Ë
    7^(hÑ ã                   óš  — d dl mZmZmZ d dlmZ d dlmZmZ d dl	m
Z
mZmZmZ d dlmZmZmZ d dlmZ d dlmZmZmZ d dlmZmZmZ d d	lmZmZmZ d d
l m!Z!m"Z"m#Z# d dl$m%Z% d dl&m'Z' d dl(m)Z)m*Z*m+Z+m,Z,m-Z-m.Z.m/Z/m0Z0m1Z1m2Z2m3Z3 d dl4m5Z5 d„ Z6ed„ «       Z7ed„ «       Z8ed„ «       Z9 G d„ de«      Z:d„ Z; G d„ de:«      Z< G d„ de:«      Z= G d„ de:«      Z> G d„ de:«      Z? G d„ de:«      Z@ G d „ d!e@«      ZA G d"„ d#e@«      ZB G d$„ d%e«      ZC G d&„ d'eC«      ZD G d(„ d)eC«      ZE G d*„ d+eC«      ZF G d,„ d-eC«      ZG G d.„ d/eC«      ZH G d0„ d1eC«      ZIy2)3é    )ÚSÚsympifyÚcacheit)ÚAdd)ÚDefinedFunctionÚArgumentIndexError)Úfuzzy_orÚ	fuzzy_andÚ	fuzzy_notÚ	FuzzyBool)ÚIÚpiÚRational)ÚDummy)ÚbinomialÚ	factorialÚRisingFactorial)Ú	bernoulliÚeulerÚnC)ÚAbsÚimÚre)ÚexpÚlogÚmatch_real_imag)Úfloor)Úsqrt)ÚacosÚacotÚasinÚatanÚcosÚcotÚcscÚsecÚsinÚtanÚ_imaginary_unit_as_coefficient)Úsymmetric_polyc           	      ó˜   — | j                  | j                  t        «      D �ci c]  }||j                  t        «      “Œ c}«      S c c}w ©N)ÚxreplaceÚatomsÚHyperbolicFunctionÚrewriter   )ÚexprÚhs     úc/var/www/skyplay_api_hub/venv/lib/python3.12/site-packages/sympy/functions/elementary/hyperbolic.pyÚ_rewrite_hyperbolics_as_expr4      sE   € Ø�=‰=Ø—‘Ô.Ó/ö1Øð ˜QŸY™Y¤s›^Ñ+ò 1ó 2ð 2ùò 1s   £Ac                  ón  — i t         t        t         dt        d«      z   z  «      “t          t        t          dt        d«      z   z  «      “t        j                  t
        dz  “t        dd«      t
        t        dd«      z  “t        d«      dz  t
        dz  “t        d«       dz  t
        t        dd«      z  “dt        d«      z  t
        dz  “dt        d«      z  t
        t        dd«      z  “t        d«      dz  t
        dz  “t        d«       dz  t
        t        dd«      z  “t        d«      dz
  t        d«      z  t
        t        dd	«      z  “t        d«      dz
   t        d«      z  t
        t        d
d	«      z  “t        dt        d«      z   «      dz  t
        dz  “t        dt        d«      z   «       dz  t
        t        d
d«      z  “t        dt        d«      z
  «      dz  t
        t        dd«      z  “t        dt        d«      z
  «       dz  t
        t        dd«      z  “dt        d«      z   dt        d«      z  z  t
        d	z  “dt        d«      z    dt        d«      z  z  t
        t        dd	«      z  t        d«      dz   dz  t
        dz  t        d«      dz    dz  t
        t        dd«      z  i¥S )Né   é   é   éÿÿÿÿé   é   é   é   é   é   é   )r   r   r   r   ÚHalfr   r   © ó    r3   Ú_acosh_tablerD      sž  € ðÜ	Œ3Œq�!”d˜1“g‘+‰Óðä	
ˆŒC”��Aœ˜Q›‘KÑ Ó!ðô 	
�‰”�1‘ðô 	��Q‹œœH Q¨›NÑ*ð	ô
 	ˆQ‹�‰	”2�a‘4ðô 
ˆa‹ˆ�‰
”B”x  1“~Ñ%ðð 	
Œ$ˆq‹'‰	”2�a‘4ðð 	Œ4�‹7‰
”B”x  1“~Ñ%ðô 	ˆQ‹�‰	”2�a‘4ðô 
ˆa‹ˆ�‰
”B”x  1“~Ñ%ðô 
ˆa‹�1‰”d˜4“jÑ ¤"¤X¨a°£_Ñ"4ðô ˆq‹'�A‰+ˆ”t˜D“zÑ!¤2¤h¨q°"£oÑ#5ðô 	ˆQ”�a“‰[Ó˜!ÑœR ™Tðô 
ˆa”$�q“'‰kÓ	Ð˜1Ñœb¤¨!¨Q£Ñ/ðô 	ˆQ”�a“‰[Ó˜!ÑœR¤¨¨A£Ñ.ðô  
ˆa”$�q“'‰kÓ	Ð˜1Ñœb¤¨!¨Q£Ñ/ð!ð" 
ŒT�!‹W‰�qœ˜a›‘yÑ!¤2 b¡5ð#ð$ Œd�1‹g‰+ˆ˜œ$˜q›'™	Ñ"¤B¤x°°BÓ'7Ñ$7Ü	ˆa‹�1‰�a‰œ˜A™Ü
ˆq‹'�A‰+ˆ�qÑœ"œX a¨›^Ñ+ñ)ð rC   c                   ól  — t         t         dz  t         t        d«      t        d«      z   z  t         dz  t         dt        d«      z   z  t         dz  t         dz  t        dt        d«      z
  «      z  t         dz  t         dz  t         dz  t         t        ddt        d«      z  z   «      z  t         dz  t         t        d«      z  t         dz  t         t        d«      dz
  z  d	t        z  dz  t         dz  t        d
«      z  t         d
z  t         dz  t        dt        d«      z   «      z  d	t        z  dz  t         t        ddt        d«      z  z
  «      z  dt        z  dz  t         t        d«      t        d«      z
  z  dt        z  dz  t        d«      t          t	        dt        d«      z   dz  «      z  iS )Nr7   r;   r>   r6   r<   é
   r=   r:   éýÿÿÿr8   éþÿÿÿéûÿÿÿ)r   r   r   r   r   rB   rC   r3   Ú_acsch_tablerJ   3   sg  € ô ”ˆs�Q‰wÜŒt�A‹wœ˜a›Ñ Ñ!¤B 3¨¡8Üˆq”4˜“7‰{‰Oœb˜S 2™XÜˆa‰C”$�qœ4 ›7‘{Ó#Ñ#¤b S¨1¡WÜˆa‰C”"��q‘ÜŒd�1�qœ˜a›‘y‘=Ó!Ñ!¤B 3¨¡7ÜŒd�1‹g‰Iœ�s˜Q‘wÜŒt�A‹w�q‰y‰M˜2œb™5 2™:Üˆa‰C”$�q“'‰MœB˜3 ™7Üˆa‰C”$�qœ4 ›7‘{Ó#Ñ# R¬¡U¨Q¡YÜŒd�1�qœ˜a›‘y‘=Ó!Ñ! 2¤b¡5¨1¡9ÜŒt�A‹wœ˜a›Ñ Ñ! 2¤b¡5¨2¡:Üˆa‹D”1�"”S˜!œD ›G™) Q™Ó'Ñ'ð
ð 
rC   c                  óh  — i t         t        t         z  dz   t        dt        d«      z   «      z   “t          t        t         z  dz  t        dt        d«      z   «      z   “t        d«      t        d«      z
  t        dz  “t        d«      t        d«      z
  dt        z  dz  “t        ddt        d«      z  z
  «      t        dz  “t        ddt        d«      z  z
  «       dt        z  dz  “dt        dt        d«      z   «      z  t        d	z  “d
t        dt        d«      z   «      z  dt        z  d	z  “dt        d«      z  t        dz  “d
t        d«      z  dt        z  dz  “t        d«      dz
  t        dz  “dt        d«      z
  dt        z  dz  “t        d«      t        dz  “t        d«       dt        z  dz  “t        ddt        d«      z  z   «      dt        z  dz  “t        ddt        d«      z  z   «       dt        z  dz  “t	        d«      t        dz  “t	        d«       dt        z  dz  t        ddt        d«      z   z  «      dt        z  d	z  t        ddt        d«      z   z  «       dt        z  d	z  dt        d«      z   dt        z  dz  dt        d«      z
  dt        z  dz  t        d«      t        d«      z   dt        z  dz  t        d«       t        d«      z
  dt        z  dz  t         t        j
                  z  t         t         z  dz  t         t        j                  z  t        t         z  dz  i	¥S )Nr7   r6   r;   r>   r@   r<   rF   é	   r=   rH   r?   r8   r:   r9   )r   r   r   r   r   ÚInfinityÚNegativeInfinityrB   rC   r3   Ú_asech_tablerO   F   s  € ð
Ü”"”Q‘$˜‘(ˆ|œc !¤d¨1£g¡+Ó.Ñ.ð
äˆB””A‘˜‘œS ¤T¨!£W¡Ó-Ñ-ð
ô �!‹W”t˜A“wÑ¤ b¡ð
ô �!‹W”t˜A“wÑ ¤B¡¨¡ð	
ô
 ��Q”t˜A“w‘Y‘Ó¤ b¡ð
ô �!�aœ˜Q›‘i‘-Ó Ð  !¤B¡$¨¡)ð
ð ”�Qœ˜a›‘[Ó!Ñ!¤2¨¡6ð
ð ”�aœ$˜q›'‘kÓ"Ñ" A¤b¡D¨1¡Hð
ð ”�Q“‰Kœ˜a™ð
ð ”�a“‰L˜!œB™$ ™(ð
ô �!‹W�q‰[œ2 ™6ð
ð ”�a“‰[˜1œR™4 !™8ð
ô �‹G”R˜!‘Vð
ô �!‹WˆH�aœ‘d˜Q‘hð
ô ��Q”t˜A“w‘Y‘Ó ¤2¡¨¡ð
ô  �!�aœ˜Q›‘i‘-Ó Ð  !¤B¡$¨¡)ð!
ô" ˆa‹D”"�q‘&ð#
ô$ ˆq‹TˆE�1”R‘4˜!‘8Ü��Aœ˜Q›‘K‘Ó! 1¤R¡4¨!¡8Ü�!�Qœ˜a›‘[‘/Ó"Ð" A¤b¡D¨1¡HØ”�a“‰[˜1œR™4 !™8Ø”$�q“'‰\˜Aœb™D 1™HÜ�!‹W”t˜A“wÑ ¤2¡¨¡Ü�1‹gˆXœ˜Q›Ñ !¤B¡$¨¡)ÜŒa�j‰j‰Lœ2˜#œa™% !™)ÜŒa× Ñ Ñ ¤"¤Q¡$¨¡(ñ5
ð 	
rC   c                   ó   — e Zd ZdZdZy)r/   ze
    Base class for hyperbolic functions.

    See Also
    ========

    sinh, cosh, tanh, coth
    TN)Ú__name__Ú
__module__Ú__qualname__Ú__doc__Ú
unbranchedrB   rC   r3   r/   r/   j   s   „ ñð �JrC   r/   c                 óN  — t         t        z  }t        j                  | «      D ]M  }||k(  rt        j
                  } nH|j                  sŒ'|j                  «       \  }}||k(  sŒ@|j                  sŒM n | t        j                  fS |t        j                  z  }||z
  }| ||z  z
  |fS )aÙ  
    Split ARG into two parts, a "rest" and a multiple of $I\pi$.
    This assumes ARG to be an ``Add``.
    The multiple of $I\pi$ returned in the second position is always a ``Rational``.

    Examples
    ========

    >>> from sympy.functions.elementary.hyperbolic import _peeloff_ipi as peel
    >>> from sympy import pi, I
    >>> from sympy.abc import x, y
    >>> peel(x + I*pi/2)
    (x, 1/2)
    >>> peel(x + I*2*pi/3 + I*pi*y)
    (x + I*pi*y + I*pi/6, 1/2)
    )r   r   r   Ú	make_argsr   ÚOneÚis_MulÚas_two_termsÚis_RationalÚZerorA   )ÚargÚipiÚaÚKÚpÚm1Úm2s          r3   Ú_peeloff_ipird   w   s˜   € ô" ŒQ‰$€CÜ�]‰]˜3Óò 	ˆØ�Š8Ü—‘ˆAÙØ�X‹XØ—>‘>Ó#‰DˆAˆqØ�C‹x˜AŸM›MÙð	ð ”A—F‘Fˆ{Ðà
Œa�f‰f‰*€BØ	
ˆR‰€BØ��C‘‰<˜ÐÐrC   c                   óÄ   — e Zd ZdZdd„Zdd„Zed„ «       Zee	d„ «       «       Z
d„ Zdd„Zdd„Zdd	„Zdd„Zd„ Zd„ Zd„ Zd„ Zd„ Zd„ Zd„ Zd„ Zd„ Zd„ Zd„ Zd„ Zd„ Zd„ Zy
)Úsinha  
    ``sinh(x)`` is the hyperbolic sine of ``x``.

    The hyperbolic sine function is $\frac{e^x - e^{-x}}{2}$.

    Examples
    ========

    >>> from sympy import sinh
    >>> from sympy.abc import x
    >>> sinh(x)
    sinh(x)

