ó
    ‰*£hI:  ã                   óè   • S r SSKJrJrJrJr  SSKJrJr  SSK	J
r
Jr  SSK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\5      r " S S\5      r " S S\5      r " S S\5      rg)zElliptic Integrals. é    )ÚSÚpiÚIÚRational)ÚDefinedFunctionÚArgumentIndexError)ÚDummyÚuniquely_named_symbol)Úsign)Úatanh)Úsqrt)ÚsinÚtan)Úgamma)ÚhyperÚmeijergc                   óZ   • \ rS rSrSr\S 5       rSS jrS rSS jr	S r
S rS	 rS
 rSrg)Ú
elliptic_ké   aÖ  
The complete elliptic integral of the first kind, defined by

.. math:: K(m) = F\left(\tfrac{\pi}{2}\middle| m\right)

where $F\left(z\middle| m\right)$ is the Legendre incomplete
elliptic integral of the first kind.

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

The function $K(m)$ is a single-valued function on the complex
plane with branch cut along the interval $(1, \infty)$.

Note that our notation defines the incomplete elliptic integral
in terms of the parameter $m$ instead of the elliptic modulus
(eccentricity) $k$.
In this case, the parameter $m$ is defined as $m=k^2$.

Examples
========

>>> from sympy import elliptic_k, I
>>> from sympy.abc import m
>>> elliptic_k(0)
pi/2
>>> elliptic_k(1.0 + I)
1.50923695405127 + 0.625146415202697*I
>>> elliptic_k(m).series(n=3)
pi/2 + pi*m/8 + 9*pi*m**2/128 + O(m**3)

See Also
========

elliptic_f

References
==========

.. [1] https://en.wikipedia.org/wiki/Elliptic_integrals
.. [2] https://functions.wolfram.com/EllipticIntegrals/EllipticK

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  [        U5      -  -
  SU-  SU-
  -  -  $ )Nr   r   r   )ÚargsÚ
elliptic_er   )ÚselfÚargindexr&   s      r'   ÚfdiffÚelliptic_k.fdiffH   s<   € Ø�I‰I�a‰LˆÜ˜1“  Q¡¬
°1«Ñ 5Ñ5¸¸!¹¸QÀ¹U¹ÑDÐDr*   c                 óª   • U R                   S   nUR                  =(       a    US-
  R                  SL a  U R                  UR	                  5       5      $ g )Nr   r   F©r,   Úis_realÚis_positiveÚfuncÚ	conjugate©r.   r&   s     r'   Ú_eval_conjugateÚelliptic_k._eval_conjugateL   sD   € Ø�I‰I�a‰LˆØ�I‰I×-˜1˜q™5×-Ñ-°%Ò7Ø—9‘9˜QŸ[™[›]Ó+Ð+ð 8r*   c                 ó`   • SSK Jn  U" U R                  [        5      R	                  XUS95      $ )Nr   ©Úhyperexpand©ÚnÚlogx)Úsympy.simplifyr=   Úrewriter   Ú_eval_nseries)r.   Úxr?   r@   Úcdirr=   s         r'   rC   Úelliptic_k._eval_nseriesQ   s)   € Ý.Ù˜4Ÿ<™<¬Ó.×<Ñ<¸QÈ$Ð<ÐOÓPÐPr*   c                 ó¦   • [         [        R                  -  [        [        R                  [        R                  4[        R                  4U5      -  $ ©N)r   r   r   r   r   ©r.   r&   Úkwargss      r'   Ú_eval_rewrite_as_hyperÚ!elliptic_k._eval_rewrite_as_hyperU   s1   € Ü”!—&‘&‰yœ¤§¡¬¯©Ð/´!·%±%°¸1Ó=Ñ=Ð=r*   c                 ó¤   • [        [        R                  [        R                  4/ 4[        R                  4[        R                  44U* 5      S-  $ ©Nr   )r   r   r   r$   rI   s      r'   Ú_eval_rewrite_as_meijergÚ#elliptic_k._eval_rewrite_as_meijergX   s;   € ÜœŸ™¤§¡Ð(¨"Ð-´·±°	¼A¿F¹F¸9Ð/EÈÀrÓJÈ1ÑLÐLr*   c                 óF   • U R                   S   nUR                  (       a  gg )Nr   T)r,   Úis_infiniter8   s     r'   Ú_eval_is_zeroÚelliptic_k._eval_is_zero[   s   € Ø�I‰I�a‰LˆØ�=�=Øð r*   c           
      óÌ   • SSK Jn  [        [        SU5      R                  5      nU R
                  S   nU" S[        SU[        U5      S-  -  -
  5      -  US[        S-  45      $ ©Nr   ©ÚIntegralÚtr   r   )	Úsympy.integrals.integralsrX   r	   r
   Únamer,   r   r   r   )r.   r,   rJ   rX   rY   r&   s         r'   Ú_eval_rewrite_as_IntegralÚ$elliptic_k._eval_rewrite_as_Integral`   s[   € Ý6ÜÔ'¨¨TÓ2×7Ñ7Ó8ˆØ�I‰I�a‰LˆÙ˜œ$˜q 1¤S¨£V¨Q¡Y¡;™Ó/Ñ/°!°Q¼¸1¹°Ó>Ð>r*   © N©r   ©r   )Ú__name__Ú
__module__Ú__qualname__Ú__firstlineno__Ú__doc__Úclassmethodr(   r0   r9   rC   rK   rO   rS   r\   Ú__static_attributes__r^   r*   r'   r   r      sB   † ñ*ðX ñó ðôEò,ô
Qò>òMòõ
?r*   r   c                   óD   • \ rS rSrSr\S 5       rS
S jrS rS r	S r
Srg	)Ú
elliptic_fég   aŠ  
The Legendre incomplete elliptic integral of the first
kind, defined by

