S
S5      -  [        SS
5      -  -  -  U [        S
S5      -  [        SS
5      -  [        [        SS
5      SU [        S
S5      -  -  S
-  5      -  S
-  -
  :X  d   e[        U5      R                  [        5      U[        [        SS
5      SU[        S
S5      -  -  S
-  5      * [        [        SS
5      SU[        S
S5      -  -  S
-  5      -   -  S
-  :X  d   e[        [        SS
U S-  -  [        SS
5      -  -  5      5      [        S
5      U [        SS
5      -  U S-  [        SS
5      -  -  S-
  -  [!        SS
[        SS
5      -  -  U [        SS
5      -  -  5      -  S-  U [        SS
5      -  U S-  [        SS
5      -  -  S-   [        SS
[        SS
5      -  -  U [        SS
5      -  -  5      -  S-  -   :X  d   eg )Nr5   Fr   r   Tr   r   r   r   rK   rJ   rE   rc   rF   rZ   rx   rT   rS   )r   r   r+   r   r#   r	   r   r)   r(   r'   r   r%   r   r   r   r   r,   r  s      r@   test_airyaiprimer    s   sAsT"AsT"Ak!nk2222q>a!Q//5!Q3H1HIIIIr?aA"aq	k111+a.!Q*	
HQN	AeHQN334q(1a.7HA7MqQVW_`acdWeQfOf7ggjklmoplpjqqs s s q>!!%(	8Aq>!Q$uR(1a.):AqDFCCQuXVWYZ^G\E\]	8Aq>%Xa^$5q!tAv>>%QRTUBW@WX	YZ Z Z k!n,,W5{CCCC!9W%aR'(2q/1qb8Aq>-A+A!+CD(1a.!aR(1a.,@*@*BCD 	EEF	GH H H q>!!'*	1WXa^Qq(1a.'8%8%:;;Q8Aq>@QT\]^`aTb?b=bc	
HQN	Xa^+GHROQq(STVW.GXEXYZEZ,[[\]]	^_ _ _ q>!!'*	GHROQq(1a.'8%8%:;;ghqRSnVWXY[cdegh[iXiVijkVk>llmnooq q q {1a1fx1~%=#=>?QHQN"AqD8Aq>#99A=>{1QPXYZ\]P^M^K^_`bjklnobp_pKp?qqrss	
HQN	AqD8Aq>1	1A	5{1QQRTUEVCVWXZbcdfgZhWhCh7iijkk	lm m mrB   c                  	   [        SSS9n [        SSS9n[        SSS9n[        [        U 5      [        5      (       d   e[        S	5      S
[        SS5      -  [	        [        SS
5      5      -  :X  d   e[        [
        5      [
        L d   e[        [
        * 5      S	:X  d   e[        [        U 5      U 5      U [        U 5      -  :X  d   e[        [        U 5      U S	S
5      S
[        SS5      -  [	        [        SS
5      5      -  S
[        SS5      -  U S-  -  S[	        [        SS
5      5      -  -  -   [        U S
-  5      -   :X  d   e[        U 5      R                  [        5      S
[        SS5      -  U S-  -  [        S[        SS
5      4U S
-  S-  5      -  S[	        [        SS
5      5      -  -  S
[        SS5      -  [        S[        SS
5      4U S
-  S-  5      -  [	        [        SS
5      5      -  -   :X  d   e[        [        U 5      R                  [        5      [        5      (       d   e[        U5      R                  [        5      [        U* 5      [        [        SS
5      SU* [        S
S5      -  -  S
-  5      [        [        SS
5      SU* [        S
S5      -  -  S
-  5      -   -  S
-  :X  d   e[        U 5      R                  [        5      [        S
5      U S-  [        [        SS
5      SU [        S
S5      -  -  S
-  5      -  U [        S
S5      -  [        SS
5      -  -  U [        S
S5      -  [        SS
5      -  [        [        SS
5      SU [        S
S5      -  -  S
-  5      -  -   -  S