    See Also
    ========

    cosh, tanh, asinh
    c                 óT   — |dk(  rt        | j                  d   «      S t        | |«      ‚)z@
        Returns the first derivative of this function.
        r6   r   )ÚcoshÚargsr   ©ÚselfÚargindexs     r3   Úfdiffz
sinh.fdiff­   s+   € ð �qŠ=Ü˜Ÿ	™	 !™Ó%Ð%ä$ T¨8Ó4Ð4rC   c                 ó   — t         S ©z7
        Returns the inverse of this function.
        ©Úasinhrj   s     r3   Úinversezsinh.inverse¶   ó	   € ô ˆrC   c                 ó¤  — |j                   r™|t        j                  u rt        j                  S |t        j                  u rt        j                  S |t        j                  u rt        j                  S |j
                  rt        j                  S |j                  r
 | | «       S y |t        j                  u rt        j                  S t        |«      }|�t        t        |«      z  S |j                  «       r
 | | «       S |j                  rOt        |«      \  }}|r?|t        z  t        z  }t!        |«      t#        |«      z  t#        |«      t!        |«      z  z   S |j
                  rt        j                  S |j$                  t&        k(  r|j(                  d   S |j$                  t*        k(  r,|j(                  d   }t-        |dz
  «      t-        |dz   «      z  S |j$                  t.        k(  r#|j(                  d   }|t-        d|dz  z
  «      z  S |j$                  t0        k(  r/|j(                  d   }dt-        |dz
  «      t-        |dz   «      z  z  S y ©Nr   r6   r7   )Ú	is_Numberr   ÚNaNrM   rN   Úis_zeror\   Úis_negativeÚComplexInfinityr)   r   r'   Úcould_extract_minus_signÚis_Addrd   r   rf   rh   Úfuncrq   ri   Úacoshr   ÚatanhÚacoth)Úclsr]   Úi_coeffÚxÚms        r3   Úevalz	sinh.eval¼   sÝ  € à�=Š=Ø”a—e‘e‰|Ü—u‘u�ØœŸ
™
Ñ"Ü—z‘zÐ!Øœ×*Ñ*Ñ*Ü×)Ñ)Ð)Ø—’Ü—v‘v�Ø—’Ù˜S˜D›	�zÐ!ð !ð ”a×'Ñ'Ñ'Ü—u‘u�ä4°SÓ9ˆGàÐ"Üœ3˜w›<Ñ'Ð'à×/Ñ/Ô1Ù  ›I˜:Ð%à�zŠzÜ# CÓ(‘��1ÙØœ"™œQ™�AÜ ›7¤4¨£7™?¬T°!«W´T¸!³W©_Ñ<Ð<à�{Š{Ü—v‘v�à�x‰xœ5Ò Ø—x‘x ‘{Ð"à�x‰xœ5Ò Ø—H‘H˜Q‘K�Ü˜A ™E“{¤T¨!¨a©%£[Ñ0Ð0à�x‰xœ5Ò Ø—H‘H˜Q‘K�Øœ˜a ! Q¡$™h›Ñ'Ð'à�x‰xœ5Ò Ø—H‘H˜Q‘K�Øœ$˜q 1™u›+¬¨Q°©U«Ñ3Ñ4Ð4ð !rC   c                 ó¼   — | dk  s| dz  dk(  rt         j                  S t        |«      }t        |«      dkD  r|d   }||dz  z  | | dz
  z  z  S || z  t	        | «      z  S )zG
        Returns the next term in the Taylor series expansion.
        r   r7   rH   r6   ©r   r\   r   Úlenr   ©Únrƒ   Úprevious_termsra   s       r3   Útaylor_termzsinh.taylor_termí   sl   € ð ˆqŠ5�A˜‘E˜Q’JÜ—6‘6ˆMä˜“
ˆAä�>Ó" QÒ&Ø" 2Ñ&�Ø˜1˜a™4‘x 1 a¨!¡e¡9Ñ-Ð-à˜1‘v¤	¨!£Ñ,Ð,rC   c                 óZ   — | j                  | j                  d   j                  «       «      S ©Nr   ©r}   ri   Ú	conjugate©rk   s    r3   Ú_eval_conjugatezsinh._eval_conjugateþ   ó"   € Ø�y‰y˜Ÿ™ 1™×/Ñ/Ó1Ó2Ð2rC   c                 ó°  — | j                   d   j                  r<|r(d|d<    | j                  |fi |¤Žt        j                  fS | t        j                  fS |r2 | j                   d   j                  |fi |¤Žj                  «       \  }}n | j                   d   j                  «       \  }}t        |«      t        |«      z  t        |«      t        |«      z  fS )z@
        Returns this function as a complex coordinate.
        r   FÚcomplex©
ri   Úis_extended_realÚexpandr   r\   Úas_real_imagrf   r#   rh   r'   ©rk   ÚdeepÚhintsr   r   s        r3   r™   zsinh.as_real_imag  s½   € ð �9‰9�Q‰<×(Ò(ÙØ#(��iÑ Ø#˜Ÿ™ DÑ2¨EÑ2´A·F±FÐ;Ð;àœaŸf™f�~Ð%ÙØ(�T—Y‘Y˜q‘\×(Ñ(¨Ñ7°Ñ7×DÑDÓF‰FˆB‘à—Y‘Y˜q‘\×.Ñ.Ó0‰FˆB�Ü�R“œ˜R›Ñ ¤$ r£(¬3¨r«7Ñ"2Ð3Ð3rC   c                 óH   —  | j                   dd|i|¤Ž\  }}||t        z  z   S ©Nr›   rB   ©r™   r   ©rk   r›   rœ   Úre_partÚim_parts        r3   Ú_eval_expand_complexzsinh._eval_expand_complex  ó0   € Ø,˜4×,Ñ,Ñ@°$Ð@¸%Ñ@Ñˆ�Ø˜¤™Ñ"Ð"rC   c                 óà  — |r! | j                   d   j                  |fi |¤Ž}n| j                   d   }d }|j                  r|j                  «       \  }}nO|j	                  d¬«      \  }}|t
        j                  ur(|j                  r|t
        j                  ur
|}|dz
  |z  }|�?t        |«      t        «      z  t        |«      t        |«      z  z   j                  d¬«      S t        |«      S ©Nr   T©Úrationalr6   )Útrig)
ri   r˜   r|   rZ   Úas_coeff_Mulr   rX   Ú
is_Integerrf   rh   ©rk   r›   rœ   r]   rƒ   ÚyÚcoeffÚtermss           r3   Ú_eval_expand_trigzsinh._eval_expand_trig  ó×   € ÙØ%�$—)‘)˜A‘,×%Ñ% dÑ4¨eÑ4‰Cà—)‘)˜A‘,ˆCØˆØ�:Š:Ø×#Ñ#Ó%‰DˆA‰qà×+Ñ+°TÐ+Ó:‰LˆE�5ØœAŸE™EÑ! e×&6Ò&6¸5ÌÏÉÑ;MØ�Ø˜Q‘Y ‘M�Øˆ=Ü˜“GœD ›G‘O¤d¨1£g¬d°1«g¡oÑ5×=Ñ=À4Ð=ÓHÐHÜ�C‹yÐrC   Nc                 ó8   — t        |«      t        | «      z
  dz  S ©Nr7   ©r   ©rk   r]   ÚlimitvarÚkwargss       r3   Ú_eval_rewrite_as_tractablezsinh._eval_rewrite_as_tractable&  ó   € Ü�C“œ3 ˜t›9Ñ$¨Ñ)Ð)rC   c                 ó8   — t        |«      t        | «      z
  dz  S r³   r´   ©rk   r]   r·   s      r3   Ú_eval_rewrite_as_expzsinh._eval_rewrite_as_exp)  r¹   rC   c                 ó6   — t          t        t         |z  «      z  S r,   ©r   r'   r»   s      r3   Ú_eval_rewrite_as_sinzsinh._eval_rewrite_as_sin,  ó   € Üˆr”Cœ˜C™“LÑ Ð rC   c                 ó6   — t          t        t         |z  «      z  S r,   ©r   r%   r»   s      r3   Ú_eval_rewrite_as_csczsinh._eval_rewrite_as_csc/  rÀ   rC   c                 óJ   — t          t        |t        t         z  dz  z   «      z  S r³   ©r   rh   r   r»   s      r3   Ú_eval_rewrite_as_coshzsinh._eval_rewrite_as_cosh2  s    € Üˆr”$�sœR¤™T !™V‘|Ó$Ñ$Ð$rC   c                 óV   — t        t        j                  |z  «      }d|z  d|dz  z
  z  S ©Nr7   r6   ©Útanhr   rA   ©rk   r]   r·   Ú	tanh_halfs       r3   Ú_eval_rewrite_as_tanhzsinh._eval_rewrite_as_tanh5  s,   € ÜœŸ™ ™Ó$ˆ	Ø�‰{˜A 	¨1¡Ñ,Ñ-Ð-rC   c                 óV   — t        t        j                  |z  «      }d|z  |dz  dz
  z  S rÈ   ©Úcothr   rA   ©rk   r]   r·   Ú	coth_halfs       r3   Ú_eval_rewrite_as_cothzsinh._eval_rewrite_as_coth9  s,   € ÜœŸ™ ™Ó$ˆ	Ø�‰{˜I q™L¨1Ñ,Ñ-Ð-rC   c                 ó   — dt        |«      z  S ©Nr6   ©Úcschr»   s      r3   Ú_eval_rewrite_as_cschzsinh._eval_rewrite_as_csch=  ó   € Ø”4˜“9‰}ÐrC   c                 ó*  — | j                   d   j                  |||¬«      }|j                  |d«      }|t        j                  u r"|j                  |d|j                  rdnd¬«      }|j                  r|S |j                  r| j                  |«      S | S ©Nr   ©ÚlogxÚcdirú-ú+)Údir)
ri   Úas_leading_termÚsubsr   rw   Úlimitry   rx   Ú	is_finiter}   ©rk   rƒ   rÝ   rÞ   r]   Úarg0s         r3   Ú_eval_as_leading_termzsinh._eval_as_leading_term@  s   € Ø�i‰i˜‰l×*Ñ*¨1°4¸dÐ*ÓCˆØ�x‰x˜˜1‹~ˆà”1—5‘5‰=Ø—9‘9˜Q ¨d×.>Ò.>¡sÀC�9ÓHˆDØ�<Š<ØˆJØ�^Š^Ø—9‘9˜T“?Ð"àˆKrC   c                 ó†   — | j                   d   }|j                  ry|j                  «       \  }}|t        z  j                  S ©Nr   T©ri   Úis_realr™   r   rx   ©rk   r]   r   r   s       r3   Ú_eval_is_realzsinh._eval_is_realM  s;   € Ø�i‰i˜‰lˆØ�;Š;Øð ×!Ñ!Ó#‰ˆˆBØ”2‘�‰ÐrC   c                 ó8   — | j                   d   j                  ryy rê   ©ri   r—   r‘   s    r3   Ú_eval_is_extended_realzsinh._eval_is_extended_realW  ó   € Ø�9‰9�Q‰<×(Ò(Øð )rC   c                 óh   — | j                   d   j                  r| j                   d   j                  S y rŽ   ©ri   r—   Úis_positiver‘   s    r3   Ú_eval_is_positivezsinh._eval_is_positive[  ó,   € Ø�9‰9�Q‰<×(Ò(Ø—9‘9˜Q‘<×+Ñ+Ð+ð )rC   c                 óh   — | j                   d   j                  r| j                   d   j                  S y rŽ   ©ri   r—   ry   r‘   s    r3   Ú_eval_is_negativezsinh._eval_is_negative_  r÷   rC   c                 ó8   — | j                   d   }|j                  S rŽ   ©ri   rå   ©rk   r]   s     r3   Ú_eval_is_finitezsinh._eval_is_finitec  ó   € Ø�i‰i˜‰lˆØ�}‰}ÐrC   c                 ój   — t        | j                  d   «      \  }}|j                  r|j                  S y rŽ   )rd   ri   rx   Ú
is_integer©rk   ÚrestÚipi_mults      r3   Ú_eval_is_zerozsinh._eval_is_zerog  s0   € Ü% d§i¡i°¡lÓ3‰ˆˆhØ�<Š<Ø×&Ñ&Ð&ð rC   ©r6   ©Tr,   )rQ   rR   rS   rT   rm   rr   Úclassmethodr…   Ústaticmethodr   rŒ   r’   r™   r£   r°   r¸   r¼   r¿   rÃ   rÆ   rÍ   rÓ   rØ   rè   rî   rñ   rö   rú   rþ   r  rB   rC   r3   rf   rf   ™   s¡   „ ñó&5óð ñ.5ó ð.5ð` Øñ-ó ó ð-ò3ó4ó #óó"*ò*ò!ò!ò%ò.ò.òòòòò,ò,òó'rC   rf   c                   ó¶   — e Zd ZdZdd„Zed„ «       Zeed„ «       «       Z	d„ Z
dd„Zdd„Zdd„Zdd
„Zd„ Zd„ Zd„ Zd„ Zd„ Zd„ Zd„ Zd„ Zd„ Zd„ Zd„ Zd„ Zd„ Zy	)rh   a"  
    ``cosh(x)`` is the hyperbolic cosine of ``x``.

    The hyperbolic cosine function is $\frac{e^x + e^{-x}}{2}$.

    Examples
    ========

    >>> from sympy import cosh
    >>> from sympy.abc import x
    >>> cosh(x)
    cosh(x)