.. math:: F\left(z\middle| m\right) =
          \int_0^z \frac{dt}{\sqrt{1 - m \sin^2 t}}

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

This function reduces to a complete elliptic integral of
the first kind, $K(m)$, when $z = \pi/2$.

Note that our notation defines the incomplete elliptic integral
in terms of the parameter $m$ instead of the elliptic modulus
(eccentricity) $k$.
In this case, the parameter $m$ is defined as $m=k^2$.

Examples
========

>>> from sympy import elliptic_f, I
>>> from sympy.abc import z, m
>>> elliptic_f(z, m).series(z)
z + z**5*(3*m**2/40 - m/30) + m*z**3/6 + O(z**6)
>>> elliptic_f(3.0 + I/2, 1.0 + I)
2.909449841483 + 1.74720545502474*I

See Also
========

elliptic_k

References
==========

.. [1] https://en.wikipedia.org/wiki/Elliptic_integrals
.. [2] https://functions.wolfram.com/EllipticIntegrals/EllipticF

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r   r   r$   r   Ú
is_integerr   r"   r#   Úcould_extract_minus_signri   )r%   Úzr&   Úks       r'   r(   Úelliptic_f.eval‘   s†   € à�9�9Ü—6‘6ˆMØ�9�9ØˆHØˆa‰C”‰FˆØ�<�<Ø”Z “]‘?Ð"Ø”1—:‘:œq×1Ñ1Ð2Ó2Ü—6‘6ˆMØ×'Ñ'×)Ñ)Ü ˜r 1Ó%Ð%Ð%ð *r*   c                 ó  • U R                   u  p#[        SU[        U5      S-  -  -
  5      nUS:X  a  SU-  $ US:X  aD  [        X#5      SU-  SU-
  -  -  [	        X#5      SU-  -  -
  [        SU-  5      SSU-
  -  U-  -  -
  $ [        X5      e)Nr   r   r   )r,   r   r   r-   ri   r   )r.   r/   rn   r&   Úfms        r'   r0   Úelliptic_f.fdiffŸ   s›   € Ø�y‰y‰ˆÜ�!�aœ˜A› ™	‘k‘/Ó"ˆØ�q‹=Ø�R‘4ˆKØ˜‹]Ü˜qÓ$ a¨¡c¨1¨q©5¡kÑ2´ZÀÓ5EÀqÈÁsÑ5KÑKÜ˜˜!™“H˜a  Q¡™i¨™lÑ+ñ,ð -ä  Ó0Ð0r*   c                 óÆ   • U R                   u  pUR                  =(       a    US-
  R                  SL a.  U R                  UR	                  5       UR	                  5       5      $ g )Nr   Fr3   ©r.   rn   r&   s      r'   r9   Úelliptic_f._eval_conjugate©   sJ   € Ø�y‰y‰ˆØ�I‰I×-˜1˜q™5×-Ñ-°%Ò7Ø—9‘9˜QŸ[™[›]¨A¯K©K«MÓ:Ð:ð 8r*   c           
      óÚ   • SSK Jn  [        [        SU5      R                  5      nU R
                  S   U R
                  S   peU" S[        SU[        U5      S-  -  -
  5      -  USU45      $ rV   )rZ   rX   r	   r
   r[   r,   r   r   )r.   r,   rJ   rX   rY   rn   r&   s          r'   r\   Ú$elliptic_f._eval_rewrite_as_Integral®   sa   € Ý6ÜÔ'¨¨TÓ2×7Ñ7Ó8ˆØ�y‰y˜‰|˜TŸY™Y q™\ˆ1Ù˜œ4  A¤c¨!£f¨a¡i¡K¡Ó0Ñ1°A°q¸!°9Ó=Ð=r*   c                 óŒ   • U R                   u  pUR                  (       a  gUR                  (       a  UR                  (       a  gg g )NT)r,   r   Úis_extended_realrR   ru   s      r'   rS   Úelliptic_f._eval_is_zero´   s1   € Ø�y‰y‰ˆØ�9�9ØØ×× !§-§-Øð #0Ðr*   r^   Nr_   )ra   rb   rc   rd   re   rf   r(   r0   r9   r\   rS   rg   r^   r*   r'   ri   ri   g   s0   † ñ'ðR ñ&ó ð&ô1ò;ò
>õr*   ri   c                   óf   ^ • \ rS rSrSr\SS j5       rSS jrS rSU 4S jjr	S r
S rS	 rS
rU =r$ )r-   é¼   a(  
Called with two arguments $z$ and $m$, evaluates the
incomplete elliptic integral of the second kind, defined by

.. math:: E\left(z\middle| m\right) = \int_0^z \sqrt{1 - m \sin^2 t} dt

Called with a single argument $m$, evaluates the Legendre complete
elliptic integral of the second kind

.. math:: E(m) = E\left(\tfrac{\pi}{2}\middle| m\right)

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

The function $E(m)$ is a single-valued function on the complex
plane with branch cut along the interval $(1, \infty)$.

Note that our notation defines the incomplete elliptic integral
in terms of the parameter $m$ instead of the elliptic modulus
(eccentricity) $k$.
In this case, the parameter $m$ is defined as $m=k^2$.

Examples
========

>>> from sympy import elliptic_e, I
>>> from sympy.abc import z, m
>>> elliptic_e(z, m).series(z)
z + z**5*(-m**2/40 + m/30) - m*z**3/6 + O(z**6)
>>> elliptic_e(m).series(n=4)
pi/2 - pi*m/8 - 3*pi*m**2/128 - 5*pi*m**3/512 + O(m**4)
>>> elliptic_e(1 + I, 2 - I/2).n()
1.55203744279187 + 0.290764986058437*I
>>> elliptic_e(0)
pi/2
>>> elliptic_e(2.0 - I)
0.991052601328069 + 0.81879421395609*I

References
==========

.. [1] https://en.wikipedia.org/wiki/Elliptic_integrals
.. [2] https://functions.wolfram.com/EllipticIntegrals/EllipticE2
.. [3] https://functions.wolfram.com/EllipticIntegrals/EllipticE

