Ë
    7^(hI:  ã                   óè   — d Z ddlmZmZmZmZ ddlmZmZ ddl	m
Z
mZ ddlmZ ddlmZ ddlmZ ddlmZmZ dd	lmZ dd
lmZmZ  G d„ de«      Z G d„ de«      Z G d„ de«      Z G d„ de«      Zy)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                   óN   — e Zd ZdZed„ «       Zdd„Zd„ Zdd„Zd„ Z	d„ Z
d„ Zd	„ Zy
)Ú
elliptic_kaN  
    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

    c                 ój  — |j                   rt        t        j                  z  S |t        j                  u r/dt        t	        dd«      z  z  t        t	        dd«      «      dz  z  S |t        j                  u rt        j                  S |t        j                  u r.t        t	        dd«      «      dz  dt        dt        z  «      z  z  S |t        j                  t        j                  t        t        j                  z  t        t        j                  z  t        j                  fv rt        j                  S y )Né   é   é   éÿÿÿÿé   é   )Úis_zeror   r   ÚHalfr   r   ÚOneÚComplexInfinityÚNegativeOner   ÚInfinityÚNegativeInfinityr   ÚZero)ÚclsÚms     úh/var/www/skyplay_api_hub/venv/lib/python3.12/site-packages/sympy/functions/special/elliptic_integrals.pyÚevalzelliptic_k.eval:   sà   € à�9Š9Ü”a—f‘f‘9ÐØ”!—&‘&‰[Ø”Rœ ! Q›Ñ'Ñ'¬¬h°r¸1«oÓ(>ÀÑ(AÑAÐAØ”!—%‘%‰ZÜ×$Ñ$Ð$Ø”!—-‘-ÑÜœ ! Q›Ó(¨!Ñ+¨Q¬t°A´b±D«z©\Ñ:Ð:Ø”1—:‘:œq×1Ñ1´1´Q·Z±Z±<Ü”Q×'Ñ'Ñ'¬×):Ñ):ð<ñ <ä—6‘6ˆMð<ó    c                 ór   — | j                   d   }t        |«      d|z
  t        |«      z  z
  d|z  d|z
  z  z  S )Nr   r   r   )ÚargsÚ
elliptic_er   )ÚselfÚargindexr%   s      r&   Úfdiffzelliptic_k.fdiffH   s<   € Ø�I‰I�a‰LˆÜ˜1“  Q¡¬
°1«Ñ 5Ñ5¸¸!¹¸QÀ¹U¹ÑDÐDr(   c                 óž   — | j                   d   }|j                  xr |dz
  j                  du r| j                  |j	                  «       «      S y )Nr   r   F©r*   Úis_realÚis_positiveÚfuncÚ	conjugate©r,   r%   s     r&   Ú_eval_conjugatezelliptic_k._eval_conjugateL   sD   € Ø�I‰I�a‰LˆØ�I‰IÒ-˜1˜q™5×-Ñ-°%Ñ7Ø—9‘9˜QŸ[™[›]Ó+Ð+ð 8r(   c                 óh   — ddl m}  || j                  t        «      j	                  |||¬«      «      S )Nr   ©Úhyperexpand©ÚnÚlogx)Úsympy.simplifyr9   Úrewriter   Ú_eval_nseries)r,   Úxr;   r<   Úcdirr9   s         r&   r?   zelliptic_k._eval_nseriesQ   s+   € Ý.Ù˜4Ÿ<™<¬Ó.×<Ñ<¸QÀ!È$Ð<ÓOÓPÐPr(   c                 ó¦   — t         t        j                  z  t        t        j                  t        j                  ft        j                  f|«      z  S ©N)r   r   r   r   r   ©r,   r%   Úkwargss      r&   Ú_eval_rewrite_as_hyperz!elliptic_k._eval_rewrite_as_hyperU   s1   € Ü”!