-  :X  d   e[        U5      R                  [        5      [        S
5      U-  [        [        SS
5      SU[        S
S5      -  -  S
-  5      [        [        SS
5      SU[        S
S5      -  -  S
-  5      -   -  S
-  :X  d   e[!        [        SS
U S-  -  [        SS
5      -  -  5      5      [        S
5      U [        SS
5      -  U S-  [        SS
5      -  -  S-
  -  [#        SS
[        SS
5      -  -  U [        SS
5      -  -  5      -  S-  U [        SS
5      -  U S-  [        SS
5      -  -  S-   [        SS
[        SS
5      -  -  U [        SS
5      -  -  5      -  S-  -   :X  d   eg )Nr5   Fr   r   Tr   r   r   r   rK   rE   rc   rZ   rJ   rF   rx   rT   rS   )r   r   r,   r   r#   r	   r   r*   r(   r'   r   r%   r   r)   r   r   r   r+   r  s      r@   test_airybiprimer    s   sAsT"AsT"Ak!nk2222q>QA.uXa^/DDDDDr?b   sq   A"aq	k111+a.!Q*	8Aq>%A//!Xa^2CAqD2H!ERZ[\^_R`LaJa2bbefghjkgkelln n n q>!!%(	8Aq>!Q$uR(1a.):AqDFCCQuXVWYZ^G\E\]	8Aq>%Xa^$5q!tAv>>uXaQR^?TT	UV V V k!n,,W5{CCCC!9W%aR'(2q/1qb8Aq>-A+A!+CD(1a.!aR(1a.,@*@*BCD 	EEF	GH H H q>!!'*QAghq!na8Aq>0A.A!.CDDaRSUVFWZbcdfgZhEhhXa^#hq!n4WXb!_aPQS[\]_`SaPaNabcNc5dde 	ffg	hi i i q>!!'*Q	78B?Aa!Q.?,?,ABWXVWYZ^]^_`bjklnobp_p]pqr]rEsstuvvx x x {1a1fx1~%=#=>?QHQN"AqD8Aq>#99A=>{1QPXYZ\]P^M^K^_`bjklnobp_pKp?qqrss	
HQN	AqD8Aq>1	1A	5{1QQRTUEVCVWXZbcdfgZhWhCh7iijkk	lm m mrB   c                     [        S5      n [        S5      n[        S5      n[        SSS5      S:X  d   e[        U SU5      [        XS-  S-  5      [        U 5      -  :X  d   e[        SSS5      S[	        [        SS5      5      -  S-  :X  d   e[        SUS5      S	[	        US-  * S-  5      -
  :X  d   e[        S[        S5      S	[	        [        S-  * S-  5      -
  :X  d   e[        S	X5      [        R                  [	        US-  * 5      [        SUS-  5      -  S-  -   :X  d   e[        SX5      [        R                  [	        US-  * 5      [        SUS-  5      -  S-  -   [	        US-  * 5      [        S	US-  5      -  -   :X  d   e[        [        S	US
5      U5      U[        S	US
5      * [        SUS
5      -   -  :X  d   e[        [        SS
U5      U5      US-  * [	        US-  * S-  [        SS5      -
  5      -  [        S	S
U-  5      -  S
-  :X  d   e[        S5      n[        SS
S5      R                  [        US9[        US-  [	        US-  * S-  [        SS5      -
  5      -  [        S	S
U-  5      -  US[        45      S
-  :X  d   e[        [        SSS
5      R                  [        5      R                  S5      /S/5      (       d   e[        S5      n[        SSS5      R                  [         US9[	        S5      [!        [        SS5      U-  [        US5      -  US[        45      -  :X  d   e[        [        S	S
S	5      R                  [         5      R                  S5      /S/5      (       d   e[        S	XSS9R                  [        5      [        R                  [	        US-  * 5      [        SUS-  5      -  S-  -   :X  d   e[        SXSS9R                  [        5      [        R                  [	        US-  * 5      [        SUS-  5      -  S-  -   [	        US-  * 5      [        S	US-  5      -  -   :X  d   e[        S
