
    shO                        S SK Jr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  S SK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  S S
KJr  S SKJrJrJ r!J"r#  S SK$J%r&J'r(J)r*  S SK+J,r,J-r-J.r.  \
" SSS9r/\
" SSSS9r0\	" S5      r1S r2S r3S r4S r5g)    )IRationaloopizoo)S)DummySymbol)atanh)sqrt)sintan)gamma)hypermeijerg)Integral)O)
elliptic_k
elliptic_f
elliptic_eelliptic_pi)test_derivative_numericallyrandom_complex_numberverify_numerically)zmniT)integerk)r   positivetc                   
   [        S5      [        S-  :X  d   e[        [        R                  5      S[        [	        SS5      -  -  [        [	        SS5      5      S-  -  :X  d   e[        S5      [        L d   e[        S5      [        [	        SS5      5      S-  S[        S[        -  5      -  -  :X  d   e[        [        5      S:X  d   e[        [        * 5      S:X  d   e[        [        [        -  5      S:X  d   e[        [        * [        -  5      S:X  d   e[        [        5      S:X  d   e[        [        5      R                  [        5      [        [        5      S[        -
  [        [        5      -  -
  S[        -  S[        -
  -  -  :X  d   e[        [        [        5      [        5      (       d   e[        SS	S
9n [        U 5      R                  5       [        U R                  5       5      :X  d   e[        SSS9n[        U5      R                  5       [        U5      :X  d   e[        [        5      R!                  ["        5      [        S-  [#        [        R                  [        R                  4[        R$                  4[        5      -  :X  d   e['        [        [        5      [        S-  [#        [        R                  [        R                  4[        R$                  4[        5      -  5      (       d   e[        [        5      R!                  [(        5      [)        [        R                  [        R                  4/ 4[        R*                  4[        R*                  44[        * 5      S-  :X  d   e['        [        [        5      [)        [        R                  [        R                  4/ 4[        R*                  4[        R*                  44[        * 5      S-  5      (       d   e[        [        5      R-                  [        5      [        S-  [        [        -  S-  -   S[        -  [        S-  -  S-  -   S[        -  [        S-  -  S-  -   S[        -  [        S-  -  S-  -   S[        -  [        S-  -  S-  -   [/        [        S-  5      -   :X  d   e[        [0        5      R!                  [2        5      R5                  [3        S[        S[0        [7        [8        5      S-  -  -
  5      -  [8        S[        S-  45      5      (       d   eg )Nr                  r   FrealTnegative	            i     i           )Kr   r   Halfr   r   r   r   r   r   r   diffEtdr
   	conjugaterewriter   Onetnr   Zeroseriesr   r   r   dummy_eqr   r"   )zizrs     ډ/Users/tiagomarins/Projetos/claudeai/copy_bank/venv/lib/python3.13/site-packages/sympy/functions/special/tests/test_elliptic_integrals.pytest_KrE      s   Q42a4<<QVV9"hq!n,,U8B?-CQ-FFFFFQ43;;R5E(1a.)1,aQrT
l;;;;R5A::bS6Q;;QrT7a<<aRU8q==S6Q;;Q499Q<AaDAE1Q4</!A#q1u+>>>>adA;;;	%	 BR5??",,. 1111	d	#BR5??"%%%Q4<<	Auaffaff%x334 4 4adRT5!&&!&&!1AEE8Q??@@@@Q4<< !&&!&&!2&!&&QVVI(>CAEF F FadGaffaff-r2affY	4JQBOPQQRRRRQ4;;q>RTBqDF]QrT!Q$Ys]:
2ad
3bAe+,.22gadl6.ABDEadGL L L L Q4<<!**4Ac!faiK((1aA,79 9 9 9    c                     [        [        S5      [        :X  d   e[        S[        5      S:X  d   e[        [        [        -  S-  [        5      [        [        [        5      -  :X  d   e[        [        [        5      S:X  d   e[        [        [        * 5      S:X  d   e[        [        * [        5      [        [        [        5      * :X  d   e[        [        [        5      R                  [        5      S[        S[        [        [        5      S-  -  -
  5      -  :X  d   e[        [        [        5      R                  [        5      [        [        [        5      S[        -  S[        -
  -  -  [        [        [        5      S[        -  -  -
  [        S[        -  5      SS[        -
  -  [        S[        [        [        5      S-  -  -
  5      -  -  -
  :X  d   e[        5       n [        [        [        U 5      [        5      (       d   e[        [        U [        5      [        5      (       d   e[        SSS9n[        [        U5      R                  5       [        [        R                  " 5       UR                  5       5      :X  d   e[        SSS	9n[        [        U5      R                  5       [        [        R                  " 5       U5      :X  d   e[        [        [        5      R                  [        5      [        [        S
-  S[        S-  -  S-  [        S-  -