c                 óÖ  • Ub¹  XpSU-  [         -  nUR                  (       a  U$ UR                  (       a  [        R                  $ UR                  (       a  U[        U5      -  $ U[        R                  [        R                  4;   a  [        R                  $ UR                  5       (       a  [        U* U5      * $ g UR                  (       a	  [         S-  $ U[        R                  L a  [        R                  $ U[        R                  L a  [        [        R                  -  $ U[        R                  L a  [        R                  $ U[        R                  L a  [        R                  $ g rN   )r   r   r   r$   rl   r-   r"   r#   r    rm   r   r   )r%   r&   rn   ro   s       r'   r(   Úelliptic_e.evalì   s	  € à‰=ØˆqØ�!‘”B‘ˆAØ�y�yØ�Ø�y�yÜ—v‘v�Ø——Øœ A›‘Ð&Ø”q—z‘z¤1×#5Ñ#5Ð6Ó6Ü×(Ñ(Ð(Ø×+Ñ+×-Ñ-Ü" A 2 qÓ)Ð)Ð)ð .ð �y�yÜ˜!‘t�Ø”a—e‘e’Ü—u‘u�Ø”a—j‘j’ÜœŸ™‘|Ð#Ø”a×(Ñ(Ò(Ü—z‘zÐ!Ø”a×'Ñ'Ò'Ü×(Ñ(Ð(ð (r*   c                 óX  • [        U R                  5      S:X  aU  U R                  u  p#US:X  a  [        SU[        U5      S-  -  -
  5      $ US:X  a  [	        X#5      [        X#5      -
  SU-  -  $ O2U R                  S   nUS:X  a  [	        U5      [        U5      -
  SU-  -  $ [        X5      e)Nr   r   r   )Úlenr,   r   r   r-   ri   r   r   )r.   r/   rn   r&   s       r'   r0   Úelliptic_e.fdiff  s¦   € Üˆt�y‰y‹>˜QÓØ—9‘9‰DˆAØ˜1‹}Ü˜A ¤# a£&¨!¡)¡™OÓ,Ð,Ø˜Q“Ü" 1Ó(¬:°aÓ+;Ñ;¸aÀ¹cÑBÐBð ð —	‘	˜!‘ˆAØ˜1‹}Ü" 1›¬
°1«Ñ5¸¸!¹Ñ<Ð<Ü  Ó0Ð0r*   c                 ó   • [        U R                  5      S:X  ab  U R                  u  pUR                  =(       a    US-
  R                  SL a.  U R	                  UR                  5       UR                  5       5      $ g U R                  S   nUR                  =(       a    US-
  R                  SL a  U R	                  UR                  5       5      $ g )Nr   r   Fr   ©r�   r,   r4   r5   r6   r7   ru   s      r'   r9   Úelliptic_e._eval_conjugate  sž   € Üˆt�y‰y‹>˜QÓØ—9‘9‰DˆAØ—	‘	×1˜q 1™u×1Ñ1°eÒ;Ø—y‘y §¡£°·±³Ó>Ð>ð <ð —	‘	˜!‘ˆAØ—	‘	×1˜q 1™u×1Ñ1°eÒ;Ø—y‘y §¡£Ó/Ð/ð <r*   c                 ó°   >• SSK Jn  [        U R                  5      S:X  a)  U" U R	                  [