—&‘&‰yœ¤§¡¬¯©Ð/´!·%±%°¸1Ó=Ñ=Ð=r(   c                 ó¤   — t        t        j                  t        j                  fg ft        j                  ft        j                  ff| «      dz  S ©Nr   )r   r   r   r#   rD   s      r&   Ú_eval_rewrite_as_meijergz#elliptic_k._eval_rewrite_as_meijergX   s;   € ÜœŸ™¤§¡Ð(¨"Ð-´·±°	¼A¿F¹F¸9Ð/EÈÀrÓJÈ1ÑLÐLr(   c                 ó<   — | j                   d   }|j                  ryy )Nr   T)r*   Úis_infiniter5   s     r&   Ú_eval_is_zerozelliptic_k._eval_is_zero[   s   € Ø�I‰I�a‰LˆØ�=Š=Øð r(   c           
      óÌ   — ddl m} t        t        d|«      j                  «      }| j
                  d   } |dt        d|t        |«      dz  z  z
  «      z  |dt        dz  f«      S ©Nr   ©ÚIntegralÚtr   r   )	Úsympy.integrals.integralsrP   r	   r
   Únamer*   r   r   r   )r,   r*   rE   rP   rQ   r%   s         r&   Ú_eval_rewrite_as_Integralz$elliptic_k._eval_rewrite_as_Integral`   s[   € Ý6ÜÔ'¨¨TÓ2×7Ñ7Ó8ˆØ�I‰I�a‰LˆÙ˜œ$˜q 1¤S¨£V¨Q¡Y¡;™Ó/Ñ/°!°Q¼¸1¹°Ó>Ð>r(   N©r   ©r   )Ú__name__Ú
__module__Ú__qualname__Ú__doc__Úclassmethodr'   r.   r6   r?   rF   rI   rL   rT   © r(   r&   r   r      sB   „ ñ*ðX ñó ðóEò,ó
Qò>òMòó
?r(   r   c                   ó:   — e Zd ZdZed„ «       Zdd„Zd„ Zd„ Zd„ Z	y)	Ú
elliptic_faú  
    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

    c                 óD  — |j                   rt        j                  S |j                   r|S d|z  t        z  }|j                  r|t        |«      z  S |t        j                  t        j                  fv rt        j                  S |j                  «       rt        | |«       S y rH   )
r   r   r#   r   Ú
is_integerr   r!   r"   Úcould_extract_minus_signr^   )r$   Úzr%   Úks       r&   r'   zelliptic_f.eval‘   sƒ   € à�9Š9Ü—6‘6ˆMØ�9Š9ØˆHØˆa‰C”‰FˆØ�<Š<Ø”Z “]‘?Ð"Ø”1—:‘:œq×1Ñ1Ð2Ñ2Ü—6‘6ˆMØ×'Ñ'Ô)Ü ˜r 1Ó%Ð%Ð%ð *r(   c                 ó  — | j                   \  }}t        d|t        |«      dz  z  z
  «      }|dk(  rd|z  S |dk(  rFt        ||«      d|z  d|z
  z  z  t	        ||«      d|z  z  z
  t        d|z  «      dd|z
  z  |z  z  z
  S t        | |«      ‚)Nr   r   r   )r*   r   r   r+   r^   r   )r,   r-   rb   r%   Úfms        r&   r.   zelliptic_f.fdiffŸ   s£   € Ø�y‰y‰ˆˆ1Ü�!�aœ˜A› ™	‘k‘/Ó"ˆØ�qŠ=Ø�R‘4ˆKØ˜Š]Ü˜q !Ó$ a¨¡c¨1¨q©5¡kÑ2´ZÀÀ1Ó5EÀqÈÁsÑ5KÑKÜ˜˜!™“H˜a  Q¡™i¨™lÑ+ñ,ð -ä   xÓ0Ð0r(   c                 ó¼   — | j                   \  }}|j                  xr |dz