  -  -   [        [        S-  -  S-  -   [!        [        S-  5      -   :X  d   e[        [        [        5      R#                  [$        5      R'                  [%        S[        S[        [        [(        5      S-  -  -
  5      -  [(        S[        45      5      (       d   eg )Nr   r$   r)   r(   r   Fr*   Tr,   r3   r&   (      r5   )Fr   r   r   r   r6   r   r8   r   r   r9   randcplxr:   r
   r;   r@   r   r<   r   rA   r"   rmimrs      rD   test_FrP   6   s   Q7a<<Q7a<<RT!VQ<1QqT6!!!Q8q==Q9>>aR8!QxQ7<<?aQ3q619_ 55555Q7<<?a1gqsAE{3a1gqsmCAaC!QU)DQs1vqy[1123 3 3 3
Aa1gq>>>a1gq>>>	%	 BQ81Q[[]BLLN#CCCC	d	#BQ81Q[[]B#7777Q7>>!	AqD!AqD&)ad"##a1fQh.1a489 9 9 Q7??8$--4Ac!faiK((1a)46 6 6 6rF   c                     [        [        S5      [        :X  d   e[        S[        5      S:X  d   e[        [        [        -  S-  [        5      [        [        [        5      -  :X  d   e[        [        [
        5      [        L d   e[        [        [
        * 5      [        L d   e[        S5      [        S-  :X  d   e[        S5      S:X  d   e[        [
        5      [        [
        -  :X  d   e[        [
        * 5      [
        L d   e[        [        5      [        L d   e[        [        * [        5      [        [        [        5      * :X  d   e[        [        [        5      R                  [        5      [        S[        [        [        5      S-  -  -
  5      :X  d   e[        [        [        5      R                  [        5      [        [        [        5      [        [        [        5      -
  S[        -  -  :X  d   e[        [        5      R                  [        5      [        [        5      [        [        5      -
  S[        -  -  :X  d   e[        5       n [        [        U [        5      [        5      (       d   e[        [        [        U 5      [        5      (       d   e[        [        [        5      [        5      (       d   e[        SSS9n[        [        U5      R!                  5       [        [        R                   " 5       UR!                  5       5      :X  d   e[        U5      R!                  5       [        UR!                  5       5      :X  d   e[        SSS9n[        [        U5      R!                  5       [        [        R                   " 5       U5      :X  d   e[        U5      R!                  5       [        U5      :X  d   e[        [        5      R#                  [$        5      [        S-  [%        ['        S	S5      [(        R*                  4[(        R,                  4[        5      -  :X  d   e[/        [        [        5      [        S-  [%        ['        S	S5      [(        R*                  4[(        R,                  4[        5      -  5      (       d   e[        [        5      R#                  [0        5      [1        [(        R*                  ['        S
S5      4/ 4[(        R2                  4[(        R2                  44[        * 5      * S-  :X  d   e[/        [        [        5      [1        [(        R*                  ['        S
S5      4/ 4[(        R2                  4[(        R2                  44[        * 5      * S-  5      (       d   e[        [        [        5      R5                  [        5      [        [        S-  [        S-  * S-  [        S-  -   -  -   [        [        S
-  -  S-  -
  [7        [        S-  5      -   :X  d   e[        [        5      R5                  [        5      [        S-  [        [        -  S-  -
  S
[        -  [        S-  -  S-  -
  S[        -  [        S
-  -  S-  -
  S[        -  [        S-  -  S-  -
  S[        -  [        S-  -  S-  -
  [7        [        S-  5      -   :X  d   e[        S[        -  [        S-   -  5      R5                  [        5      [        S-  [        [        -  S-  -
  [        [        S-  -  S-  -   S
[        -  [        S
-  -  S-  -
  S[        -  [        S-  -  S-  -
  S[        -  [        S-  -  S-  -
  [7        [        S-  5      -   :X  d   e[        [        [        5      R#                  [8        5      R;                  [9        [        S[        [        [<        5      S-  -  -
  5      [<        S[        45      5      (       d   e[        [        5      R#                  [8        5      R;                  [9        [        S[        [        [<        5      S-  -  -
  5      [<        S[        S-  45      5      (       d   eg )Nr   r$   r)   r   Fr*   Tr,   r'   r&   r(   r3   rH   rI   r5   r%   r/   r1      r2   i  r4      ]   )r9   r   r   r   r   r   r   r   r8   r   r   rJ   r6   rK   r:   r
   r;   r<   r   r   r   r7   r=   r>   r   r?   r@   r   r   rA   r"   rL   s      rD   test_ErU   Q   s   Q7a<<Q7a<<QrT!VQ<1QqT6!!!Q8s??Q9Q42a4<<Q4199R5AbD==bS6R<<S6S==aR8!QxQ7<<?d1qQ{?3333Q7<<?qAw1a01Q37777Q499Q<AaD1Q4K!A#....