        5      R                  XUS95      $ [        TU ]  XUS9$ )Nr   r<   r   r>   )rA   r=   r�   r,   rB   r   rC   Úsuper)r.   rD   r?   r@   rE   r=   Ú	__class__s         €r'   rC   Úelliptic_e._eval_nseries  sO   ø€ Ý.Üˆt�y‰y‹>˜QÓÙ˜tŸ|™|¬EÓ2×@Ñ@ÀÈdÐ@ÐSÓTÐTÜ‰wÑ$ Q°$Ð$Ð7Ð7r*   c                 ó¬   • [        U5      S:X  aE  US   n[        S-  [        [        SS5      [        R
                  4[        R                  4U5      -  $ g )Nr   r   r   r   )r�   r   r   r   r   r   r   ©r.   r,   rJ   r&   s       r'   rK   Ú!elliptic_e._eval_rewrite_as_hyper$  sH   € Üˆt‹9˜‹>Ø�Q‘ˆAÜ�q‘Dœ%¤¨"¨a£´!·&±&Ð 9¼A¿E¹E¸8ÀQÓGÑGÐGð r*   c                 óÈ   • [        U5      S:X  aS  US   n[        [        R                  [	        SS5      4/ 4[        R
                  4[        R
                  44U* 5      * S-  $ g )Nr   r   r   r   r   )r�   r   r   r   r   r$   r‹   s       r'   rO   Ú#elliptic_e._eval_rewrite_as_meijerg)  sc   € Üˆt‹9˜‹>Ø�Q‘ˆAÜœaŸf™f¤h¨q°!£nÐ5°rÐ:ÜŸf™f˜Y¬¯©¨	Ð2°Q°Bó8ð 8Ø89ñ:ð :ð r*   c           	      ó  • SSK Jn  [        U R                  5      S:X  a  [        S-  U R                  S   4OU R                  u  pE[        [        SU5      R                  5      nU" [        SU[        U5      S-  -  -
  5      USU45      $ )Nr   rW   r   r   rY   )
rZ   rX   r�   r,   r   r	   r
   r[   r   r   )r.   r,   rJ   rX   rn   r&   rY   s          r'   r\   Ú$elliptic_e._eval_rewrite_as_Integral/  sr   € Ý6Ü'*¨4¯9©9£~¸Ó':”�1‘�d—i‘i ‘lÑ#ÀÇ	Á	‰ˆÜÔ'¨¨TÓ2×7Ñ7Ó8ˆÙœ˜Q ¤3 q£6¨1¡9¡™_Ó-°°1°a¨yÓ9Ð9r*   r^   rH   r_   r`   )ra   rb   rc   rd   re   rf   r(   r0   r9   rC   rK   rO   r\   rg   Ú__classcell__)rˆ   s   @r'   r-   r-   ¼   sA   ø† ñ-ð^ ó)ó ð)ô41ò0÷8òHò
:÷:ð :r*   r-   c                   óB   • \ rS rSrSr\S	S j5       rS rS
S jrS r	Sr
g)Úelliptic_pii6  aÏ  
Called with three arguments $n$, $z$ and $m$, evaluates the
Legendre incomplete elliptic integral of the third kind, defined by