  j                  du r.| j                  |j	                  «       |j	                  «       «      S y )Nr   Fr0   ©r,   rb   r%   s      r&   r6   zelliptic_f._eval_conjugate©   sL   € Ø�y‰y‰ˆˆ1Ø�I‰IÒ-˜1˜q™5×-Ñ-°%Ñ7Ø—9‘9˜QŸ[™[›]¨A¯K©K«MÓ:Ð:ð 8r(   c           
      óÜ   — ddl m} t        t        d|«      j                  «      }| j
                  d   | j
                  d   }} |dt        d|t        |«      dz  z  z
  «      z  |d|f«      S rN   )rR   rP   r	   r
   rS   r*   r   r   )r,   r*   rE   rP   rQ   rb   r%   s          r&   rT   z$elliptic_f._eval_rewrite_as_Integral®   sc   € Ý6ÜÔ'¨¨TÓ2×7Ñ7Ó8ˆØ�y‰y˜‰|˜TŸY™Y q™\ˆ1ˆÙ˜œ4  A¤c¨!£f¨a¡i¡K¡Ó0Ñ1°A°q¸!°9Ó=Ð=r(   c                 óp   — | j                   \  }}|j                  ry|j                  r|j                  ryy y )NT)r*   r   Úis_extended_realrK   rg   s      r&   rL   zelliptic_f._eval_is_zero´   s3   € Ø�y‰y‰ˆˆ1Ø�9Š9ØØ×Ò !§-¢-Øð #0Ðr(   NrU   )
rW   rX   rY   rZ   r[   r'   r.   r6   rT   rL   r\   r(   r&   r^   r^   g   s0   „ ñ'ðR ñ&ó ð&ó1ò;ò
>ór(   r^   c                   óV   ‡ — e Zd ZdZed	d„«       Zd
d„Zd„ Zdˆ fd„	Zd„ Z	d„ Z
d„ Zˆ xZS )r+   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                 óš  — |�¥||}}d|z  t         z  }|j                  r|S |j                  rt        j                  S |j                  r|t        |«      z  S |t        j                  t        j                  fv rt        j                  S |j                  «       rt        | |«       S y |j                  r	t         dz  S |t        j                  u rt        j                  S |t        j                  u rt        t        j                  z  S |t        j                  u rt        j                  S |t        j                  u rt        j                  S y rH   )r   r   r   r#   r`   r+   r!   r"   r   ra   r   r   )r$   r%   rb   rc   s       r&   r'   zelliptic_e.evalì   s
  € àˆ=Ø�aˆqˆAØ�!‘”B‘ˆAØ�yŠyØ�Ø�yŠyÜ—v‘v�Ø—’Øœ A›‘Ð&Ø”q—z‘z¤1×#5Ñ#5Ð6Ñ6Ü×(Ñ(Ð(Ø×+Ñ+Ô-Ü" A 2 qÓ)Ð)Ð)ð .ð �yŠyÜ˜!‘t�Ø”a—e‘e‘Ü—u‘u�Ø”a—j‘j‘ÜœŸ™‘|Ð#Ø”a×(Ñ(Ñ(Ü—z‘zÐ!Ø”a×'Ñ'Ñ'Ü×(Ñ(Ð(ð (r(   c                 óV  — t        | j                  «      dk(  rU| j                  \  }}|dk(  rt        d|t        |«      dz  z  z
  «      S |dk(  rPt	        ||«      t        ||«      z
  d|z  z  S | j                  d   }|dk(  rt	        |«      t        |«      z
  d|z  z  S t        | |«      ‚)Nr   r   r   )Úlenr*   r   r   r+   r^   r   r   )r,   r-   rb   r%   s       r&   r.   zelliptic_e.fdiff  s§   € Üˆt�y‰y‹>˜QÒØ—9‘9‰DˆAˆqØ˜1Š}Ü˜A ¤# a£&¨!¡)¡™OÓ,Ð,Ø˜Q’Ü" 1 aÓ(¬:°a¸Ó+;Ñ;¸aÀ¹cÑBÐBà—	‘	˜!‘ˆAØ˜1Š}Ü" 1›¬