    See Also
    ========

    sinh, tanh, acosh
    c                 óT   — |dk(  rt        | j                  d   «      S t        | |«      ‚©Nr6   r   ©rf   ri   r   rj   s     r3   rm   z
cosh.fdiff�  s)   € Ø�qŠ=Ü˜Ÿ	™	 !™Ó%Ð%ä$ T¨8Ó4Ð4rC   c                 óx  — ddl m} |j                  r˜|t        j                  u rt        j                  S |t        j
                  u rt        j
                  S |t        j                  u rt        j
                  S |j                  rt        j                  S |j                  r	 | | «      S y |t        j                  u rt        j                  S t        |«      }|� ||«      S |j                  «       r	 | | «      S |j                  rOt        |«      \  }}|r?|t        z  t         z  }t#        |«      t#        |«      z  t%        |«      t%        |«      z  z   S |j                  rt        j                  S |j&                  t(        k(  rt+        d|j,                  d   dz  z   «      S |j&                  t.        k(  r|j,                  d   S |j&                  t0        k(  r!dt+        d|j,                  d   dz  z
  «      z  S |j&                  t2        k(  r/|j,                  d   }|t+        |dz
  «      t+        |dz   «      z  z  S y )Nr   )r#   r6   r7   )Ú(sympy.functions.elementary.trigonometricr#   rv   r   rw   rM   rN   rx   rX   ry   rz   r)   r{   r|   rd   r   r   rh   rf   r}   rq   r   ri   r~   r   r€   )r�   r]   r#   r‚   rƒ   r„   s         r3   r…   z	cosh.eval‡  sÆ  € å@Ø�=Š=Ø”a—e‘e‰|Ü—u‘u�ØœŸ
™
Ñ"Ü—z‘zÐ!Øœ×*Ñ*Ñ*Ü—z‘zÐ!Ø—’Ü—u‘u�Ø—’Ù˜C˜4“yÐ ð !ð ”a×'Ñ'Ñ'Ü—u‘u�ä4°SÓ9ˆGàÐ"Ù˜7“|Ð#à×/Ñ/Ô1Ù ˜t›9Ð$à�zŠzÜ# CÓ(‘��1ÙØœ"™œQ™�AÜ ›7¤4¨£7™?¬T°!«W´T¸!³W©_Ñ<Ð<à�{Š{Ü—u‘u�à�x‰xœ5Ò Ü˜A §¡¨¡¨Q¡Ñ.Ó/Ð/à�x‰xœ5Ò Ø—x‘x ‘{Ð"à�x‰xœ5Ò Øœ˜a #§(¡(¨1¡+¨q¡.Ñ0Ó1Ñ1Ð1à�x‰xœ5Ò Ø—H‘H˜Q‘K�Øœ$˜q 1™u›+¬¨Q°©U«Ñ3Ñ4Ð4ð !rC   c                 ó¼   — | dk  s| dz  dk(  rt         j                  S t        |«      }t        |«      dkD  r|d   }||dz  z  | | dz
  z  z  S || z  t	        | «      z  S )Nr   r7   r6   rH   r‡   r‰   s       r3   rŒ   zcosh.taylor_term·  sl   € ð ˆqŠ5�A˜‘E˜Q’JÜ—6‘6ˆMä˜“
ˆAä�>Ó" QÒ&Ø" 2Ñ&�Ø˜1˜a™4‘x 1 a¨!¡e¡9Ñ-Ð-à˜1‘vœi¨›lÑ*Ð*rC   c                 óZ   — | j                  | j                  d   j                  «       «      S rŽ   r�   r‘   s    r3   r’   zcosh._eval_conjugateÅ  r“   rC   c                 ó°  — | j                   d   j                  r<|r(d|d<    | j                  |fi |¤Žt        j                  fS | t        j                  fS |r2 | j                   d   j                  |fi |¤Žj                  «       \  }}n | j                   d   j                  «       \  }}t        |«      t        |«      z  t        |«      t        |«      z  fS )Nr   Fr•   )
ri   r—   r˜   r   r\   r™   rh   r#   rf   r'   rš   s        r3   r™   zcosh.as_real_imagÈ  s»   € Ø�9‰9�Q‰<×(Ò(ÙØ#(��iÑ Ø#˜Ÿ™ DÑ2¨EÑ2´A·F±FÐ;Ð;àœaŸf™f�~Ð%ÙØ(�T—Y‘Y˜q‘\×(Ñ(¨Ñ7°Ñ7×DÑDÓF‰FˆB‘à—Y‘Y˜q‘\×.Ñ.Ó0‰FˆB�ä�R“œ˜R›Ñ ¤$ r£(¬3¨r«7Ñ"2Ð3Ð3rC   c                 óH   —  | j                   dd|i|¤Ž\  }}||t        z  z   S rž   rŸ   r    s        r3   r£   zcosh._eval_expand_complexÖ  r¤   rC   c                 óà  — |r! | j                   d   j                  |fi |¤Ž}n| j                   d   }d }|j                  r|j                  «       \  }}nO|j	                  d¬«      \  }}|t
        j                  ur(|j                  r|t
        j                  ur
|}|dz
  |z  }|�?t        |«      t        «      z  t        |«      t        |«      z  z   j                  d¬«      S t        |«      S r¦   )
ri   r˜   r|   rZ   rª   r   rX   r«   rh   rf   r¬   s           r3   r°   zcosh._eval_expand_trigÚ  r±   rC   Nc                 ó8   — t        |«      t        | «      z   dz  S r³   r´   rµ   s       r3   r¸   zcosh._eval_rewrite_as_tractableë  r¹   rC   c                 ó8   — t        |«      t        | «      z   dz  S r³   r´   r»   s      r3   r¼   zcosh._eval_rewrite_as_expî  r¹   rC   c                 ó*   — t        t        |z  d¬«      S ©NF©Úevaluate©r#   r   r»   s      r3   Ú_eval_rewrite_as_coszcosh._eval_rewrite_as_cosñ  ó   € Ü”1�s‘7 UÔ+Ð+rC   c                 ó0   — dt        t        |z  d¬«      z  S ©Nr6   Fr  ©r&   r   r»   s      r3   Ú_eval_rewrite_as_seczcosh._eval_rewrite_as_secô  ó   € Ø”3”q˜3‘w¨Ô/Ñ/Ð/rC   c                 óN   — t          t        |t        t         z  dz  z   d¬«      z  S ©Nr7   Fr  ©r   rf   r   r»   s      r3   Ú_eval_rewrite_as_sinhzcosh._eval_rewrite_as_sinh÷  s"   € Üˆr”$�sœR¤™T !™V‘|¨eÔ4Ñ4Ð4rC   c                 óV   — t        t        j                  |z  «      dz  }d|z   d|z
  z  S rÈ   rÉ   rË   s       r3   rÍ   zcosh._eval_rewrite_as_tanhú  s,   € ÜœŸ™ ™Ó$ aÑ'ˆ	Ø�I‘  I¡Ñ.Ð.rC   c                 óV   — t        t        j                  |z  «      dz  }|dz   |dz
  z  S rÈ   rÏ   rÑ   s       r3   rÓ   zcosh._eval_rewrite_as_cothþ  s,   € ÜœŸ™ ™Ó$ aÑ'ˆ	Ø˜A‘ 	¨A¡Ñ.Ð.rC   c                 ó   — dt        |«      z  S rÕ   ©Úsechr»   s      r3   Ú_eval_rewrite_as_sechzcosh._eval_rewrite_as_sech  rÙ   rC   c                 óF  — | j                   d   j                  |||¬«      }|j                  |d«      }|t        j                  u r"|j                  |d|j                  rdnd¬«      }|j                  rt        j                  S |j                  r| j                  |«      S | S rÛ   )ri   râ   rã   r   rw   rä   ry   rx   rX   rå   r}   ræ   s         r3   rè   zcosh._eval_as_leading_term  sƒ   € Ø�i‰i˜‰l×*Ñ*¨1°4¸dÐ*ÓCˆØ�x‰x˜˜1‹~ˆà”1—5‘5‰=Ø—9‘9˜Q ¨d×.>Ò.>¡sÀC�9ÓHˆDØ�<Š<Ü—5‘5ˆLØ�^Š^Ø—9‘9˜T“?Ð"àˆKrC   c                 óž   — | j                   d   }|j                  s|j                  ry|j                  «       \  }}|t        z  j
                  S rê   )ri   rì   Úis_imaginaryr™   r   rx   rí   s       r3   rî   zcosh._eval_is_real  sE   € Ø�i‰i˜‰lˆð �;Š;˜#×*Ò*Øð
 ×!Ñ!Ó#‰ˆˆBØ”2‘�‰ÐrC   c                 ó  — | j                   d   }|j                  «       \  }}|dt        z  z  }|j                  }|ry|j                  }|du r|S t	        |t        |t	        |t        dz  k  |dt        z  dz  kD  g«      g«      g«      S ©Nr   r7   TFr8   ©ri   r™   r   rx   r	   r
   ©rk   Úzrƒ   r­   ÚymodÚyzeroÚxzeros          r3   rö   zcosh._eval_is_positive  s˜   € ð �I‰I�a‰Lˆà�~‰~Ó‰ˆˆ1Ø�A”b‘D‰zˆà—‘ˆáØà—	‘	ˆà�E‰>ØˆLäàäØÜ˜d¤R¨¡T™k¨4°!´B±$°q±&©=Ð9Ó:ðó ð	ó ð 	rC   c                 ó  — | j                   d   }|j                  «       \  }}|dt        z  z  }|j                  }|ry|j                  }|du r|S t	        |t        |t	        |t        dz  k  |dt        z  dz  k\  g«      g«      g«      S r1  r2  r3  s          r3   Ú_eval_is_nonnegativezcosh._eval_is_nonnegative?  s–   € Ø�I‰I�a‰Lˆà�~‰~Ó‰ˆˆ1Ø�A”b‘D‰zˆà—‘ˆáØà—	‘	ˆà�E‰>ØˆLäàäØÜ˜d¤b¨¡d™l¨D°A´b±D¸±F©NÐ;Ó<ðó ð	ó ð 	rC   c                 ó8   — | j                   d   }|j                  S rŽ   rü   rý   s     r3   rþ   zcosh._eval_is_finiteY  rÿ   rC   c                 ó’   — t        | j                  d   «      \  }}|r*|j                  r|t        j                  z
  j
                  S y y rŽ   )rd   ri   rx   r   rA   r  r  s      r3   r  zcosh._eval_is_zero]  s=   € Ü% d§i¡i°¡lÓ3‰ˆˆhÙ˜ŸšØœqŸv™vÑ%×1Ñ1Ð1ð %ˆ8rC   r  r  r,   )rQ   rR   rS   rT   rm   r  r…   r	  r   rŒ   r’   r™   r£   r°   r¸   r¼   r  r!  r&  rÍ   rÓ   r,  rè   rî   rö   r9  rþ   r  rB   rC   r3   rh   rh   m  s˜   „ ñó&5ð ñ-5ó ð-5ð^ Øñ
+ó ó ð
+ò3ó4ó#óó"*ò*ò,ò0ò5ò/ò/òòòòò@ò4ó2rC   rh   c                   ó´   — e Zd ZdZdd„Zdd„Zed„ «       Zee	d„ «       «       Z
d„ Zdd„Zd„ Zdd
„Zd„ Zd„ Zd„ Zd„ Zd„ Zd„ Zd„ Zd„ Zd„ Zd„ Zd„ Zd„ Zd„ Zy	)rÊ   a'  
    ``tanh(x)`` is the hyperbolic tangent of ``x``.

    The hyperbolic tangent function is $\frac{\sinh(x)}{\cosh(x)}$.

    Examples
    ========

    >>> from sympy import tanh
    >>> from sympy.abc import x
    >>> tanh(x)
    tanh(x)

    See Also
    ========

    sinh, cosh, atanh
    c                 ó|   — |dk(  r,t         j                  t        | j                  d   «      dz  z
  S t	        | |«      ‚©Nr6   r   r7   )r   rX   rÊ   ri   r   rj   s     r3   rm   z
tanh.fdiffw  s7   € Ø�qŠ=Ü—5‘5œ4 §	¡	¨!¡Ó-¨qÑ0Ñ0Ð0ä$ T¨8Ó4Ð4rC   c                 ó   — t         S ro   ©r   rj   s     r3   rr   ztanh.inverse}  rs   rC   c                 ó¼  — |j                   r™|t        j                  u rt        j                  S |t        j                  u rt        j                  S |t        j
                  u rt        j                  S |j                  rt        j                  S |j                  r
 | | «       S y |t        j                  u rt        j                  S t        |«      }|�6|j                  «       rt         t        | «      z  S t        t        |«      z  S |j                  «       r
 | | «       S |j                  rQt!        |«      \  }}|rAt#        |t$        z  t        z  «      }|t        j                  u rt'        |«      S t#        |«      S |j                  rt        j                  S |j(                  t*        k(  r#|j,                  d   }|t/        d|dz  z   «      z  S |j(                  t0        k(  r/|j,                  d   }t/        |dz
  «      t/        |dz   «      z  |z  S |j(                  t2        k(  r|j,                  d   S |j(                  t4        k(  rd|j,                  d   z  S y ru   )rv   r   rw   rM   rX   rN   ÚNegativeOnerx   r\   ry   rz   r)   r{   r   r(   r|   rd   rÊ   r   rÐ   r}   rq   ri   r   r~   r   r€   )r�   r]   r‚   rƒ   r„   Útanhms         r3   r…   z	tanh.evalƒ  sä  € à�=Š=Ø”a—e‘e‰|Ü—u‘u�ØœŸ
™
Ñ"Ü—u‘u�Øœ×*Ñ*Ñ*Ü—}‘}Ð$Ø—’Ü—v‘v�Ø—’Ù˜S˜D›	�zÐ!ð !ð ”a×'Ñ'Ñ'Ü—u‘u�ä4°SÓ9ˆGàÐ"Ø×3Ñ3Ô5Ü˜2¤ W H£Ñ-Ð-Üœ3˜w›<Ñ'Ð'à×/Ñ/Ô1Ù  ›I˜:Ð%à�zŠzÜ# CÓ(‘��1ÙÜ  ¤2¡¤a¡›L�EØ¤× 1Ñ 1Ñ1Ü# A›w˜ä# A›w˜à�{Š{Ü—v‘v�à�x‰xœ5Ò Ø—H‘H˜Q‘K�Øœ˜a ! Q¡$™h›Ñ'Ð'à�x‰xœ5Ò Ø—H‘H˜Q‘K�Ü˜A ™E“{¤T¨!¨a©%£[Ñ0°1Ñ4Ð4à�x‰xœ5Ò Ø—x‘x ‘{Ð"à�x‰xœ5Ò Ø˜Ÿ™ !™‘}Ð$ð !rC   c                 óÂ   — | dk  s| dz  dk(  rt         j                  S t        |«      }d| dz   z  }t        | dz   «      }t	        | dz   «      }||dz
  z  |z  |z  || z  z  S ©Nr   r7   r6   )r   r\   r   r   r   )rŠ   rƒ   r‹   r_   ÚBÚFs         r3   rŒ   ztanh.taylor_term¸  so   € ð ˆqŠ5�A˜‘E˜Q’JÜ—6‘6ˆMä˜“
ˆAà�A˜‘E‘
ˆAä˜!˜a™%Ó ˆAÜ˜!˜a™%Ó ˆAà�a˜!‘e‘9˜q‘= ‘? Q¨¡TÑ)Ð)rC   c                 óZ   — | j                  | j                  d   j                  «       «      S rŽ   r�   r‘   s    r3   r’   ztanh._eval_conjugateÇ  r“   rC   c                 óö  — | j                   d   j                  r<|r(d|d<    | j                  |fi |¤Žt        j                  fS | t        j                  fS |r2 | j                   d   j                  |fi |¤Žj                  «       \  }}n | j                   d   j                  «       \  }}t        |«      dz  t        |«      dz  z   }t        |«      t        |«      z  |z  t        |«      t        |«      z  |z  fS )Nr   Fr•   r7   r–   )rk   r›   rœ   r   r   Údenoms         r3   r™   ztanh.as_real_imagÊ  sÞ   € Ø�9‰9�Q‰<×(Ò(ÙØ#(��iÑ Ø#˜Ÿ™ DÑ2¨EÑ2´A·F±FÐ;Ð;àœaŸf™f�~Ð%ÙØ(�T—Y‘Y˜q‘\×(Ñ(¨Ñ7°Ñ7×DÑDÓF‰FˆB‘à—Y‘Y˜q‘\×.Ñ.Ó0‰FˆB�Ü�R“˜!‘œc "›g q™jÑ(ˆÜ�R“œ˜b›Ñ! %Ñ'¬¨R«´°R³©¸Ñ)>Ð?Ð?rC   c                 óâ  — | j                   d   }|j                  rƒt        |j                   «      }|j                   D �cg c]  }t        |d¬«      j	                  «       ‘Œ }}ddg}t        |dz   «      D ]  }||dz  xx   t        ||«      z  cc<   Œ |d   |d   z  S |j                  r¬|j                  «       \  }}	|j                  r�|dkD  rˆt        |	«      }
t        d|dz   d«      D �cg c]  }t        t        |«      |«      |
|z  z  ‘Œ }}t        d|dz   d«      D �cg c]  }t        t        |«      |«      |
|z  z  ‘Œ }}t        |Ž t        |Ž z  S t        |«      S c c}w c c}w c c}w )Nr   Fr  r6   r7   )ri   r|   rˆ   rÊ   r°   Úranger*   rY   rª   r«   r   r   )rk   rœ   r]   rŠ   rƒ   ÚTXra   Úir®   r¯   ÚTÚkÚds                r3   r°   ztanh._eval_expand_trigØ  sb  € Ø�i‰i˜‰lˆØ�:Š:Ü�C—H‘H“ˆAàŸ™ö#Øô �q 5Ô)×;Ñ;Õ=ð #ˆBð #à�A�ˆAÜ˜1˜q™5“\ò 2�Ø�!�a‘%“œN¨1¨bÓ1Ñ1”ð2à�Q‘4˜˜!™‘9ÐØ�ZŠZØ×+Ñ+Ó-‰LˆE�5Ø×Ò E¨A¢IÜ˜“K�Ü7<¸QÀÈÁ	È1Ó7MÖN°!”Rœ˜e› aÓ(¨¨A©Ó-ÐN�ÐNÜ7<¸QÀÈÁ	È1Ó7MÖN°!”Rœ˜e› aÓ(¨¨A©Ó-ÐN�ÐNÜ˜A�wœs A˜w‘Ð&Ü�C‹yÐùò#ùò OùÚNs   ¿"E"Ã,"E'Ä""E,Nc                 óF   — t        | «      t        |«      }}||z
  ||z   z  S r,   r´   ©rk   r]   r¶   r·   Úneg_expÚpos_exps         r3   r¸   ztanh._eval_rewrite_as_tractableë  ó*   € Ü ˜t›9¤c¨#£h�ˆØ˜'Ñ! G¨gÑ$5Ñ6Ð6rC   c                 óF   — t        | «      t        |«      }}||z
  ||z   z  S r,   r´   ©rk   r]   r·   rT  rU  s        r3   r¼   ztanh._eval_rewrite_as_expï  rV  rC   c                 ó:   — t          t        t         |z  d¬«      z  S r  )r   r(   r»   s      r3   Ú_eval_rewrite_as_tanztanh._eval_rewrite_as_tanó  ó   € Üˆr”Cœ˜C™¨%Ô0Ñ0Ð0rC   c                 ó:   — t          t        t         |z  d¬«      z  S r  )r   r$   r»   s      r3   Ú_eval_rewrite_as_cotztanh._eval_rewrite_as_cotö  r[  rC   c                 ód   — t         t        |«      z  t        t        t         z  dz  |z
  d¬«      z  S r$  r%  r»   s      r3   r&  ztanh._eval_rewrite_as_sinhù  s(   € Ü”�c“‰{œ4¤¤1¡ Q¡¨¡°uÔ=Ñ=Ð=rC   c                 ód   — t         t        t        t         z  dz  |z
  d¬«      z  t        |«      z  S r$  rÅ   r»   s      r3   rÆ   ztanh._eval_rewrite_as_coshü  s)   € Ü””bœ‘d˜1‘f˜s‘l¨UÔ3Ñ3´D¸³IÑ=Ð=rC   c                 ó   — dt        |«      z  S rÕ   ©rÐ   r»   s      r3   rÓ   ztanh._eval_rewrite_as_cothÿ  ó   € Ø”�c“‰{ÐrC   c                 ó¼   — ddl m} | j                  d   j                  |«      }||j                  v r |d|«      j                  |«      r|S | j                  |«      S ©Nr   )ÚOrderr6   ©Úsympy.series.orderre  ri   râ   Úfree_symbolsÚcontainsr}   ©rk   rƒ   rÝ   rÞ   re  r]   s         r3   rè   ztanh._eval_as_leading_term  sQ   € Ý,Ø�i‰i˜‰l×*Ñ*¨1Ó-ˆà�× Ñ Ñ ¡U¨1¨a£[×%9Ñ%9¸#Ô%>ØˆJà—9‘9˜S“>Ð!rC   c                 ó¾   — | j                   d   }|j                  ry|j                  «       \  }}|dk(  r|t        z  t        dz  k(  ry |t        dz  z  j                  S )Nr   Tr7   rë   rí   s       r3   rî   ztanh._eval_is_real  s[   € Ø�i‰i˜‰lˆØ�;Š;Øà×!Ñ!Ó#‰ˆˆBð �Š7�rœB‘w¤" Q¡$’Øð ”b˜‘d‘×$Ñ$Ð$rC   c                 ó8   — | j                   d   j                  ryy rê   rð   r‘   s    r3   rñ   ztanh._eval_is_extended_real  rò   rC   c                 óh   — | j                   d   j                  r| j                   d   j                  S y rŽ   rô   r‘   s    r3   rö   ztanh._eval_is_positive  r÷   rC   c                 óh   — | j                   d   j                  r| j                   d   j                  S y rŽ   rù   r‘   s    r3   rú   ztanh._eval_is_negative"  r÷   rC   c                 óÂ   — | j                   d   }|j                  «       \  }}t        |«      dz  t        |«      dz  z   }|dk(  ry|j                  ry|j
                  ryy )Nr   r7   FT)ri   r™   r#   rf   Ú	is_numberr—   )rk   r]   r   r   rJ  s        r3   rþ   ztanh._eval_is_finite&  s`   € Ø�i‰i˜‰lˆà×!Ñ!Ó#‰ˆˆBÜ�B“˜‘
œT "›X q™[Ñ(ˆØ�AŠ:ØØ�_Š_ØØ×ÒØð  rC   c                 ó<   — | j                   d   }|j                  ryy rê   ©ri   rx   rý   s     r3   r  ztanh._eval_is_zero2  s   € Ø�i‰i˜‰lˆØ�;Š;Øð rC   r  r  r,   )rQ   rR   rS   rT   rm   rr   r  r…   r	  r   rŒ   r’   r™   r°   r¸   r¼   rZ  r]  r&  rÆ   rÓ   rè   rî   rñ   rö   rú   rþ   r  rB   rC   r3   rÊ   rÊ   c  s˜   „ ñó&5óð ñ2%ó ð2%ðh Øñ*ó ó ð*ò3ó@òó&7ò7ò1ò1ò>ò>òò"ò%òò,ò,ò
órC   rÊ   c                   ó�   — e Zd ZdZdd„Zdd„Zed„ «       Zee	d„ «       «       Z
d„ Zdd„Zdd	„Zd
„ Zd„ Zd„ Zd„ Zd„ Zd„ Zd„ Zd„ Zy)rÐ   a+  
    ``coth(x)`` is the hyperbolic cotangent of ``x``.