.. math:: \Pi\left(n; z\middle| m\right) = \int_0^z \frac{dt}
          {\left(1 - n \sin^2 t\right) \sqrt{1 - m \sin^2 t}}

Called with two arguments $n$ and $m$, evaluates the complete
elliptic integral of the third kind:

.. math:: \Pi\left(n\middle| m\right) =
          \Pi\left(n; \tfrac{\pi}{2}\middle| m\right)

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

Note that our notation defines the incomplete elliptic integral
in terms of the parameter $m$ instead of the elliptic modulus
(eccentricity) $k$.
In this case, the parameter $m$ is defined as $m=k^2$.

Examples
========

>>> from sympy import elliptic_pi, I
>>> from sympy.abc import z, n, m
>>> elliptic_pi(n, z, m).series(z, n=4)
z + z**3*(m/6 + n/3) + O(z**4)
>>> elliptic_pi(0.5 + I, 1.0 - I, 1.2)
2.50232379629182 - 0.760939574180767*I
>>> elliptic_pi(0, 0)
pi/2
>>> elliptic_pi(1.0 - I/3, 2.0 + I)
3.29136443417283 + 0.32555634906645*I

References
==========

.. [1] https://en.wikipedia.org/wiki/Elliptic_integrals
.. [2] https://functions.wolfram.com/EllipticIntegrals/EllipticPi3
.. [3] https://functions.wolfram.com/EllipticIntegrals/EllipticPi