Aa1gq>>>a1gq>>>adA;;;	%	 BQ81Q[[]BLLN#CCCCR5??",,. 1111	d	#BQ81Q[[]B#7777R5??"%%%Q4<<2a4Q/H155(TU)V"VVVVadRT5(2q/166!:QUUHaHHIIIIQ4<< 	1668Aq>*B/166)affY1G!	LLQNO O OadWqvvx1~6;qvvi!&&=SVWUWXXYZZ[[[[Q7>>!	AqD1a4%(QrT/""Qq!tVAX-!Q$78 8 8Q4;;q>RTBqDF]QrT!Q$Ys]:	"QT	#Bq!tE)*,/F1a4K,>?AB1a4I I I IQqS!A#Y<q!
1r!tAv1a4	!AbDAIaK/"R%1*S.@2b5A:c>QTUVWYZVZT[[\ \ \ Q7??8$--a!CFAI+o&Aq	24 4 4 4Q4<<!**a!CFAI+o&Ar!t57 7 7 7rF   c                  b   [        S[        [        5      [        [        [        5      :X  d   e[        S[        [        5      [        [        [        5      [	        S[        [        [        5      S-  -  -
  5      [        [        5      -  [        [        [        5      -
  S[        -
  -  -   :X  d   e[        [        [        [        -  S-  [        5      [        [        [        [        5      -  :X  d   e[        [        [        S5      [        [	        [        S-
  5      [        [        5      -  5      [	        [        S-
  5      -  :X  d   e[        [        [        [        5      [        [        [        5      [        S[        [        5      -
  [        [        5      [	        S[        [        [        5      S-  -  -
  5      -  -   :X  d   e[        [        [        [        5      S:X  d   e[        [        * [        [        5      S:X  d   e[        [        [        [        5      S:X  d   e[        [        [        [        * 5      S:X  d   e[        S[        5      [        [        5      :X  d   e[        S[        5      [        L d   e[        [        S5      [        S[	        S[        -
  5      -  -  :X  d   e[        SS5      [        * L d   e[        SS5      [        L d   e[        [        [        5      [        [        5      S[        -
  -  :X  d   e[        [        [        * [        5      [        [        [        [        5      * :X  d   e[        SSS9[        SSS9p[        U [        U5      R!                  5       [        U R!                  5       [        R                   " 5       UR!                  5       5      :X  d   e[        SS	S
9[        SS	S
9p2[        U[        U5      R!                  5       [        U[        R                   " 5       U5      :X  d   e[        [        [        5      R!                  5       [        [        R                   " 5       [        R                   " 5       5      :X  d   e[        [        [        [        5      R#                  [        5      [        [        [        5      [        [        -
  [        [        [        5      -  [        -  -   [        S-  [        -
  [        [        [        [        5      -  [        -  -   [        [	        S[        [        [        5      S-  -  -
  5      -  [        S[        -  5      -  SS[        [        [        5      S-  -  -
  -  -  -
  S[        [        -
  -  [        S-
  -  -  :X  d   e[        [        [        [        5      R#                  [        5      S[	        S[        [        [        5      S-  -  -
  5      S[        [        [        5      S-  -  -