°1«Ñ5¸¸!¹Ñ<Ð<Ü   xÓ0Ð0r(   c                 óˆ  — t        | j                  «      dk(  r]| j                  \  }}|j                  xr |dz
  j                  du r.| j	                  |j                  «       |j                  «       «      S y | j                  d   }|j                  xr |dz
  j                  du r| j	                  |j                  «       «      S y )Nr   r   Fr   ©rn   r*   r1   r2   r3   r4   rg   s      r&   r6   zelliptic_e._eval_conjugate  s    € Üˆt�y‰y‹>˜QÒØ—9‘9‰DˆAˆqØ—	‘	Ò1˜q 1™u×1Ñ1°eÑ;Ø—y‘y §¡£°·±³Ó>Ð>ð <ð —	‘	˜!‘ˆAØ—	‘	Ò1˜q 1™u×1Ñ1°eÑ;Ø—y‘y §¡£Ó/Ð/ð <r(   c                 ó¾   •— ddl m} t        | j                  «      dk(  r- || j	                  t
        «      j                  |||¬«      «      S t        ‰| �  |||¬«      S )Nr   r8   r   r:   )r=   r9   rn   r*   r>   r   r?   Úsuper)r,   r@   r;   r<   rA   r9   Ú	__class__s         €r&   r?   zelliptic_e._eval_nseries  sS   ø€ Ý.Üˆt�y‰y‹>˜QÒÙ˜tŸ|™|¬EÓ2×@Ñ@ÀÀaÈdÐ@ÓSÓTÐTÜ‰wÑ$ Q¨!°$Ð$Ó7Ð7r(   c                 óª   — t        |«      dk(  rE|d   }t        dz  t        t        dd«      t        j
                  ft        j                  f|«      z  S y )Nr   r   r   r   )rn   r   r   r   r   r   r   ©r,   r*   rE   r%   s       r&   rF   z!elliptic_e._eval_rewrite_as_hyper$  sH   € Üˆt‹9˜Š>Ø�Q‘ˆAÜ�q‘Dœ%¤¨"¨a£´!·&±&Ð 9¼A¿E¹E¸8ÀQÓGÑGÐGð r(   c                 óÆ   — t        |«      dk(  rS|d   }t        t        j                  t	        dd«      fg ft        j
                  ft        j
                  ff| «       dz  S y )Nr   r   r   r   r   )rn   r   r   r   r   r#   ru   s       r&   rI   z#elliptic_e._eval_rewrite_as_meijerg)  sc   € Üˆt‹9˜Š>Ø�Q‘ˆAÜœaŸf™f¤h¨q°!£nÐ5°rÐ:ÜŸf™f˜Y¬¯©¨	Ð2°Q°Bó8ð 8Ø89ñ:ð :ð r(   c           	      ó  — ddl m} t        | j                  «      dk(  rt        dz  | j                  d   fn| j                  \  }}t        t        d|«      j                  «      } |t        d|t        |«      dz  z  z
  «      |d|f«      S )Nr   rO   r   r   rQ   )
rR   rP   rn   r*   r   r	   r
   rS   r   r   )r,   r*   rE   rP   rb   r%   rQ   s          r&   rT   z$elliptic_e._eval_rewrite_as_Integral/  st   € Ý6Ü'*¨4¯9©9£~¸Ò':”�1‘�d—i‘i ‘lÑ#ÀÇ	Á	‰ˆˆ1ÜÔ'¨¨TÓ2×7Ñ7Ó8ˆÙœ˜Q ¤3 q£6¨1¡9¡™_Ó-°°1°a¨yÓ9Ð9r(   rC   rU   rV   )rW   rX   rY   rZ   r[   r'   r.   r6   r?   rF   rI   rT   Ú__classcell__)rs   s   @r&   r+   r+   ¼   s<   ø„ ñ-ð^ ò)ó ð)ó41ò0õ8òHò
:ö:r(   r+   c                   ó6   — e Zd ZdZedd„«       Zd„ Zdd„Zd„ Zy)	Úelliptic_piaO  