Nc           	      ót  • UGb*  X#p#UR                   (       a  [        X25      $ U[        R                  L aG  [        X25      [	        SU[        U5      S-  -  -
  5      [        U5      -  [        X25      -
  SU-
  -  -   $ SU-  [        -  nUR                  (       a  U[        X5      -  $ UR                   (       a2  [        [	        US-
  5      [        U5      -  5      [	        US-
  5      -  $ X:X  aB  [        X15      [        SX15      -
  [        U5      [	        SU[        U5      S-  -  -
  5      -  -   $ U[        R                  [        R                  4;   a  [        R                  $ U[        R                  [        R                  4;   a  [        R                  $ UR                  5       (       a  [        X* U5      * $ UR                   (       a  [        X25      $ UR                   (       a  UR"                  (       d"  UR                   (       a"  UR"                  (       a  [        R                  $ g g UR                   (       a  [%        U5      $ U[        R                  L a  [        R&                  $ UR                   (       a  [        S[	        SU-
  5      -  -  $ U[        R                  :X  a  [        R                  [)        US-
  5      -  $ X:X  a  [        U5      SU-
  -  $ U[        R                  [        R                  4;   a  [        R                  $ U[        R                  [        R                  4;   a  [        R                  $ UR                   (       a  [%        U5      $ UR                   (       a  UR"                  (       d"  UR                   (       a"  UR"                  (       a  [        R                  $ g g )Nr   r   )r   ri   r   r   r   r   r   r-   r   rl   r“   r   r"   r#   r$   rm   rz   rR   r   r    r   )r%   r?   r&   rn   ro   s        r'   r(   Úelliptic_pi.evalc  sµ  € àŠ=ØˆqØ�y�yÜ! !Ó'Ð'Ø”a—e‘e’Ü" 1Ó(Ü˜a !¤C¨£F¨A¡I¡+™oÓ.¬s°1«vÑ5Ü# AÓ)ñ*Ø,-°©Eñ3ñ3ð 4ð �!‘”B‘ˆAØ�|�|Øœ QÓ*Ñ*Ð*Ø——ÜœT ! a¡%›[¬¨Q«Ñ/Ó0´°a¸!±e³Ñ<Ð<Ø“Ü" 1Ó(¬;°q¸!Ó+?Ñ?Ü˜A›œt A¨¬#¨a«&°!©)©¡OÓ4Ñ4ñ5ð 6à”q—z‘z¤1×#5Ñ#5Ð6Ó6Ü—v‘v�Ø”q—z‘z¤1×#5Ñ#5Ð6Ó6Ü—v‘v�Ø×+Ñ+×-Ñ-Ü# A r¨1Ó-Ð-Ð-Ø�y�yÜ! !Ó'Ð'Ø×!×! a§m§mØ×&×&¨1¯=¯=Ü—v‘v�ð ,9Ð&ð �y�yÜ! !“}Ð$Ø”a—e‘e’Ü×(Ñ(Ð(Ø——Ü˜1œT ! a¡%›[™=Ñ)Ð)Ø”a—e‘e“Ü×)Ñ)¬$¨q°1©u«+Ñ5Ð5Ø“Ü! !“} a¨!¡eÑ,Ð,Ø”q—z‘z¤1×#5Ñ#5Ð6Ó6Ü—v‘v�Ø”q—z‘z¤1×#5Ñ#5Ð6Ó6Ü—v‘v�Ø�y�yÜ! !“}Ð$Ø×!×! a§m§mØ×&×&¨1¯=¯=Ü—v‘v�ð ,9Ð&r*   c                 óÜ  • [        U R                  5      S:X  a˜  U R                  u  pnUR                  =(       a    US-
  R                  SL ac  UR                  =(       a    US-