    The hyperbolic cotangent function is $\frac{\cosh(x)}{\sinh(x)}$.

    Examples
    ========

    >>> from sympy import coth
    >>> from sympy.abc import x
    >>> coth(x)
    coth(x)

    See Also
    ========

    sinh, cosh, acoth
    c                 ó`   — |dk(  rdt        | j                  d   «      dz  z  S t        | |«      ‚)Nr6   r9   r   r7   r  rj   s     r3   rm   z
coth.fdiffL  s3   € Ø�qŠ=Ø”d˜4Ÿ9™9 Q™<Ó(¨!Ñ+Ñ+Ð+ä$ T¨8Ó4Ð4rC   c                 ó   — t         S ro   )r€   rj   s     r3   rr   zcoth.inverseR  rs   rC   c                 ó¼  — |j                   r™|t        j                  u rt        j                  S |t        j                  u rt        j                  S |t        j
                  u rt        j                  S |j                  rt        j                  S |j                  r
 | | «       S y |t        j                  u rt        j                  S t        |«      }|�6|j                  «       rt        t        | «      z  S t         t        |«      z  S |j                  «       r
 | | «       S |j                  rQt        |«      \  }}|rAt!        |t"        z  t        z  «      }|t        j                  u rt!        |«      S t%        |«      S |j                  rt        j                  S |j&                  t(        k(  r#|j*                  d   }t-        d|dz  z   «      |z  S |j&                  t.        k(  r/|j*                  d   }|t-        |dz
  «      t-        |dz   «      z  z  S |j&                  t0        k(  rd|j*                  d   z  S |j&                  t2        k(  r|j*                  d   S y ru   )rv   r   rw   rM   rX   rN   rB  rx   rz   ry   r)   r{   r   r$   r|   rd   rÐ   r   rÊ   r}   rq   ri   r   r~   r   r€   )r�   r]   r‚   rƒ   r„   Úcothms         r3   r…   z	coth.evalX  sê  € à�=Š=Ø”a—e‘e‰|Ü—u‘u�ØœŸ
™
Ñ"Ü—u‘u�Øœ×*Ñ*Ñ*Ü—}‘}Ð$Ø—’Ü×(Ñ(Ð(Ø—’Ù˜S˜D›	�zÐ!ð !ð ”a×'Ñ'Ñ'Ü—u‘u�ä4°SÓ9ˆGàÐ"Ø×3Ñ3Ô5Üœs G 8›}Ñ,Ð,Ü�rœC ›LÑ(Ð(à×/Ñ/Ô1Ù  ›I˜:Ð%à�zŠzÜ# CÓ(‘��1ÙÜ  ¤2¡¤a¡›L�EØ¤× 1Ñ 1Ñ1Ü# A›w˜ä# A›w˜à�{Š{Ü×(Ñ(Ð(à�x‰xœ5Ò Ø—H‘H˜Q‘K�Ü˜A  1¡™H“~ aÑ'Ð'à�x‰xœ5Ò Ø—H‘H˜Q‘K�Øœ$˜q 1™u›+¬¨Q°©U«Ñ3Ñ4Ð4à�x‰xœ5Ò Ø˜Ÿ™ !™‘}Ð$à�x‰xœ5Ò Ø—x‘x ‘{Ð"ð !rC   c                 óØ   — | dk(  rdt        |«      z  S | dk  s| dz  dk(  rt        j                  S t        |«      }t        | dz   «      }t	        | dz   «      }d| dz   z  |z  |z  || z  z  S ru   ©r   r   r\   r   r   ©rŠ   rƒ   r‹   rF  rG  s        r3   rŒ   zcoth.taylor_term�  sx   € ð �Š6Ø”w˜q“z‘>Ð!Ø�ŠU�a˜!‘e˜q’jÜ—6‘6ˆMä˜“
ˆAä˜!˜a™%Ó ˆAÜ˜!˜a™%Ó ˆAà�q˜1‘u‘: ‘> !Ñ# a¨¡dÑ*Ð*rC   c                 óZ   — | j                  | j                  d   j                  «       «      S rŽ   r�   r‘   s    r3   r’   zcoth._eval_conjugateœ  r“   rC   c                 óö  — ddl m}m} | j                  d   j                  r<|r(d|d<    | j
                  |fi |¤Žt        j                  fS | t        j                  fS |r2 | j                  d   j
                  |fi |¤Žj                  «       \  }}n | j                  d   j                  «       \  }}t        |«      dz   ||«      dz  z   }t        |«      t        |«      z  |z   ||«        ||«      z  |z  fS )Nr   )r#   r'   Fr•   r7   )r  r#   r'   ri   r—   r˜   r   r\   r™   rf   rh   )rk   r›   rœ   r#   r'   r   r   rJ  s           r3   r™   zcoth.as_real_imagŸ  sä   € ßGØ�9‰9�Q‰<×(Ò(ÙØ#(��iÑ Ø#˜Ÿ™ DÑ2¨EÑ2´A·F±FÐ;Ð;àœaŸf™f�~Ð%ÙØ(�T—Y‘Y˜q‘\×(Ñ(¨Ñ7°Ñ7×DÑDÓF‰FˆB‘à—Y‘Y˜q‘\×.Ñ.Ó0‰FˆB�Ü�R“˜!‘™c "›g q™jÑ(ˆÜ�R“œ˜b›Ñ! %Ñ'©#¨b«'¨±#°b³'Ñ)9¸%Ñ)?Ð@Ð@rC   Nc                 óF   — t        | «      t        |«      }}||z   ||z
  z  S r,   r´   rS  s         r3   r¸   zcoth._eval_rewrite_as_tractable®  rV  rC   c                 óF   — t        | «      t        |«      }}||z   ||z
  z  S r,   r´   rX  s        r3   r¼   zcoth._eval_rewrite_as_exp²  rV  rC   c                 óf   — t          t        t        t         z  dz  |z
  d¬«      z  t        |«      z  S r$  r%  r»   s      r3   r&  zcoth._eval_rewrite_as_sinh¶  s+   € Üˆr”$”rœ!‘t˜A‘v ‘|¨eÔ4Ñ4´T¸#³YÑ>Ð>rC   c                 óf   — t          t        |«      z  t        t        t         z  dz  |z
  d¬«      z  S r$  rÅ   r»   s      r3   rÆ   zcoth._eval_rewrite_as_cosh¹  s*   € Üˆr”$�s“)‰|œD¤¤A¡ a¡¨#¡¸Ô>Ñ>Ð>rC   c                 ó   — dt        |«      z  S rÕ   ©rÊ   r»   s      r3   rÍ   zcoth._eval_rewrite_as_tanh¼  rb  rC   c                 óh   — | j                   d   j                  r| j                   d   j                  S y rŽ   rô   r‘   s    r3   rö   zcoth._eval_is_positive¿  r÷   rC   c                 óh   — | j                   d   j                  r| j                   d   j                  S y rŽ   rù   r‘   s    r3   rú   zcoth._eval_is_negativeÃ  r÷   rC   c                 óÂ   — ddl m} | j                  d   j                  |«      }||j                  v r |d|«      j                  |«      rd|z  S | j                  |«      S rd  rf  rj  s         r3   rè   zcoth._eval_as_leading_termÇ  sU   € Ý,Ø�i‰i˜‰l×*Ñ*¨1Ó-ˆà�× Ñ Ñ ¡U¨1¨a£[×%9Ñ%9¸#Ô%>Ø�S‘5ˆLà—9‘9˜S“>Ð!rC   c                 ó´  — | j                   d   }|j                  r˜|j                   D �cg c]  }t        |d¬«      j                  «       ‘Œ }}g g g}t	        |j                   «      }t        |dd«      D ]&  }|||z
  dz     j                  t        ||«      «       Œ( t        |d   Ž t        |d   Ž z  S |j                  rŠ|j                  d¬«      \  }}|j                  ri|dkD  rdt        |d¬«      }	g g g}t        |dd«      D ],  }|||z
  dz     j                  t        ||«      |	|z  z  «       Œ. t        |d   Ž t        |d   Ž z  S t        |«      S c c}w )	Nr   Fr  r9   r7   r6   Tr§   )ri   r|   rÐ   r°   rˆ   rL  Úappendr*   r   rY   rª   r«   r   )
rk   rœ   r]   rƒ   ÚCXra   rŠ   rN  r®   Úcs
             r3   r°   zcoth._eval_expand_trigÐ  sY  € Ø�i‰i˜‰lˆØ�:Š:ØGJÇxÁxÖPÀ!”$�q 5Ô)×;Ñ;Õ=ÐPˆBÐPØ�R�ˆAÜ�C—H‘H“ˆAÜ˜1˜b "Ó%ò =�Ø�1�q‘5˜A‘+‘×%Ñ%¤n°Q¸Ó&;Õ<ð=ä˜˜!™�:œc 1 Q¡4˜jÑ(Ð(Ø�ZŠZØ×'Ñ'°Ð'Ó6‰HˆE�1Ø×Ò E¨A¢IÜ˜ UÔ+�Ø˜�H�Ü˜u b¨"Ó-ò G�AØ�u˜q‘y A‘oÑ&×-Ñ-¬h°u¸aÓ.@ÀÀAÁÑ.EÕFðGä˜A˜a™D�z¤# q¨¡t *Ñ,Ð,Ü�C‹yÐùò Qs   ª"Er  r  r,   )rQ   rR   rS   rT   rm   rr   r  r…   r	  r   rŒ   r’   r™   r¸   r¼   r&  rÆ   rÍ   rö   rú   rè   r°   rB   rC   r3   rÐ   rÐ   8  sz   „ ñó&5óð ñ2#ó ð2#ðh Øñ+ó ó ð+ò3óAó7ò7ò?ò?òò,ò,ò"órC   rÐ   c                   óœ   — e Zd ZU dZdZdZeed<   dZeed<   e	d„ «       Z
d„ Zd„ Zd„ Zd	„ Zdd
„Zd„ Zd„ Zdd„Zd„ Zdd„Zd„ Zd„ Zd„ Zd„ Zy)ÚReciprocalHyperbolicFunctionz=Base class for reciprocal functions of hyperbolic functions. NÚ_is_evenÚ_is_oddc                 ó  — |j                  «       r+| j                  r	 | | «      S | j                  r
 | | «       S | j                  j	                  |«      }t        |d«      r"|j                  «       | k(  r|j                  d   S |�d|z  S |S )Nrr   r   r6   )r{   rŒ  r�  Ú_reciprocal_ofr…   Úhasattrrr   ri   )r�   r]   Úts      r3   r…   z!ReciprocalHyperbolicFunction.evalì  s„   € à×'Ñ'Ô)Ø�|Š|Ù˜C˜4“yÐ Ø�{Š{Ù˜S˜D›	�zÐ!à×Ñ×#Ñ# CÓ(ˆÜ�3˜	Ô" s§{¡{£}¸Ò';Ø—8‘8˜A‘;ÐØ�mˆq�‰sÐ*¨Ð*rC   c                 ób   — | j                  | j                  d   «      } t        ||«      |i |¤ŽS rŽ   )r�  ri   Úgetattr)rk   Úmethod_nameri   r·   Úos        r3   Ú_call_reciprocalz-ReciprocalHyperbolicFunction._call_reciprocalù  s3   € à×Ñ §	¡	¨!¡Ó-ˆØ&Œw�q˜+Ó&¨Ð7°Ñ7Ð7rC   c                 ó@   —  | j                   |g|¢­i |¤Ž}|�d|z  S |S rÕ   )r–  )rk   r”  ri   r·   r‘  s        r3   Ú_calculate_reciprocalz2ReciprocalHyperbolicFunction._calculate_reciprocalþ  s3   € ð "ˆD×!Ñ! +Ð?°Ò?¸Ñ?ˆØ�mˆq�‰sÐ*¨Ð*rC   c                 ó`   — | j                  ||«      }|�|| j                  |«      k7  rd|z  S y y rÕ   )r–  r�  )rk   r”  r]   r‘  s       r3   Ú_rewrite_reciprocalz0ReciprocalHyperbolicFunction._rewrite_reciprocal  s=   € ð ×!Ñ! +¨sÓ3ˆØˆ=˜Q $×"5Ñ"5°cÓ":Ò:Ø�Q‘3ˆJð ;ˆ=rC   c                 ó&   — | j                  d|«      S )Nr¼   ©rš  r»   s      r3   r¼   z1ReciprocalHyperbolicFunction._eval_rewrite_as_exp  s   € Ø×'Ñ'Ð(>ÀÓDÐDrC   c                 ó&   — | j                  d|«      S )Nr¸   rœ  rµ   s       r3   r¸   z7ReciprocalHyperbolicFunction._eval_rewrite_as_tractable  s   € Ø×'Ñ'Ð(DÀcÓJÐJrC   c                 ó&   — | j                  d|«      S )NrÍ   rœ  r»   s      r3   rÍ   z2ReciprocalHyperbolicFunction._eval_rewrite_as_tanh  ó   € Ø×'Ñ'Ð(?ÀÓEÐErC   c                 ó&   — | j                  d|«      S )NrÓ   rœ  r»   s      r3   rÓ   z2ReciprocalHyperbolicFunction._eval_rewrite_as_coth  rŸ  rC   c                 óf   —  d| j                  | j                  d   «      z  j                  |fi |¤ŽS r  )r�  ri   r™   )rk   r›   rœ   s      r3   r™   z)ReciprocalHyperbolicFunction.as_real_imag  s2   € ØC��D×'Ñ'¨¯	©	°!©Ó5Ñ5×CÑCÀDÑRÈEÑRÐRrC   c                 óZ   — | j                  | j                  d   j                  «       «      S rŽ   r�   r‘   s    r3   r’   z,ReciprocalHyperbolicFunction._eval_conjugate  r“   rC   c                 óH   —  | j                   dddi|¤Ž\  }}|t        |z  z   S )Nr›   TrB   rŸ   r    s        r3   r£   z1ReciprocalHyperbolicFunction._eval_expand_complex  s0   € Ø,˜4×,Ñ,Ñ@°$Ð@¸%Ñ@Ñˆ�Øœ˜7™Ñ"Ð"rC   c                 ó&   —  | j                   di |¤ŽS )N)r°   )r˜  )rk   rœ   s     r3   r°   z.ReciprocalHyperbolicFunction._eval_expand_trig!  s   € Ø)ˆt×)Ñ)ÑGÀÑGÐGrC   c                 óh   — d| j                  | j                  d   «      z  j                  |||¬«      S )Nr6   r   rÜ   )r�  ri   rè   )rk   rƒ   rÝ   rÞ   s       r3   rè   z2ReciprocalHyperbolicFunction._eval_as_leading_term$  s4   € Ø�$×%Ñ% d§i¡i°¡lÓ3Ñ3×JÑJÈ1ÐSWÐ^bÐJÓcÐcrC   c                 óR   — | j                  | j                  d   «      j                  S rŽ   )r�  ri   r—   r‘   s    r3   rñ   z3ReciprocalHyperbolicFunction._eval_is_extended_real'  s!   € Ø×"Ñ" 4§9¡9¨Q¡<Ó0×AÑAÐArC   c                 óX   — d| j                  | j                  d   «      z  j                  S r  )r�  ri   rå   r‘   s    r3   rþ   z,ReciprocalHyperbolicFunction._eval_is_finite*  s&   € Ø�$×%Ñ% d§i¡i°¡lÓ3Ñ3×>Ñ>Ð>rC   r,   r  )rQ   rR   rS   rT   r�  rŒ  r   Ú__annotations__r�  r  r…   r–  r˜  rš  r¼   r¸   rÍ   rÓ   r™   r’   r£   r°   rè   rñ   rþ   rB   rC   r3   r‹  r‹  ä  s€   … ÙGð €NØ€HˆiÓØ€GˆYÓàñ
+ó ð
+ò8ò
+òòEóKòFòFóSò3ó#òHòdòBó?rC   r‹  c                   ó^   — e Zd ZdZeZdZdd„Zee	d„ «       «       Z
d„ Zd„ Zd„ Zd„ Zd	„ Zd
„ Zy)r×   a8  
    ``csch(x)`` is the hyperbolic cosecant of ``x``.