    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           	      óÐ  — |��||}}|j                   rt        ||«      S |t        j                  u rIt        ||«      t	        d|t        |«      dz  z  z
  «      t        |«      z  t        ||«      z
  d|z
  z  z   S d|z  t        z  }|j                  r|t        ||«      z  S |j                   r2t        t	        |dz
  «      t        |«      z  «      t	        |dz
  «      z  S ||k(  rDt        ||«      t        d||«      z
  t        |«      t	        d|t        |«      dz  z  z
  «      z  z   S |t        j                  t        j                  fv rt        j                  S |t        j                  t        j                  fv rt        j                  S |j                  «       rt        || |«       S |j                   rt        ||«      S |j                   r|j"                  s|j                   r|j"                  rt        j                  S y y |j                   rt%        |«      S |t        j                  u rt        j&                  S |j                   rt        dt	        d|z
  «      z  z  S |t        j                  k(  rt        j                  t)        |dz
  «      z  S ||k(  rt        |«      d|z
  z  S |t        j                  t        j                  fv rt        j                  S |t        j                  t        j                  fv rt        j                  S |j                   rt%        |«      S |j                   r|j"                  s|j                   r|j"                  rt        j                  S y y )Nr   r   )r   r^   r   r   r   r   r   r+   r   r`   rz   r   r!   r"   r#   ra   rj   rK   r   r   r   )r$   r;   r%   rb   rc   s        r&   r'   zelliptic_pi.evalc  sÊ  € à‰=Ø�aˆqˆAØ�yŠyÜ! ! QÓ'Ð'Ø”a—e‘e‘Ü" 1 aÓ(Ü˜a !¤C¨£F¨A¡I¡+™oÓ.¬s°1«vÑ5Ü# A qÓ)ñ*Ø,-°©Eñ3ñ3ð 4ð �!‘”B‘ˆAØ�|Š|Øœ Q¨Ó*Ñ*Ð*Ø—’ÜœT ! a¡%›[¬¨Q«Ñ/Ó0´°a¸!±e³Ñ<Ð<Ø�a’Ü" 1 aÓ(¬;°q¸!¸QÓ+?Ñ?Ü˜A›œt A¨¬#¨a«&°!©)©¡OÓ4Ñ4ñ5ð 6à”q—z‘z¤1×#5Ñ#5Ð6Ñ6Ü—v‘v�Ø”q—z‘z¤1×#5Ñ#5Ð6Ñ6Ü—v‘v�Ø×+Ñ+Ô-Ü# A¨ r¨1Ó-Ð-Ð-Ø�yŠyÜ! ! QÓ'Ð'Ø×!Ò! a§m¢mØ×&Ò&¨1¯=ª=Ü—v‘v�ð ,9Ð&ð �yŠyÜ! !“}Ð$Ø”a—e‘e‘Ü×(Ñ(Ð(Ø—’Ü˜1œT ! a¡%›[™=Ñ)Ð)Ø”a—e‘e’Ü×)Ñ)¬$¨q°1©u«+Ñ5Ð5Ø�a’Ü! !“} a¨!¡eÑ,Ð,Ø”q—z‘z¤1×#5Ñ#5Ð6Ñ6Ü—v‘v�Ø”q—z‘z¤1×#5Ñ#5Ð6Ñ6Ü—v‘v�Ø�yŠyÜ! !“}Ð$Ø×!Ò! a§m¢mØ×&Ò&¨1¯=ª=Ü—v‘v�ð ,9Ð&r(   c                 óÆ  — t        | j                  «      dk(  r�| j                  \  }}}|j                  xr |dz
  j                  du r]|j                  xr |dz