  R                  SL a=  U R	                  UR                  5       UR                  5       UR                  5       5      $ g g U R                  u  pU R	                  UR                  5       UR                  5       5      $ )Nr   r   Fr„   )r.   r?   rn   r&   s       r'   r9   Úelliptic_pi._eval_conjugate•  s®   € Üˆt�y‰y‹>˜QÓØ—i‘i‰GˆA�!Ø—	‘	×1˜q 1™u×1Ñ1°eÒ;Ø—	‘	×1˜q 1™u×1Ñ1°eÒ;Ø—y‘y §¡£°·±³¸q¿{¹{»}ÓMÐMð <ð <ð —9‘9‰DˆAØ—9‘9˜QŸ[™[›]¨A¯K©K«MÓ:Ð:r*   c                 óL  • [        U R                  5      S:X  aý  U R                  u  p#n[        SU[        U5      S-  -  -
  5      SU[        U5      S-  -  -
  peUS:X  a`  [	        X45      XB-
  [        X45      -  U-  -   US-  U-
  [        X#U5      -  U-  -   X%-  [        SU-  5      -  SU-  -  -
  SXB-
  -  US-
  -  -  $ US:X  a  SXV-  -  $ US:X  aD  [	        X45      US-
  -  [        X#U5      -   U[        SU-  5      -  SUS-
  -  U-  -  -
  SX$-
  -  -  $ O„U R                  u  p$US:X  aE  [	        U5      XB-
  [        U5      -  U-  -   US-  U-
  [        X$5      -  U-  -   SXB-
  -  US-
  -  -  $ US:X  a%  [	        U5      US-
  -  [        X$5      -   SX$-
  -  -  $ [        X5      e)Nr   r   r   )	r�   r,   r   r   r-   ri   r“   r   r   )r.   r/   r?   rn   r&   rr   Úfns          r'   r0   Úelliptic_pi.fdiffŸ  sî  € Üˆt�y‰y‹>˜QÓØ—i‘i‰GˆA�!Ü˜!˜a¤ A£¨¡	™k™/Ó*¨A°´#°a³&¸!±)±©O�Ø˜1‹}Ü" 1Ó(¨A©E´:¸aÓ3CÑ+CÀAÑ+EÑEØ˜A™ ™¤;¨q°QÓ#7Ñ7¸Ñ9ñ:à™œS  1¡›X™ q¨¡tÑ,ñ-à/0°!±%©y¸!¸a¹%Ñ/@ñBð Bð ˜Q“Ø˜"™%‘yÐ Ø˜Q“Ü" 1Ó(¨!¨a©%Ñ0Ü# A¨!Ó,ñ-àœ#˜a ™c›(™
 A q¨1¡u¡I¨b¡LÑ1ñ2à45°q±u±Iñ?ð ?ð ð
 —9‘9‰DˆAØ˜1‹}Ü" 1›¨©´
¸1³Ñ(=¸aÑ(?Ñ?Ø˜A™ ™¤;¨qÓ#4Ñ4°QÑ6ñ7Ø9:¸A¹E¹ÀAÈÁEÑ9JñLð Là˜Q“Ü" 1› q¨1¡uÑ-´¸AÓ0AÑAÀAÀqÁuÁIÑNÐNÜ  Ó0Ð0r*   c                 óh  • SSK Jn  [        U R                  5      S:X  a'  U R                  S   U R                  S   [        S-  penOU R                  u  pFn[        [        SU5      R                  5      nU" SSU[        U5      S-  -  -
  [        SU[        U5      S-  -  -
  5      -  -  USU45      $ )Nr   rW   r   r   rY   )
rZ   rX   r�   r,   r   r	   r
   r[   r   r   )r.   r,   rJ   rX   r?   r&   rn   rY   s           r'   r\   Ú%elliptic_pi._eval_rewrite_as_Integral¶  s�   € Ý6Üˆt�y‰y‹>˜QÓØ—i‘i ‘l D§I¡I¨a¡L´"°Q±$�!ˆA�!à—i‘i‰GˆA�!ÜÔ'¨¨TÓ2×7Ñ7Ó8ˆÙ˜˜A ¤# a£&¨!¡)¡™O¬T°!°a¼¸A»À¹	±k±/Ó-BÑBÑCÀaÈÈAÀYÓOÐOr*   r^   rH   r_   )ra   rb   rc   rd   re   rf   r(   r9   r0   r\   rg   r^   r*   r'   r“   r“   6  s-   † ñ*ðX ó/ó ð/òb;ô1õ.Pr*   r“   N)re   Ú
sympy.corer   r   r   r   Úsympy.core.functionr   r   Úsympy.core.symbolr	   r
   Ú$sympy.functions.elementary.complexesr   Ú%sympy.functions.elementary.hyperbolicr   Ú(sympy.functions.elementary.miscellaneousr   Ú(sympy.functions.elementary.trigonometricr   r   Ú'sympy.functions.special.gamma_functionsr   Úsympy.functions.special.hyperr   r   r   ri   r-   r“   r^   r*   r'   Ú<module>r¦      sc   ðÙ ç )Ó )ß Cß 9Ý 5Ý 7Ý 9ß =Ý 9ß 8ôW?�ô W?ôtR�ô Rôjw:�ô w:ôtGP�/õ GPr*   