X5      R                  [        5      [        S	US-  5      [        SUS-  5      -   [	        US-  * 5      -  [        R                  -   [	        US-  * 5      [        SUS-  5      -  S-  -   :X  d   e[        SSS5      R                  [        5      [	        S5      [        SS5      -  S-  [        SS5      [        S
S5      -   [        SS5      -   [        S	S5      -   [	        S5      -  -   [        R                  -   :X  d   e[        XU5      R                  [        5      [        XU5      :X  d   e[        SS S!9n[        X1U5      R                  [        5      [        X1U5      :X  d   eg )"Nr   r   r   r   rJ   rZ      irE   rK   rx   r8   ra   r8   rS   r:   gZphL?r7   r   r   r   )r7   ir]   #   gyhH?Fr   r~   irb   Tr   )r   r-   r$   r#   r   r   r
   r   r   r   r   r   r&   r	   r   r   r   )r   r   r   r8   r7   s        r@   test_marcumqr    s   sAsAsA1aq   1az!T!V4U1X====1ar#hsA&6"7799991aq31uQw<////1b!CQqM 11111aqvvQTE
71ad3C(CA(EEEEE1aqvvQTE
71ad3C(CA(EEQPQTE
SZ[\^_ab^bScHccccc1a !$GAq!,<+<wq!Q?O+O(PPPP1a !$Ac1a4%'HQN2J.K(KGTUWXYZWZO([\](]]]]sA1a##H#2AqDadU1Wx1~566wq!A#FAr
STUUV V Vwq"a ((288<=n    	sA2r2&&sa&0s8C!Q!+GArN:Q2JGGH H Hwq!Q'',22267n    1aU+33G<aQRdUT[\]_`bc_cTdIdefIf@ffff1aU+33G<aQRdUT[\]_`bc_cTdIdefIf@f1u:gaA&&A' ' ' '1a##G,AqD1AGAqRStDT1TVY[\^_[_Z_V`0`661AJwq!Q$//112 2 2 21a##G,CB0G0IArNWQ^+gan<wq"~MsSVxW1XZ[Z`Z`1a a a a1##G,a0@@@@sD!A1##G,a0@@@@rB   c                     [        S5      n [        SSS5      R                  [        U S9R	                  [        U S-  [        U S-  * S-  [        SS5      -
  5      -  [        SSU -  5      -  U S[        45      S-  5      (       d   eg )Nr8   rJ   rK   ra   r  rx   rE   )	r   r-   r   r&   dummy_eqr   r   r   r	   r  s    r@   test_issue_26134r  $  s    sA1a##H#2;;Ac1a4%'HQN233GAqsOCaBZPQRRT T T TrB   N)gư>)q	itertoolsr   sympy.concrete.summationsr   sympy.core.functionr   r   sympy.core.numbersr   r   r	   r
   sympy.core.singletonr   sympy.core.symbolr   r   $sympy.functions.elementary.complexesr   r   &sympy.functions.elementary.exponentialr   r   r   %sympy.functions.elementary.hyperbolicr   r   (sympy.functions.elementary.miscellaneousr   (sympy.functions.elementary.trigonometricr   r   sympy.functions.special.besselr   r   r   r   r   r   r   r   r    r!   r"   'sympy.functions.special.gamma_functionsr#   r$   sympy.functions.special.hyperr%   sympy.integrals.integralsr&   sympy.series.orderr'   sympy.series.seriesr(   r)   r*   r+   r,   r-   r   r.   r<   r/   r   r0   r;   r1   sympy.simplifyr2   sympy.testing.pytestr3   r4   	sympy.abcr5   r6   r7   r8   r=   rA   rM   rQ   rV   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  rF   rB   r@   <module>r%     sK    ) 3 4 4 " / H H H > 9 ? } } } } G / .   &O O0 0 & -    
*-.LG?j"S&!SHW. mFIN8?vFOT K K>!!1:/,
K4/n:90"PJ:
5!iH!hHmD mF&ARTrB   