    The hyperbolic cosecant function is $\frac{2}{e^x - e^{-x}}$

    Examples
    ========

    >>> from sympy import csch
    >>> from sympy.abc import x
    >>> csch(x)
    csch(x)

    See Also
    ========

    sinh, cosh, tanh, sech, asinh, acosh
    Tc                 óˆ   — |dk(  r2t        | j                  d   «       t        | j                  d   «      z  S t        | |«      ‚)z?
        Returns the first derivative of this function
        r6   r   )rÐ   ri   r×   r   rj   s     r3   rm   z
csch.fdiffE  s@   € ð �qŠ=Ü˜Ÿ™ 1™Ó&Ð&¬¨d¯i©i¸©lÓ);Ñ;Ð;ä$ T¨8Ó4Ð4rC   c                 óÞ   — | dk(  rdt        |«      z  S | dk  s| dz  dk(  rt        j                  S t        |«      }t        | dz   «      }t	        | dz   «      }ddd| z  z
  z  |z  |z  || z  z  S )zF
        Returns the next term in the Taylor series expansion
        r   r6   r7   ry  rz  s        r3   rŒ   zcsch.taylor_termN  s}   € ð �Š6Ø”W˜Q“Z‘<ÐØ�ŠU�a˜!‘e˜q’jÜ—6‘6ˆMä˜“
ˆAä˜!˜a™%Ó ˆAÜ˜!˜a™%Ó ˆAà˜˜A˜q™D™‘> AÑ% aÑ'¨!¨Q©$Ñ.Ð.rC   c                 ó8   — t         t        t         |z  d¬«      z  S r  r¾   r»   s      r3   r¿   zcsch._eval_rewrite_as_sin`  ó   € Ü”3”q˜3‘w¨Ô/Ñ/Ð/rC   c                 ó8   — t         t        t         |z  d¬«      z  S r  rÂ   r»   s      r3   rÃ   zcsch._eval_rewrite_as_cscc  r­  rC   c                 óL   — t         t        |t         t        z  dz  z   d¬«      z  S r$  rÅ   r»   s      r3   rÆ   zcsch._eval_rewrite_as_coshf  s!   € Ü”4˜œa¤"™f q™jÑ(°5Ô9Ñ9Ð9rC   c                 ó   — dt        |«      z  S rÕ   ©rf   r»   s      r3   r&  zcsch._eval_rewrite_as_sinhi  rÙ   rC   c                 óh   — | j                   d   j                  r| j                   d   j                  S y rŽ   rô   r‘   s    r3   rö   zcsch._eval_is_positivel  r÷   rC   c                 óh   — | j                   d   j                  r| j                   d   j                  S y rŽ   rù   r‘   s    r3   rú   zcsch._eval_is_negativep  r÷   rC   Nr  )rQ   rR   rS   rT   rf   r�  r�  rm   r	  r   rŒ   r¿   rÃ   rÆ   r&  rö   rú   rB   rC   r3   r×   r×   .  sR   „ ñð& €NØ€Gó5ð Øñ/ó ó ð/ò 0ò0ò:òò,ó,rC   r×   c                   óX   — e Zd ZdZeZdZdd„Zee	d„ «       «       Z
d„ Zd„ Zd„ Zd„ Zd	„ Zy
)r+  a:  
    ``sech(x)`` is the hyperbolic secant of ``x``.

    The hyperbolic secant function is $\frac{2}{e^x + e^{-x}}$

    Examples
    ========

    >>> from sympy import sech
    >>> from sympy.abc import x
    >>> sech(x)
    sech(x)

    See Also
    ========

    sinh, cosh, tanh, coth, csch, asinh, acosh
    Tc                 óˆ   — |dk(  r2t        | j                  d   «       t        | j                  d   «      z  S t        | |«      ‚r  )rÊ   ri   r+  r   rj   s     r3   rm   z
sech.fdiffŒ  s>   € Ø�qŠ=Ü˜$Ÿ)™) A™,Ó'Ð'¬¨T¯Y©Y°q©\Ó(:Ñ:Ð:ä$ T¨8Ó4Ð4rC   c                 óŒ   — | dk  s| dz  dk(  rt         j                  S t        |«      }t        | «      t	        | «      z  || z  z  S rE  )r   r\   r   r   r   ©rŠ   rƒ   r‹   s      r3   rŒ   zsech.taylor_term’  sC   € ð ˆqŠ5�A˜‘E˜Q’JÜ—6‘6ˆMä˜“
ˆAÜ˜“8œi¨›lÑ*¨Q°©VÑ3Ð3rC   c                 ó0   — dt        t        |z  d¬«      z  S r  r  r»   s      r3   r  zsech._eval_rewrite_as_cos›  r"  rC   c                 ó*   — t        t        |z  d¬«      S r  r   r»   s      r3   r!  zsech._eval_rewrite_as_secž  r  rC   c                 óL   — t         t        |t         t        z  dz  z   d¬«      z  S r$  r%  r»   s      r3   r&  zsech._eval_rewrite_as_sinh¡  s    € Ü”4˜œa¤"™f a™i™°%Ô8Ñ8Ð8rC   c                 ó   — dt        |«      z  S rÕ   ©rh   r»   s      r3   rÆ   zsech._eval_rewrite_as_cosh¤  rÙ   rC   c                 ó8   — | j                   d   j                  ryy rê   rð   r‘   s    r3   rö   zsech._eval_is_positive§  rò   rC   Nr  )rQ   rR   rS   rT   rh   r�  rŒ  rm   r	  r   rŒ   r  r!  r&  rÆ   rö   rB   rC   r3   r+  r+  u  sM   „ ñð& €NØ€Hó5ð Øñ4ó ó ð4ò0ò,ò9òórC   r+  c                   ó   — e Zd ZdZy)ÚInverseHyperbolicFunctionz,Base class for inverse hyperbolic functions.N)rQ   rR   rS   rT   rB   rC   r3   r¿  r¿  °  s   „ Ù6àrC   r¿  c                   ó˜   ‡ — e Zd ZdZdd„Zed„ «       Zeed„ «       «       Z	d„ Z
dˆ fd„	Zd„ ZeZd„ Zd	„ Zd
„ Zd„ Zdd„Zd„ Zd„ Zd„ Zˆ xZS )rq   aM  
    ``asinh(x)`` is the inverse hyperbolic sine of ``x``.

    The inverse hyperbolic sine function.

    Examples
    ========

    >>> from sympy import asinh
    >>> from sympy.abc import x
    >>> asinh(x).diff(x)
    1/sqrt(x**2 + 1)
    >>> asinh(1)
    log(1 + sqrt(2))

    See Also
    ========

    acosh, atanh, sinh
    c                 óf   — |dk(  r!dt        | j                  d   dz  dz   «      z  S t        | |«      ‚r>  )r   ri   r   rj   s     r3   rm   zasinh.fdiffÌ  s7   € Ø�qŠ=Ø”T˜$Ÿ)™) A™,¨™/¨AÑ-Ó.Ñ.Ð.ä$ T¨8Ó4Ð4rC   c                 ó&  — |j                   rê|t        j                  u rt        j                  S |t        j                  u rt        j                  S |t        j                  u rt        j                  S |j
                  rt        j                  S |t        j                  u rt        t        d«      dz   «      S |t        j                  u rt        t        d«      dz
  «      S |j                  r� | | «       S |t        j                  u rt        j                  S |j
                  rt        j                  S t        |«      }|�t        t        |«      z  S |j!                  «       r
 | | «       S t#        |t$        «      r”|j&                  d   j(                  rz|j&                  d   }|j*                  r|S t-        |«      \  }}|�L|�It/        |t0        dz  z   t0        z  «      }|t        t0        z  |z  z
  }|j2                  }|du r|S |du r| S y y y y y )Nr7   r6   r   TF)rv   r   rw   rM   rN   rx   r\   rX   r   r   rB  ry   rz   r)   r   r!   r{   Ú
isinstancerf   ri   rp  rì   r   r   r   Úis_even)	r�   r]   r‚   r4  ÚrrN  Úfr„   Úevens	            r3   r…   z
asinh.evalÒ  sº  € à�=Š=Ø”a—e‘e‰|Ü—u‘u�ØœŸ
™
Ñ"Ü—z‘zÐ!Øœ×*Ñ*Ñ*Ü×)Ñ)Ð)Ø—’Ü—v‘v�ØœŸ™‘Üœ4 ›7 Q™;Ó'Ð'ØœŸ™Ñ%Üœ4 ›7 Q™;Ó'Ð'Ø—’Ù˜S˜D›	�zÐ!à”a×'Ñ'Ñ'Ü×(Ñ(Ð(à�{Š{Ü—v‘v�ä4°SÓ9ˆGàÐ"Üœ4 ›=Ñ(Ð(à×/Ñ/Ô1Ù  ›I˜:Ð%ä�cœ4Ô  S§X¡X¨a¡[×%:Ò%:Ø—‘˜‘ˆAØ�yŠyØ�Ü" 1Ó%‰DˆAˆqØˆ}  Ü˜1œr !™t™8¤R™-Ó(�Øœœ"™˜Q™‘J�Ø—y‘y�Ø˜4‘<Ø�HØ˜U‘]Ø˜2�Ið #ð "/ˆ}ð &;Ð rC   c                 óV  — | dk  s| dz  dk(  rt         j                  S t        |«      }t        |«      dk\  r%| dkD  r |d   }| | dz
  dz  z  | | dz
  z  z  |dz  z  S | dz
  dz  }t	        t         j
                  |«      }t        |«      }t         j                  |z  |z  |z  || z  z  | z  S ©Nr   r7   rH   r6   )r   r\   r   rˆ   r   rA   r   rB  ©rŠ   rƒ   r‹   ra   rP  ÚRrG  s          r3   rŒ   zasinh.taylor_term   s»   € ð ˆqŠ5�A˜‘E˜Q’JÜ—6‘6ˆMä˜“
ˆAÜ�>Ó" aÒ'¨A°ªEØ" 2Ñ&�Ø�r˜Q ™U Q™J‘¨¨1¨q©5©	Ñ2°Q¸±TÑ9Ð9à˜‘U˜q‘L�Ü#¤A§F¡F¨AÓ.�Ü˜a“L�Ü—}‘} aÑ'¨!Ñ+¨aÑ/°!°Q±$Ñ6¸Ñ:Ð:rC   c                 ó�  — | j                   d   }|j                  |d«      j                  «       }|j                  r|j	                  |«      S |t
        j                  u r0| j                  |j	                  |«      «      }|j                  r|S | S |t         t        t
        j                  fv r'| j                  t        «      j                  |||¬«      S d|dz  z   j                  rÑ|j                  ||r|nd«      }t!        |«      j"                  r5t%        |«      j                  r‘| j                  |«       t        t&        z  z
  S t!        |«      j                  r5t%        |«      j"                  rG| j                  |«       t        t&        z  z   S | j                  t        «      j                  |||¬«      S | j                  |«      S ©Nr   rÜ   r6   r7   )ri   rã   Úcancelrx   râ   r   rw   r}   rå   r   rz   r0   r   rè   ry   rá   r   rõ   r   r   ©rk   rƒ   rÝ   rÞ   r]   Úx0r1   Úndirs           r3   rè   zasinh._eval_as_leading_term  si  € Ø�i‰i˜‰lˆØ�X‰X�a˜‹^×"Ñ"Ó$ˆØ�:Š:Ø×&Ñ& qÓ)Ð)à”—‘‰;Ø—9‘9˜S×0Ñ0°Ó3Ó4ˆDØ�~Š~Ø�à�ð ”1�"”aœ×*Ñ*Ð+Ñ+Ø—<‘<¤Ó$×:Ñ:¸1À4ÈdÐ:ÓSÐSà��A‘‰I×"Ò"Ø—7‘7˜1¡d™d°Ó2ˆDÜ�$‹x×#Ò#Ü�b“6×%Ò%Ø ŸI™I b›M˜>¬A¬b©DÑ0Ð0Ü�D“×%Ò%Ü�b“6×%Ò%Ø ŸI™I b›M˜>¬A¬b©DÑ0Ð0à—|‘|¤CÓ(×>Ñ>¸qÀtÐRVÐ>ÓWÐWØ�y‰y˜‹}ÐrC   c                 ó˜  •— | j                   d   }|j                  |d«      }|t        t         fv r(| j                  t        «      j                  ||||¬«      S t        ‰	| �  |||¬«      }|t        j                  u r|S d|dz  z   j                  r¸|j                  ||r|nd«      }t        |«      j                  r(t        |«      j                  r| t        t        z  z
  S |S t        |«      j                  r(t        |«      j                  r| t        t        z  z   S |S | j                  t        «      j                  ||||¬«      S |S ©Nr   rÜ   ©rŠ   rÝ   r6   r7   )ri   rã   r   r0   r   Ú_eval_nseriesÚsuperr   rz   ry   rá   r   rõ   r   r   ©
rk   rƒ   rŠ   rÝ   rÞ   r]   rç   ÚresrÑ  Ú	__class__s
            €r3   rÕ  zasinh._eval_nseries-  s/  ø€ Ø�i‰i˜‰lˆØ�x‰x˜˜1‹~ˆð ”Aœ�r�7‰?Ø—<‘<¤Ó$×2Ñ2°1°a¸dÈÐ2ÓNÐNä‰gÑ# A¨°Ð#Ó6ˆØ”1×$Ñ$Ñ$ØˆJð ��a‘‰K×$Ò$Ø—7‘7˜1¡d™d°Ó2ˆDÜ�$‹x×#Ò#Ü�d“8×'Ò'Ø˜4¤!¤B¡$™;Ð&ð ˆ
ô �D“×%Ò%Ü�d“8×'Ò'Ø˜4¤!¤B¡$™;Ð&ð ˆ
ð —|‘|¤CÓ(×6Ñ6°q¸!À$ÈTÐ6ÓRÐRØˆ
rC   c                 ó<   — t        |t        |dz  dz   «      z   «      S rÈ   ©r   r   ©rk   rƒ   r·   s      r3   Ú_eval_rewrite_as_logzasinh._eval_rewrite_as_logF  s   € Ü�1”t˜A˜q™D 1™H“~Ñ%Ó&Ð&rC   c                 ó<   — t        |t        d|dz  z   «      z  «      S ©Nr6   r7   )r   r   rÜ  s      r3   Ú_eval_rewrite_as_atanhzasinh._eval_rewrite_as_atanhK  s   € Ü�Q”t˜A  1¡™H“~Ñ%Ó&Ð&rC   c                 óˆ   — t         |z  }t         t        d|z
  «      t        |dz
  «      z  t        |«      z  t        dz  z
  z  S rß  )r   r   r~   r   )rk   rƒ   r·   Úixs       r3   Ú_eval_rewrite_as_acoshzasinh._eval_rewrite_as_acoshN  s=   € Üˆq‰SˆÜ”$�q˜2‘v“,œt B¨¡F›|Ñ+¬e°B«iÑ7¼"¸Q¹$Ñ>Ñ?Ð?rC   c                 ó:   — t          t        t         |z  d¬«      z  S r  )r   r!   rÜ  s      r3   Ú_eval_rewrite_as_asinzasinh._eval_rewrite_as_asinR  s   € Üˆr”Dœ˜Q™¨Ô/Ñ/Ð/rC   c                 óZ   — t         t        t         |z  d¬«      z  t         t        z  dz  z
  S )NFr  r7   )r   r   r   rÜ  s      r3   Ú_eval_rewrite_as_acoszasinh._eval_rewrite_as_acosU  s%   € Ü”4œ˜A™¨Ô.Ñ.´´2±°a±Ñ7Ð7rC   c                 ó   — t         S ro   r±  rj   s     r3   rr   zasinh.inverseX  ó	   € ô ˆrC   c                 ó4   — | j                   d   j                  S rŽ   rr  r‘   s    r3   r  zasinh._eval_is_zero^  s   € Ø�y‰y˜‰|×#Ñ#Ð#rC   c                 ó4   — | j                   d   j                  S rŽ   rð   r‘   s    r3   rñ   zasinh._eval_is_extended_reala  ó   € Ø�y‰y˜‰|×,Ñ,Ð,rC   c                 ó4   — | j                   d   j                  S rŽ   rü   r‘   s    r3   rþ   zasinh._eval_is_finited  ó   € Ø�y‰y˜‰|×%Ñ%Ð%rC   r  ©r   )rQ   rR   rS   rT   rm   r  r…   r	  r   rŒ   rè   rÕ  rÝ  r¸   rà  rã  rå  rç  rr   r  rñ   rþ   Ú__classcell__©rÙ  s   @r3   rq   rq   ¶  s~   ø„ ñó*5ð ñ+ó ð+ðZ Øñ;ó ó ð;òõ:ò2'ð "6Ðò'ò@ò0ò8óò$ò-ö&rC   rq   c                   ó˜   ‡ — e Zd ZdZdd„Zed„ «       Zeed„ «       «       Z	d„ Z
dˆ fd„	Zd„ ZeZd„ Zd	„ Zd
„ Zd„ Zdd„Zd„ Zd„ Zd„ Zˆ xZS )r~   aM  
    ``acosh(x)`` is the inverse hyperbolic cosine of ``x``.