  j                  du r=| j	                  |j                  «       |j                  «       |j                  «       «      S y y | j                  \  }}| j	                  |j                  «       |j                  «       «      S )Nr   r   Frp   )r,   r;   rb   r%   s       r&   r6   zelliptic_pi._eval_conjugate•  s²   € Üˆt�y‰y‹>˜QÒØ—i‘i‰GˆAˆq�!Ø—	‘	Ò1˜q 1™u×1Ñ1°eÑ;Ø—	‘	Ò1˜q 1™u×1Ñ1°eÑ;Ø—y‘y §¡£°·±³¸q¿{¹{»}ÓMÐMð <ð <ð —9‘9‰DˆAˆqØ—9‘9˜QŸ[™[›]¨A¯K©K«MÓ:Ð:r(   c                 óf  — t        | j                  «      dk(  �r| j                  \  }}}t        d|t        |«      dz  z  z
  «      d|t        |«      dz  z  z
  }}|dk(  rft	        ||«      ||z
  t        ||«      z  |z  z   |dz  |z
  t        |||«      z  |z  z   ||z  t        d|z  «      z  d|z  z  z
  d||z
  z  |dz
  z  z  S |dk(  rd||z  z  S |dk(  rÏt	        ||«      |dz
  z  t        |||«      z   |t        d|z  «      z  d|dz
  z  |z  z  z
  d||z
  z  z  S | j                  \  }}|dk(  rHt	        |«      ||z
  t        |«      z  |z  z   |dz  |z
  t        ||«      z  |z  z   d||z
  z  |dz
  z  z  S |dk(  r't	        |«      |dz
  z  t        ||«      z   d||z
  z  z  S t        | |«      ‚)Nr   r   r   )	rn   r*   r   r   r+   r^   rz   r   r   )r,   r-   r;   rb   r%   re   Úfns          r&   r.   zelliptic_pi.fdiffŸ  s  € Üˆt�y‰y‹>˜QÓØ—i‘i‰GˆAˆq�!Ü˜!˜a¤ A£¨¡	™k™/Ó*¨A°´#°a³&¸!±)±©O�ˆBØ˜1Š}Ü" 1 aÓ(¨A°©E´:¸aÀÓ3CÑ+CÀAÑ+EÑEØ˜A™ ™¤;¨q°!°QÓ#7Ñ7¸Ñ9ñ:à˜"™œS  1¡›X™ q¨¡tÑ,ñ-à/0°!°a±%©y¸!¸a¹%Ñ/@ñBð Bð ˜Q’Ø˜"˜R™%‘yÐ Ø˜Q’Ü" 1 aÓ(¨!¨a©%Ñ0Ü# A q¨!Ó,ñ-àœ#˜a ™c›(™
 A q¨1¡u¡I¨b¡LÑ1ñ2à45°q¸1±u±Iñ?ð ?ð —9‘9‰DˆAˆqØ˜1Š}Ü" 1›¨¨Q©´
¸1³Ñ(=¸aÑ(?Ñ?Ø˜A™ ™¤;¨q°!Ó#4Ñ4°QÑ6ñ7Ø9:¸AÀ¹E¹ÀAÈÁEÑ9JñLð Là˜Q’Ü" 1› q¨1¡uÑ-´¸A¸qÓ0AÑAÀAÀqÈ1ÁuÁIÑNÐNÜ   xÓ0Ð0r(   c                 ój  — ddl m} t        | j                  «      dk(  r(| j                  d   | j                  d   t        dz  }}}n| j                  \  }}}t        t        d|«      j                  «      } |dd|t        |«      dz  z  z
  t        d|t        |«      dz  z  z
  «      z  z  |d|f«      S )Nr   rO   r   r   rQ   )
rR   rP   rn   r*   r   r	   r
   rS   r   r   )r,   r*   rE   rP   r;   r%   rb   rQ   s           r&   rT   z%elliptic_pi._eval_rewrite_as_Integral¶  sŸ   € Ý6Üˆt�y‰y‹>˜QÒØ—i‘i ‘l D§I¡I¨a¡L´"°Q±$�!ˆq‰Aà—i‘i‰GˆAˆq�!ÜÔ'¨¨TÓ2×7Ñ7Ó8ˆÙ˜˜A ¤# a£&¨!¡)¡™O¬T°!°a¼¸A»À¹	±k±/Ó-BÑBÑCÀaÈÈAÀYÓOÐOr(   rC   rU   )	rW   rX   rY   rZ   r[   r'   r6   r.   rT   r\   r(   r&   rz   rz   6  s-   „ ñ*ðX ò/ó ð/òb;ó1ó.Pr(   rz   N)rZ   Ú
sympy.corer   r   r   r   Úsympy.core.functionr   r   Úsympy.core.symbolr	   r
   Ú$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   r^   r+   rz   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(   