  -  -  :X  d   e[        [        [        [        5      R#                  [        5      [        [        [        5      [        S-
  -  [        [        [        [        5      -   [        [        S[        -  5      -  S[        S-
  -  [	        S[        [        [        5      S-  -  -
  5      -  -  -
  S[        [        -
  -  -  :X  d   e[        [        [        5      R#                  [        5      [        [        5      [        [        -
  [        [        5      -  [        -  -   [        S-  [        -
  [        [        [        5      -  [        -  -   S[        [        -
  -  [        S-
  -  -  :X  d   e[        [        [        5      R#                  [        5      [        [        5      [        S-
  -  [        [        [        5      -   S[        [        -
  -  -  :X  d   e[        [        [        [        5      R%                  [        5      [        [        S-  [        S-  [        S-  -   -  -   [        S-  S[        S-  -  S-  [        [        -  S-  -   [        S-  -
  [        S-  S-  -   [        S-  -
  -  -   ['        [        S-  5      -   :X  d   e[        [        [        [        5      R)                  [*        5      R-                  [+        SS[        [        [.        5      S-  -  -
  [	        S[        [        [.        5      S-  -  -
  5      -  -  [.        S[        45      5      (       d   e[        [        [        5      R)                  [*        5      R-                  [+        SS[        [        [.        5      S-  -  -
  [	        S[        [        [.        5      S-  -  -
  5      -  -  [.        S[        S-  45      5      (       d   eg )Nr   r)   r$   r'   r   Fr*   r   Tr,   r&   r5   r3   rH   
   rI   rS   )Pr   r   rJ   r   r   r   r9   r   r   r   r   r   r6   r   r
   r;   r8   r@   r   r<   r   rA   r"   )nirN   nrrO   s       rD   test_Pr[      s   Q1:1a   Q1:1a	a!CFAI+o	s1v	%!Q	/!a%89 9 9 9Q"Q?a!Qi'''Q1:tAE{3q6124A;>>>>Q1:1a1Q1:-AtA#a&!)O7L0LLLLLRA;!bS!Q<1Q2;!QB3<1Q7ad??Q7c>>Q7b!DQK-((((Q7rc>>R8r>>Q7adAEl"""QA;1Q1:+%%%Ce$fSu&=RB<!!#	",,.!++-89 9 9C$'sT" 	RB<!!#qQ[[]B'????Q7!AKKM1;;="AAAAQ1:??1!Aq'QUAaGOA,="=	
A1Q1:a# "#Dc!faiK* % #QqS#"#$a!CFAI+o#6#8#8:;QU)QU:K"M M M M Q1:??1DQs1vqy[$91qQ{?$K!LLLLQ1:??1!Aq'1q5/AaAJ">	#ac(
Aq1uId1qQ{?334#578!a%y"B B B BQ7<<?qtq1uadl1n4	
A1Q71  !a%y!a%02 2 2 2Q7<<?qtQU|a1g51q5	BBBB Q1:Q1q!tQqS1Q3Y'7#7	1a1fRi!A#b& 1R4'!Q$q&01R478$9;<QT7$C C C C Q1:h'00Q3q619_d1qQ{?&;;<q!QiHJ J J JQ7??8$--Q3q619_d1qQ{?&;;<q!RTlKM M M MrF   N)6sympy.core.numbersr   r   r   r   r   sympy.core.singletonr   sympy.core.symbolr	   r
   %sympy.functions.elementary.hyperbolicr   (sympy.functions.elementary.miscellaneousr   (sympy.functions.elementary.trigonometricr   r   'sympy.functions.special.gamma_functionsr   sympy.functions.special.hyperr   r   sympy.integrals.integralsr   sympy.series.orderr   *sympy.functions.special.elliptic_integralsr   r6   r   rJ   r   r9   r   rX   sympy.core.randomr   r:   r   rK   r   r>   	sympy.abcr   r   r   r   jr"   rE   rP   rU   r[    rF   rD   <module>rk      s~    9 9 " - 7 9 ? 9 : .  8 8@ @  
3
3t,	#J9@66-7^5MrF   