    The inverse hyperbolic cosine function.

    Examples
    ========

    >>> from sympy import acosh
    >>> from sympy.abc import x
    >>> acosh(x).diff(x)
    1/(sqrt(x - 1)*sqrt(x + 1))
    >>> acosh(1)
    0

    See Also
    ========

    asinh, atanh, cosh
    c                 ó‚   — |dk(  r/| j                   d   }dt        |dz
  «      t        |dz   «      z  z  S t        | |«      ‚r  ©ri   r   r   )rk   rl   r]   s      r3   rm   zacosh.fdiff~  sC   € Ø�qŠ=Ø—)‘)˜A‘,ˆCØ”d˜3 ™7“m¤D¨¨q©£MÑ1Ñ2Ð2ä$ T¨8Ó4Ð4rC   c                 ój  — |j                   rÃ|t        j                  u rt        j                  S |t        j                  u rt        j                  S |t        j                  u rt        j                  S |j
                  rt        t        z  dz  S |t        j                  u rt        j                  S |t        j                  u rt        t        z  S |j                  r+t        «       }||v r|j                  r||   t        z  S ||   S |t        j                  u rt        j                  S |t        t        j                  z  k(  r!t        j                  t        t        z  dz  z   S |t         t        j                  z  k(  r!t        j                  t        t        z  dz  z
  S |j
                  rt        t        z  t        j                  z  S t!        |t"        «      rÚ|j$                  d   j                  rÀ|j$                  d   }|j&                  rt)        |«      S t+        |«      \  }}|�‰|�†t-        |t        z  «      }|t        t        z  |z  z
  }|j.                  }|du r|j0                  r|S |j2                  r| S y |du r.|t        t        z  z  }|j4                  r| S |j6                  r|S y y y y y y )Nr7   r   TF)rv   r   rw   rM   rN   rx   r   r   rX   r\   rB  rp  rD   r—   rz   rA   rÃ  rh   ri   rì   r   r   r   rÄ  Úis_nonnegativery   Úis_nonpositiverõ   )	r�   r]   Ú	cst_tabler4  rÅ  rN  rÆ  r„   rÇ  s	            r3   r…   z
acosh.eval…  s)  € à�=Š=Ø”a—e‘e‰|Ü—u‘u�ØœŸ
™
Ñ"Ü—z‘zÐ!Øœ×*Ñ*Ñ*Ü—z‘zÐ!Ø—’Üœ!‘t˜a‘x�ØœŸ™‘Ü—v‘v�ØœŸ™Ñ%Üœ!‘t�à�=Š=Ü$›ˆIà�iÑØ×'Ò'Ø$ S™>¬!Ñ+Ð+Ø  ‘~Ð%à”!×#Ñ#Ñ#Ü×$Ñ$Ð$Ø”!”A—J‘J‘,ÒÜ—:‘:¤¤"¡ Q¡Ñ&Ð&Ø”1�"”Q—Z‘Z‘-ÒÜ—:‘:¤¤"¡ Q¡Ñ&Ð&à�;Š;Ü”a‘4œŸ™‘;Ðä�cœ4Ô  S§X¡X¨a¡[×%:Ò%:Ø—‘˜‘ˆAØ�yŠyÜ˜1“v�Ü" 1Ó%‰DˆAˆqØˆ}  Ü˜!œB™$“K�Øœœ"™˜Q™‘J�Ø—y‘y�Ø˜4‘<Ø×'Ò'Ø ˜ØŸšØ !˜r˜	ð 'à˜U‘]Øœœ2™‘I�AØ×'Ò'Ø !˜r˜	ØŸšØ ˜ð 'ð	 #ð "/ˆ}ð &;Ð rC   c                 óf  — | dk(  rt         t        z  dz  S | dk  s| dz  dk(  rt        j                  S t	        |«      }t        |«      dk\  r$| dkD  r|d   }|| dz
  dz  z  | | dz
  z  z  |dz  z  S | dz
  dz  }t        t        j                  |«      }t        |«      }| |z  t         z  || z  z  | z  S rÉ  )	r   r   r   r\   r   rˆ   r   rA   r   rÊ  s          r3   rŒ   zacosh.taylor_term¼  sÄ   € ð �Š6Ü”R‘4˜‘6ˆMØ�ŠU�a˜!‘e˜q’jÜ—6‘6ˆMä˜“
ˆAÜ�>Ó" aÒ'¨A°ªEØ" 2Ñ&�Ø˜A ™E A™:‘~ q¨!¨a©%¡yÑ1°A°q±DÑ8Ð8à˜‘U˜q‘L�Ü#¤A§F¡F¨AÓ.�Ü˜a“L�Ø�r˜A‘v¤‘z A q¡DÑ(¨1Ñ,Ð,rC   c                 óH  — | j                   d   }|j                  |d«      j                  «       }|t        j                   t        j
                  t        j                  t        j                  fv r'| j                  t        «      j                  |||¬«      S |t        j                  u r0| j                  |j                  |«      «      }|j                  r|S | S |dz
  j                  rª|j                  ||r|nd«      }t!        |«      j                  rC|dz   j                  r"| j                  |«      dt"        z  t$        z  z
  S | j                  |«       S t!        |«      j&                  s'| j                  t        «      j                  |||¬«      S | j                  |«      S rÍ  )ri   rã   rÎ  r   rX   r\   rz   r0   r   rè   rw   r}   râ   rå   ry   rá   r   r   r   rõ   rÏ  s           r3   rè   zacosh._eval_as_leading_termÎ  sG  € Ø�i‰i˜‰lˆØ�X‰X�a˜‹^×"Ñ"Ó$ˆà”1—5‘5�&œ!Ÿ&™&¤!§%¡%¬×):Ñ):Ð;Ñ;Ø—<‘<¤Ó$×:Ñ:¸1À4ÈdÐ:ÓSÐSà”—‘‰;Ø—9‘9˜S×0Ñ0°Ó3Ó4ˆDØ�~Š~Ø�à�ð �‰F×ÒØ—7‘7˜1¡d™d°Ó2ˆDÜ�$‹x×#Ò#Ø˜‘F×'Ò'ØŸ9™9 R›=¨1¬Q©3¬r©6Ñ1Ð1ØŸ	™	 "›�~Ð%Ü˜“X×)Ò)Ø—|‘|¤CÓ(×>Ñ>¸qÀtÐRVÐ>ÓWÐWØ�y‰y˜‹}ÐrC   c                 ób  •— | j                   d   }|j                  |d«      }|t        j                  t        j                  fv r(| j                  t        «      j                  ||||¬«      S t        ‰	| �  |||¬«      }|t        j                  u r|S |dz
  j                  r�|j                  ||r|nd«      }t        |«      j                  r%|dz   j                  r|dt        z  t        z  z
  S | S t        |«      j                  s(| j                  t        «      j                  ||||¬«      S |S rÓ  ©ri   rã   r   rX   rB  r0   r   rÕ  rÖ  rz   ry   rá   r   r   r   rõ   r×  s
            €r3   rÕ  zacosh._eval_nseriesç  s  ø€ Ø�i‰i˜‰lˆØ�x‰x˜˜1‹~ˆð ”A—E‘Eœ1Ÿ=™=Ð)Ñ)Ø—<‘<¤Ó$×2Ñ2°1°a¸dÈÐ2ÓNÐNä‰gÑ# A¨°Ð#Ó6ˆØ”1×$Ñ$Ñ$ØˆJð �1‰H×!Ò!Ø—7‘7˜1¡d™d°Ó2ˆDÜ�$‹x×#Ò#Ø˜1‘H×)Ò)Ø ¤1¡¤R¡™<Ð'Ø�t�Ü˜“X×)Ò)Ø—|‘|¤CÓ(×6Ñ6°q¸!À$ÈTÐ6ÓRÐRØˆ
rC   c                 óT   — t        |t        |dz   «      t        |dz
  «      z  z   «      S rÕ   rÛ  rÜ  s      r3   rÝ  zacosh._eval_rewrite_as_logþ  s'   € Ü�1”t˜A ™E“{¤T¨!¨a©%£[Ñ0Ñ0Ó1Ð1rC   c                 óT   — t        |dz
  «      t        d|z
  «      z  t        |«      z  S rÕ   )r   r   rÜ  s      r3   rç  zacosh._eval_rewrite_as_acos  s&   € Ü�A˜‘E‹{œ4  A¡›;Ñ&¬¨a«Ñ0Ð0rC   c                 óh   — t        |dz
  «      t        d|z
  «      z  t        dz  t        |«      z
  z  S rß  )r   r   r!   rÜ  s      r3   rå  zacosh._eval_rewrite_as_asin  s.   € Ü�A˜‘E‹{œ4  A¡›;Ñ&¬"¨Q©$´°a³©.Ñ9Ð9rC   c                 óˆ   — t        |dz
  «      t        d|z
  «      z  t        dz  t        t        t        |z  d¬«      z  z   z  S ©Nr6   r7   Fr  )r   r   r   rq   rÜ  s      r3   Ú_eval_rewrite_as_asinhzacosh._eval_rewrite_as_asinh	  s;   € Ü�A˜‘E‹{œ4  A¡›;Ñ&¬"¨Q©$´´5¼¸1¹ÀuÔ3MÑ1MÑ*MÑNÐNrC   c                 óò   — t        |dz
  «      }t        d|z
  «      }t        |dz  dz
  «      }t        dz  |z  |z  d|t        d|dz  z  «      z  z
  z  |t        |dz   «      z  |z  t        ||z  «      z  z   S rß  )r   r   r   )rk   rƒ   r·   Úsxm1Ús1mxÚsx2m1s         r3   rà  zacosh._eval_rewrite_as_atanh  sƒ   € Ü�A˜‘E‹{ˆÜ�A˜‘E‹{ˆÜ�Q˜‘T˜A‘X“ˆÜ�1‘�T‘	˜$‘  A¬¨Q¨q°!©t©V«Ñ$4Ñ 4Ñ5Ø”T˜!˜a™%“[Ñ  Ñ&¬¨u°Q©w«Ñ7ñ8ð 	9rC   c                 ó   — t         S ro   r¼  rj   s     r3   rr   zacosh.inverse  ré  rC   c                 ó>   — | j                   d   dz
  j                  ryy )Nr   r6   Trr  r‘   s    r3   r  zacosh._eval_is_zero  s    € Ø�I‰I�a‰L˜1Ñ×%Ò%Øð &rC   c                 ó~   — t        | j                  d   j                  | j                  d   dz
  j                  g«      S ©Nr   r6   )r
   ri   r—   Úis_extended_nonnegativer‘   s    r3   rñ   zacosh._eval_is_extended_real  s3   € Ü˜$Ÿ)™) A™,×7Ñ7¸$¿)¹)ÀA¹,ÈÑ:J×9cÑ9cÐdÓeÐerC   c                 ó4   — | j                   d   j                  S rŽ   rü   r‘   s    r3   rþ   zacosh._eval_is_finite   rî  rC   r  rï  )rQ   rR   rS   rT   rm   r  r…   r	  r   rŒ   rè   rÕ  rÝ  r¸   rç  rå  r  rà  rr   r  rñ   rþ   rð  rñ  s   @r3   r~   r~   h  s   ø„ ñó*5ð ñ4!ó ð4!ðl Øñ-ó ó ð-ò õ2ò.2ð "6Ðò1ò:òOò9óòòfö&rC   r~   c                   óŒ   ‡ — e Zd ZdZdd„Zed„ «       Zeed„ «       «       Z	d„ Z
dˆ fd„	Zd„ ZeZd„ Zd	„ Zd
„ Zd„ Zd„ Zdd„Zˆ xZS )r   a)  
    ``atanh(x)`` is the inverse hyperbolic tangent of ``x``.

    The inverse hyperbolic tangent function.

    Examples
    ========

    >>> from sympy import atanh
    >>> from sympy.abc import x
    >>> atanh(x).diff(x)
    1/(1 - x**2)

    See Also
    ========

    asinh, acosh, tanh
    c                 óT   — |dk(  rdd| j                   d   dz  z
  z  S t        | |«      ‚r>  ©ri   r   rj   s     r3   rm   zatanh.fdiff8  ó2   € Ø�qŠ=Ø�a˜$Ÿ)™) A™,¨™/Ñ)Ñ*Ð*ä$ T¨8Ó4Ð4rC   c                 óX  — |j                   râ|t        j                  u rt        j                  S |j                  rt        j                  S |t        j
                  u rt        j                  S |t        j                  u rt        j                  S |t        j                  u rt         t        |«      z  S |t        j                  u rt        t        | «      z  S |j                  rz | | «       S |t        j                  u r%ddlm} t         |t         dz  t        dz  «      z  S t!        |«      }|�t        t        |«      z  S |j#                  «       r
 | | «       S |j                  rt        j                  S t%        |t&        «      r |j(                  d   j*                  r†|j(                  d   }|j,                  r|S t/        |«      \  }}|�X|�Ut1        d|z  t        z  «      }|j2                  }|t        |z  t        z  dz  z
  }	|du r|	S |du r|	t        t        z  dz  z
  S y y y y y )Nr   ©ÚAccumBoundsr7   TF)rv   r   rw   rx   r\   rX   rM   rB  rN   r   r"   ry   rz   Ú!sympy.calculus.accumulationboundsr  r   r)   r{   rÃ  rÊ   ri   rp  rì   r   r   rÄ  )
r�   r]   r  r‚   r4  rÅ  rN  rÆ  rÇ  r„   s
             r3   r…   z
atanh.eval>  sÑ  € à�=Š=Ø”a—e‘e‰|Ü—u‘u�Ø—’Ü—v‘v�ØœŸ™‘Ü—z‘zÐ!ØœŸ™Ñ%Ü×)Ñ)Ð)ØœŸ
™
Ñ"Ü�rœD ›I‘~Ð%Øœ×*Ñ*Ñ*Üœ4  ›:‘~Ð%Ø—’Ù˜S˜D›	�zÐ!à”a×'Ñ'Ñ'ÝIÜ™¤b S¨¡U¬B¨q©DÓ1Ñ1Ð1ä4°SÓ9ˆGàÐ"Üœ4 ›=Ñ(Ð(à×/Ñ/Ô1Ù  ›I˜:Ð%à�;Š;Ü—6‘6ˆMä�cœ4Ô  S§X¡X¨a¡[×%:Ò%:Ø—‘˜‘ˆAØ�yŠyØ�Ü" 1Ó%‰DˆAˆqØˆ}  Ü˜!˜A™#œb™&“M�Ø—y‘y�Øœ˜!™œB™˜q™‘L�Ø˜4‘<Ø�HØ˜U‘]Øœq¤™t A™v™:Ð%ð #ð "/ˆ}ð &;Ð rC   c                 ób   — | dk  s| dz  dk(  rt         j                  S t        |«      }|| z  | z  S ©Nr   r7   )r   r\   r   r·  s      r3   rŒ   zatanh.taylor_termm  s4   € ð ˆqŠ5�A˜‘E˜Q’JÜ—6‘6ˆMä˜“
ˆAØ�a‘4˜!‘8ˆOrC   c                 ó�  — | j                   d   }|j                  |d«      j                  «       }|j                  r|j	                  |«      S |t
        j                  u r0| j                  |j	                  |«      «      }|j                  r|S | S |t
        j                   t
        j                  t
        j                  fv r'| j                  t        «      j                  |||¬«      S d|dz  z
  j                  r½|j                  ||r|nd«      }t!        |«      j                  r+|j                  r†| j                  |«      t"        t$        z  z
  S t!        |«      j&                  r+|j&                  rF| j                  |«      t"        t$        z  z   S | j                  t        «      j                  |||¬«      S | j                  |«      S rÍ  )ri   rã   rÎ  rx   râ   r   rw   r}   rå   rX   rz   r0   r   rè   ry   rá   r   r   r   rõ   rÏ  s           r3   rè   zatanh._eval_as_leading_termv  sa  € Ø�i‰i˜‰lˆØ�X‰X�a˜‹^×"Ñ"Ó$ˆØ�:Š:Ø×&Ñ& qÓ)Ð)Ø”—‘‰;Ø—9‘9˜S×0Ñ0°Ó3Ó4ˆDØ�~Š~Ø�à�ð ”1—5‘5�&œ!Ÿ%™%¤×!2Ñ!2Ð3Ñ3Ø—<‘<¤Ó$×:Ñ:¸1À4ÈdÐ:ÓSÐSà��A‘‰I×"Ò"Ø—7‘7˜1¡d™d°Ó2ˆDÜ�$‹x×#Ò#Ø—>’>ØŸ9™9 R›=¬1¬R©4Ñ/Ð/Ü�D“×%Ò%Ø—>’>ØŸ9™9 R›=¬1¬R©4Ñ/Ð/à—|‘|¤CÓ(×>Ñ>¸qÀtÐRVÐ>ÓWÐWØ�y‰y˜‹}ÐrC   c                 ó–  •— | j                   d   }|j                  |d«      }|t        j                  t        j                  fv r(| j                  t        «      j                  ||||¬«      S t        ‰	| �  |||¬«      }|t        j                  u r|S d|dz  z
  j                  r¤|j                  ||r|nd«      }t        |«      j                  r|j                  r|t        t        z  z
  S |S t        |«      j                  r|j                  r|t        t        z  z   S |S | j                  t        «      j                  ||||¬«      S |S rÓ  rü  r×  s
            €r3   rÕ  zatanh._eval_nseries’  s+  ø€ Ø�i‰i˜‰lˆØ�x‰x˜˜1‹~ˆð ”A—E‘Eœ1Ÿ=™=Ð)Ñ)Ø—<‘<¤Ó$×2Ñ2°1°a¸dÈÐ2ÓNÐNä‰gÑ# A¨°Ð#Ó6ˆØ”1×$Ñ$Ñ$ØˆJð ��a‘‰K×$Ò$Ø—7‘7˜1¡d™d°Ó2ˆDÜ�$‹x×#Ò#Ø×#Ò#Ø¤¤2¡™:Ð%ð ˆ
ô �D“×%Ò%Ø×#Ò#Ø¤¤2¡™:Ð%ð ˆ
ð —|‘|¤CÓ(×6Ñ6°q¸!À$ÈTÐ6ÓRÐRØˆ
rC   c                 óB   — t        d|z   «      t        d|z
  «      z
  dz  S rß  ©r   rÜ  s      r3   rÝ  zatanh._eval_rewrite_as_log«  s"   € Ü�A˜‘E“
œS  Q¡›ZÑ'¨1Ñ,Ð,rC   c                 óÖ   — t        d|dz  dz
  z  «      }t        |z  dt        |dz   «      z  z  t        | «      t        d|dz  z
  «      z  t        |«      z  |z  t        |«      z  z
  S rß  )r   r   rq   )rk   rƒ   r·   rÆ  s       r3   r  zatanh._eval_rewrite_as_asinh°  sm   € Ü��A�q‘D˜1‘H‘ÓˆÜ�1‘�aœ˜a ™d˜U›‘mÑ$Ü�a�R“œ˜a ! Q¡$™h›Ñ'¬¨Q«Ñ/°Ñ1´%¸³(Ñ:ñ;ð 	<rC   c                 ó8   — | j                   d   j                  ryy rê   rr  r‘   s    r3   r  zatanh._eval_is_zeroµ  s   € Ø�9‰9�Q‰<×ÒØð  rC   c                 ó´   — t        | j                  d   j                  d| j                  d   z
  j                  | j                  d   dz   j                  g«      S r
  ©r
   ri   r—   rö  r‘   s    r3   rñ   zatanh._eval_is_extended_real¹  sN   € Ü˜$Ÿ)™) A™,×7Ñ7¸!¸d¿i¹iÈ¹lÑ:J×9ZÑ9ZÐ]a×]fÑ]fÐghÑ]iÐlmÑ]m×\}Ñ\}Ð~ÓÐrC   c                 ó–   — t        t        | j                  d   dz
  j                  | j                  d   dz   j                  g«      «      S r
  ©r   r	   ri   rx   r‘   s    r3   rþ   zatanh._eval_is_finite¼  ó=   € Üœ D§I¡I¨a¡L°1Ñ$4×#=Ñ#=ÀÇ	Á	È!ÁÈqÑ@P×?YÑ?YÐ"ZÓ[Ó\Ð\rC   c                 ó4   — | j                   d   j                  S rŽ   )ri   r/  r‘   s    r3   Ú_eval_is_imaginaryzatanh._eval_is_imaginary¿  s   € Ø�y‰y˜‰|×(Ñ(Ð(rC   c                 ó   — t         S ro   r‚  rj   s     r3   rr   zatanh.inverseÂ  ré  rC   r  rï  )rQ   rR   rS   rT   rm   r  r…   r	  r   rŒ   rè   rÕ  rÝ  r¸   r  r  rñ   rþ   r#  rr   rð  rñ  s   @r3   r   r   $  su   ø„ ñó&5ð ñ,&ó ð,&ð\ Øñó ó ðòõ8ò2-ð "6Ðò<ò
ò@ò]ò)÷rC   r   c                   ó†   ‡ — e Zd ZdZdd„Zed„ «       Zeed„ «       «       Z	d„ Z
dˆ fd„	Zd„ ZeZd„ Zd	„ Zdd
„Zd„ Zd„ Zˆ xZS )r€   a-  
    ``acoth(x)`` is the inverse hyperbolic cotangent of ``x``.

    The inverse hyperbolic cotangent function.

    Examples
    ========

    >>> from sympy import acoth
    >>> from sympy.abc import x
    >>> acoth(x).diff(x)
    1/(1 - x**2)

    See Also
    ========

    asinh, acosh, coth
    c                 óT   — |dk(  rdd| j                   d   dz  z
  z  S t        | |«      ‚r>  r  rj   s     r3   rm   zacoth.fdiffÝ  r  rC   c                 óà  — |j                   rÜ|t        j                  u rt        j                  S |t        j                  u rt        j                  S |t        j
                  u rt        j                  S |j                  rt        t        z  dz  S |t        j                  u rt        j                  S |t        j                  u rt        j
                  S |j                  rf | | «       S |t        j                  u rt        j                  S t        |«      }|�t         t        |«      z  S |j                  «       r
 | | «       S |j                  rt        t        z  t        j                   z  S y r³   )rv   r   rw   rM   r\   rN   rx   r   r   rX   rB  ry   rz   r)   r    r{   rA   )r�   r]   r‚   s      r3   r…   z
acoth.evalã  s  € à�=Š=Ø”a—e‘e‰|Ü—u‘u�ØœŸ
™
Ñ"Ü—v‘v�Øœ×*Ñ*Ñ*Ü—v‘v�Ø—’Üœ!‘t˜a‘x�ØœŸ™‘Ü—z‘zÐ!ØœŸ™Ñ%Ü×)Ñ)Ð)Ø—’Ù˜S˜D›	�zÐ!à”a×'Ñ'Ñ'Ü—v‘v�ä4°SÓ9ˆGàÐ"Ü�rœD ›MÑ)Ð)à×/Ñ/Ô1Ù  ›I˜:Ð%à�;Š;Ü”a‘4œŸ™‘;Ðð rC   c                 óŽ   — | dk(  rt          t        z  dz  S | dk  s| dz  dk(  rt        j                  S t	        |«      }|| z  | z  S r  )r   r   r   r\   r   r·  s      r3   rŒ   zacoth.taylor_term  sJ   € ð �Š6Ü�2”b‘5˜‘7ˆNØ�ŠU�a˜!‘e˜q’jÜ—6‘6ˆMä˜“
ˆAØ�a‘4˜!‘8ˆOrC   c                 óº  — | j                   d   }|j                  |d«      j                  «       }|t        j                  u rd|z  j                  |«      S |t        j                  u r0| j                  |j                  |«      «      }|j                  r|S | S |t        j                   t        j                  t        j                  fv r'| j                  t        «      j                  |||¬«      S |j                  rÏd|dz  z
  j                  r½|j!                  ||r|nd«      }t#        |«      j$                  r+|j                  r†| j                  |«      t&        t(        z  z   S t#        |«      j                  r+|j$                  rF| j                  |«      t&        t(        z  z
  S | j                  t        «      j                  |||¬«      S | j                  |«      S )Nr   r6   rÜ   r7   )ri   rã   rÎ  r   rz   râ   rw   r}   rå   rX   r\   r0   r   rè   rì   rõ   rá   r   ry   r   r   rÏ  s           r3   rè   zacoth._eval_as_leading_term  sp  € Ø�i‰i˜‰lˆØ�X‰X�a˜‹^×"Ñ"Ó$ˆØ”×"Ñ"Ñ"Ø�c‘E×*Ñ*¨1Ó-Ð-Ø”—‘‰;Ø—9‘9˜S×0Ñ0°Ó3Ó4ˆDØ�~Š~Ø�à�ð ”1—5‘5�&œ!Ÿ%™%¤§¡Ð(Ñ(Ø—<‘<¤Ó$×:Ñ:¸1À4ÈdÐ:ÓSÐSà�:Š:˜1˜r 1™u™9×1Ò1Ø—7‘7˜1¡d™d°Ó2ˆDÜ�$‹x×#Ò#Ø—>’>ØŸ9™9 R›=¬1¬R©4Ñ/Ð/Ü�D“×%Ò%Ø—>’>ØŸ9™9 R›=¬1¬R©4Ñ/Ð/à—|‘|¤CÓ(×>Ñ>¸qÀtÐRVÐ>ÓWÐWØ�y‰y˜‹}ÐrC   c                 ó®  •— | j                   d   }|j                  |d«      }|t        j                  t        j                  fv r(| j                  t        «      j                  ||||¬«      S t        ‰	| �  |||¬«      }|t        j                  u r|S |j                  r¶d|dz  z
  j                  r¤|j                  ||r|nd«      }t        |«      j                  r|j                  r|t        t         z  z   S |S t        |«      j                  r|j                  r|t        t         z  z
  S |S | j                  t        «      j                  ||||¬«      S |S rÓ  )ri   rã   r   rX   rB  r0   r   rÕ  rÖ  rz   rì   rõ   rá   r   ry   r   r   r×  s
            €r3   rÕ  zacoth._eval_nseries*  s1  ø€ Ø�i‰i˜‰lˆØ�x‰x˜˜1‹~ˆð ”A—E‘Eœ1Ÿ=™=Ð)Ñ)Ø—<‘<¤Ó$×2Ñ2°1°a¸dÈÐ2ÓNÐNä‰gÑ# A¨°Ð#Ó6ˆØ”1×$Ñ$Ñ$ØˆJð �<Š<˜Q  q¡™[×5Ò5Ø—7‘7˜1¡d™d°Ó2ˆDÜ�$‹x×#Ò#Ø×#Ò#Ø¤¤2¡™:Ð%ð ˆ
ô �D“×%Ò%Ø×#Ò#Ø¤¤2¡™:Ð%ð ˆ
ð —|‘|¤CÓ(×6Ñ6°q¸!À$ÈTÐ6ÓRÐRØˆ
rC   c                 óN   — t        dd|z  z   «      t        dd|z  z
  «      z
  dz  S rß  r  rÜ  s      r3   rÝ  zacoth._eval_rewrite_as_logC  s*   € Ü�A˜˜!™‘G“œs 1 q¨¡s¡7›|Ñ+¨qÑ0Ð0rC   c                 ó   — t        d|z  «      S rÕ   r@  rÜ  s      r3   rà  zacoth._eval_rewrite_as_atanhH  s   € Ü�Q�q‘S‹zÐrC   c           	      ó  — t         t        z  dz  t        |dz
  |z  «      t        ||dz
  z  «      z  t        dd|z  z   «      t        ||dz   z  «      z  z
  z  |t        d|dz  z  «      z  t        t        d|dz  dz
  z  «      «      z  z   S rÈ   )r   r   r   rq   rÜ  s      r3   r  zacoth._eval_rewrite_as_asinhK  sŒ   € Ü”1‘�Q‘œ˜a !™e Q™Y›¬¨Q°°A±©Y«Ñ7¼$¸qÀ1ÀQÁ3¹w»-ÌÈQÐPQÐTUÑPUÉYËÑ:WÑWÑXØ”$�q˜˜A™‘v“,‘œu¤T¨!¨Q°©T°A©X©,Ó%7Ó8Ñ8ñ9ð 	:rC   c                 ó   — t         S ro   ra  rj   s     r3   rr   zacoth.inverseO  ré  rC   c                 óÈ   — t        | j                  d   j                  t        | j                  d   dz
  j                  | j                  d   dz   j
                  g«      g«      S r
  )r
   ri   r—   r	   r  Úis_extended_nonpositiver‘   s    r3   rñ   zacoth._eval_is_extended_realU  sw   € Ü˜$Ÿ)™) A™,×7Ñ7¼ÀDÇIÁIÈaÁLÐSTÑDT×CmÑCmÐpt×pyÑpyÐz{Ñp|ð  @Añ  qA÷  pZñ  pZð  C[ó  :\ð  ]ó  ^ð  	^rC   c                 ó–   — t        t        | j                  d   dz
  j                  | j                  d   dz   j                  g«      «      S r
  r   r‘   s    r3   rþ   zacoth._eval_is_finiteX  r!  rC   r  rï  )rQ   rR   rS   rT   rm   r  r…   r	  r   rŒ   rè   rÕ  rÝ  r¸   rà  r  rr   rñ   rþ   rð  rñ  s   @r3   r€   r€   É  so   ø„ ñó&5ð ñó ðð> Øñó ó ðòõ8ò21ð "6Ðòò:óò^ö]rC   r€   c                   ó’   ‡ — e Zd ZdZdd„Zed„ «       Zeed„ «       «       Z	d„ Z
dˆ fd„	Zdd„Zd„ ZeZd	„ Zd
„ Zd„ Zd„ Zd„ Zd„ Zˆ xZS )Úasecha³  
    ``asech(x)`` is the inverse hyperbolic secant of ``x``.

    The inverse hyperbolic secant function.

    Examples
    ========

    >>> from sympy import asech, sqrt, S
    >>> from sympy.abc import x
    >>> asech(x).diff(x)
    -1/(x*sqrt(1 - x**2))
    >>> asech(1).diff(x)
    0
    >>> asech(1)
    0
    >>> asech(S(2))
    I*pi/3
    >>> asech(-sqrt(2))
    3*I*pi/4
    >>> asech((sqrt(6) - sqrt(2)))
    I*pi/12

    See Also
    ========

    asinh, atanh, cosh, acoth

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Hyperbolic_function
    .. [2] https://dlmf.nist.gov/4.37
    .. [3] https://functions.wolfram.com/ElementaryFunctions/ArcSech/

    c                 óp   — |dk(  r&| j                   d   }d|t        d|dz  z
  «      z  z  S t        | |«      ‚©Nr6   r   r9   r7   rô  ©rk   rl   r4  s      r3   rm   zasech.fdiff‚  s?   € Ø�qŠ=Ø—	‘	˜!‘ˆAØ�qœ˜a ! Q¡$™h›Ñ'Ñ(Ð(ä$ T¨8Ó4Ð4rC   c                 ó¶  — |j                   rÃ|t        j                  u rt        j                  S |t        j                  u rt        t
        z  dz  S |t        j                  u rt        t
        z  dz  S |j                  rt        j                  S |t        j                  u rt        j                  S |t        j                  u rt        t
        z  S |j                  r+t        «       }||v r|j                  r||   t
        z  S ||   S |t        j                  u r%ddlm} t
         |t         dz  t        dz  «      z  S |j                  rt        j                  S y )Nr7   r   r  )rv   r   rw   rM   r   r   rN   rx   rX   r\   rB  rp  rO   r—   rz   r  r  )r�   r]   rø  r  s       r3   r…   z
asech.eval‰  s  € à�=Š=Ø”a—e‘e‰|Ü—u‘u�ØœŸ
™
Ñ"Üœ!‘t˜a‘x�Øœ×*Ñ*Ñ*Üœ!‘t˜a‘x�Ø—’Ü—z‘zÐ!ØœŸ™‘Ü—v‘v�ØœŸ™Ñ%Üœ!‘t�à�=Š=Ü$›ˆIà�iÑØ×'Ò'Ø$ S™>¬!Ñ+Ð+Ø  ‘~Ð%à”!×#Ñ#Ñ#ÝEÜ‘[¤"  Q¡¬¨1©Ó-Ñ-Ð-à�;Š;Ü—:‘:Ðð rC   c                 ó|  — | dk(  rt        d|z  «      S | dk  s| dz  dk(  rt        j                  S t        |«      }t	        |«      dkD  r*| dkD  r%|d   }|| dz
  | dz
  z  z  |dz  z  d| dz  dz  z  z  S | dz  }t        t        j                  |«      | z  }t        |«      | z  dz  | z  dz  }d|z  |z  || z  z  dz  S )Nr   r7   r6   rH   r:   r9   )r   r   r\   r   rˆ   r   rA   r   rÊ  s          r3   rŒ   zasech.taylor_term¨  sá   € ð �Š6Ü�q˜1‘u“:ÐØ�ŠU�a˜!‘e˜q’jÜ—6‘6ˆMä˜“
ˆAÜ�>Ó" QÒ&¨1¨qª5Ø" 2Ñ&�Ø˜Q ™U Q q¡S™MÑ*¨Q°©TÑ1°1¸¸1¹¸q±y±=ÑAÐAà˜‘F�Ü#¤A§F¡F¨AÓ.°Ñ2�Ü˜a“L 1Ñ$¨Ñ)¨AÑ-°Ñ2�Ø˜A‘v ‘z A q¡DÑ(¨1Ñ,Ð,rC   c                 óx  — | j                   d   }|j                  |d«      j                  «       }|t        j                   t        j
                  t        j                  t        j                  fv r'| j                  t        «      j                  |||¬«      S |t        j                  u r0| j                  |j                  |«      «      }|j                  r|S | S |j                  sd|z
  j                  r¶|j                  ||r|nd«      }t!        |«      j"                  rO|j"                  s|dz   j                  r| j                  |«       S | j                  |«      dt$        z  t&        z  z
  S t!        |«      j                  s'| j                  t        «      j                  |||¬«      S | j                  |«      S rÍ  )ri   rã   rÎ  r   rX   r\   rz   r0   r   rè   rw   r}   râ   rå   ry   rá   r   rõ   r   r   rÏ  s           r3   rè   zasech._eval_as_leading_termº  sS  € Ø�i‰i˜‰lˆØ�X‰X�a˜‹^×"Ñ"Ó$ˆà”1—5‘5�&œ!Ÿ&™&¤!§%¡%¬×):Ñ):Ð;Ñ;Ø—<‘<¤Ó$×:Ñ:¸1À4ÈdÐ:ÓSÐSà”—‘‰;Ø—9‘9˜S×0Ñ0°Ó3Ó4ˆDØ�~Š~Ø�à�ð �>Š>˜a "™f×1Ò1Ø—7‘7˜1¡d™d°Ó2ˆDÜ�$‹x×#Ò#Ø—>’> b¨1¡f×%9Ò%9Ø ŸI™I b›M˜>Ð)Ø—y‘y “} q¬¡s¬2¡vÑ-Ð-Ü˜“X×)Ò)Ø—|‘|¤CÓ(×>Ñ>¸qÀtÐRVÐ>ÓWÐWØ�y‰y˜‹}ÐrC   c                 óæ  •— ddl m} | j                  d   }|j                  |d«      }|t        j
                  u �r]t        dd¬«      }t        t        j
                  |dz  z
  «      j                  t        «      j                  |dd|z  «      }	t        j
                  | j                  d   z
  }
|
j                  |«      }|
|z
  |z  }|j                  |d«      s|dk(  r |d«      S  |t        |«      «      S t        t        j
                  |z   «      j                  |||¬«      }|j                  «       t        |«      z  j!                  «       }|	j                  «       j                  ||«      j!                  «       j#                  «        |||z  |«      z   S |t        j$                  u �rkt        dd¬«      }t        t        j$                  |dz  z   «      j                  t        «      j                  |dd|z  «      }	t        j
                  | j                  d   z   }
|
j                  |«      }|
|z
  |z  }|j                  |d«      s,|dk(  r |d«      S t&        t(        z   |t        |«      «      z   S t        t        j
                  |z   «      j                  |||¬«      }|j                  «       t        |«      z  j!                  «       }|	j                  «       j                  ||«      j!                  «       j#                  «        |||z  |«      z   S t*        ‰| �9  |||¬«      }|t        j,                  u r|S |j.                  sd|z
  j.                  r™|j1                  ||r|nd«      }t3        |«      j4                  r1|j4                  s|dz   j.                  r| S |dt&        z  t(        z  z
  S t3        |«      j.                  s(| j                  t        «      j                  ||||¬	«      S |S ©
Nr   )ÚOr‘  T)Úpositiver7   r6   rÔ  rÜ   )rg  r<  ri   rã   r   rX   r   r3  r0   r   Únseriesrâ   Úis_meromorphicr   rÕ  ÚremoveOr˜   ÚpowsimprB  r   r   rÖ  rz   ry   rá   r   rõ   ©rk   rƒ   rŠ   rÝ   rÞ   r<  r]   rç   r‘  ÚserÚarg1rÆ  ÚgÚres1rØ  rÑ  rÙ  s                   €r3   rÕ  zasech._eval_nseriesÓ  s=  ø€ Ý(Ø�i‰i˜‰lˆØ�x‰x˜˜1‹~ˆð ”1—5‘5Š=Ü�c DÔ)ˆAÜœŸ™  1¡™Ó%×-Ñ-¬cÓ2×:Ñ:¸1¸aÀÀ1ÁÓEˆCÜ—5‘5˜4Ÿ9™9 Q™<Ñ'ˆDØ×$Ñ$ QÓ'ˆAØ˜‘˜A‘ˆAØ×#Ñ# A qÔ)Ø  Ašv‘q˜“tÐ5©1¬T°!«W«:Ð5ÜœŸ™ ™	“?×0Ñ0°°a¸dÐ0ÓCˆDØ—<‘<“>¤$ q£'Ñ)×1Ñ1Ó3ˆCØ—;‘;“=×%Ñ% a¨Ó-×4Ñ4Ó6×>Ñ>Ó@Á1ÀQÈÁTÈ1Ã:ÑMÐMà”1—=‘=Ò Ü�c DÔ)ˆAÜœŸ™¨¨1©Ñ,Ó-×5Ñ5´cÓ:×BÑBÀ1ÀaÈÈ1ÉÓMˆCÜ—5‘5˜4Ÿ9™9 Q™<Ñ'ˆDØ×$Ñ$ QÓ'ˆAØ˜‘˜A‘ˆAØ×#Ñ# A qÔ)Ø  Ašv‘q˜“tÐ<¬1¬R©4±!´D¸³G³*Ñ+<Ð<ÜœŸ™ ™	“?×0Ñ0°°a¸dÐ0ÓCˆDØ—<‘<“>¤$ q£'Ñ)×1Ñ1Ó3ˆCØ—;‘;“=×%Ñ% a¨Ó-×4Ñ4Ó6×>Ñ>Ó@Á1ÀQÈÁTÈ1Ã:ÑMÐMä‰gÑ# A¨°Ð#Ó6ˆØ”1×$Ñ$Ñ$ØˆJð ×Ò  D¡×5Ò5Ø—7‘7˜1¡d™d°Ó2ˆDÜ�$‹x×#Ò#Ø×#Ò#¨¨q©×'=Ò'=Ø˜4�KØ˜Qœq™S¤™V‘|Ð#Ü˜“X×)Ò)Ø—|‘|¤CÓ(×6Ñ6°q¸!À$ÈTÐ6ÓRÐRØˆ
rC   c                 ó   — t         S ro   r*  rj   s     r3   rr   zasech.inverse   ré  rC   c                 óf   — t        d|z  t        d|z  dz
  «      t        d|z  dz   «      z  z   «      S rÕ   rÛ  r»   s      r3   rÝ  zasech._eval_rewrite_as_log  s3   € Ü�1�S‘5œ4  #¡¨¡	›?¬T°!°C±%¸!±)«_Ñ<Ñ<Ó=Ð=rC   c                 ó   — t        d|z  «      S rÕ   )r~   r»   s      r3   rã  zasech._eval_rewrite_as_acosh  ó   € Ü�Q�s‘U‹|ÐrC   c                 ó°   — t        d|z  dz
  «      t        dd|z  z
  «      z  t        t        t        |z  d¬«      z  t        t        j
                  z  z   z  S r  )r   r   rq   r   r   rA   r»   s      r3   r  zasech._eval_rewrite_as_asinh  sN   € Ü�A�c‘E˜A‘I‹œt A¨¨#©¡I›Ñ.´´%¼¸#¹ÈÔ2NÑ0NÜ24´Q·V±V±)ñ1<ñ =ð 	=rC   c           	      óh  — t         t        z  dt        |«      t        d|z  «      z  z
  t         dz  t        | «      z  t        |«      z  z
  t         dz  t        |dz  «      z  t        |dz   «      z  z
  z  t        d|dz   z  «      t        |dz   «      z  t        t        d|dz  z
  «      «      z  z   S rß  )r   r   r   r   rÜ  s      r3   rà  zasech._eval_rewrite_as_atanh  s©   € Ü”"‘�aœ$˜q›'¤$ q¨¡s£)Ñ+Ñ+¬a°©c´$¸°r³(©l¼4À»7Ñ.BÑBÄQÀqÁSÌÈaÐQRÉdËÁ^ÔTXÐZ[Ð]^ÑZ^ÐY^ÓT_ÑE_Ñ_Ñ`Ü�q˜!˜a™%‘y“/¤$ q¨1¡u£+Ñ-¬e´D¸¸QÀ¹T¹³NÓ.CÑCñDð 	ErC   c                 ó”   — t        d|z  dz
  «      t        dd|z  z
  «      z  t        dz  t        t        t        |z  d¬«      z  z
  z  S r  )r   r   r   ÚacschrÜ  s      r3   Ú_eval_rewrite_as_acschzasech._eval_rewrite_as_acsch  sC   € Ü�A�a‘C˜!‘G‹}œT ! a¨¡c¡'›]Ñ*¬B¨q©D´1´U¼1¸Q¹3ÈÔ5OÑ3OÑ,OÑPÐPrC   c                 ó®   — t        | j                  d   j                  | j                  d   j                  d| j                  d   z
  j                  g«      S r
  r  r‘   s    r3   rñ   zasech._eval_is_extended_real  sI   € Ü˜$Ÿ)™) A™,×7Ñ7¸¿¹À1¹×9TÑ9TÐWXÐ[_×[dÑ[dÐefÑ[gÑWg×VwÑVwÐxÓyÐyrC   c                 óF   — t        | j                  d   j                  «      S rŽ   ©r   ri   rx   r‘   s    r3   rþ   zasech._eval_is_finite  ó   € Ü˜Ÿ™ 1™×-Ñ-Ó.Ð.rC   r  rï  )rQ   rR   rS   rT   rm   r  r…   r	  r   rŒ   rè   rÕ  rr   rÝ  r¸   rã  r  rà  rO  rñ   rþ   rð  rñ  s   @r3   r3  r3  \  s|   ø„ ñ#óJ5ð ñó ðð< Øñ-ó ó ð-ò õ2+óZò>ð "6Ðòò=òEòQòzö/rC   r3  c                   ó’   ‡ — e Zd ZdZdd„Zed„ «       Zeed„ «       «       Z	d„ Z
dˆ fd„	Zdd„Zd„ ZeZd	„ Zd
„ Zd„ Zd„ Zd„ Zd„ Zˆ xZS )rN  aµ  
    ``acsch(x)`` is the inverse hyperbolic cosecant of ``x``.

    The inverse hyperbolic cosecant function.

    Examples
    ========

    >>> from sympy import acsch, sqrt, I
    >>> from sympy.abc import x
    >>> acsch(x).diff(x)
    -1/(x**2*sqrt(1 + x**(-2)))
    >>> acsch(1).diff(x)
    0
    >>> acsch(1)
    log(1 + sqrt(2))
    >>> acsch(I)
    -I*pi/2
    >>> acsch(-2*I)
    I*pi/6
    >>> acsch(I*(sqrt(6) - sqrt(2)))
    -5*I*pi/12

    See Also
    ========

    asinh

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Hyperbolic_function
    .. [2] https://dlmf.nist.gov/4.37
    .. [3] https://functions.wolfram.com/ElementaryFunctions/ArcCsch/

    c                 ó|   — |dk(  r,| j                   d   }d|dz  t        dd|dz  z  z   «      z  z  S t        | |«      ‚r5  rô  r6  s      r3   rm   zacsch.fdiffF  sH   € Ø�qŠ=Ø—	‘	˜!‘ˆAØ�q˜!‘tœD  Q q¨!¡t¡V¡Ó,Ñ,Ñ-Ð-ä$ T¨8Ó4Ð4rC   c                 óú  — |j                   rÕ|t        j                  u rt        j                  S |t        j                  u rt        j                  S |t        j
                  u rt        j                  S |j                  rt        j                  S |t        j                  u rt        dt        d«      z   «      S |t        j                  u rt        dt        d«      z   «       S |j                  rt        «       }||v r||   t        z  S |t        j                  u rt        j                  S |j                  rt        j                  S |j                  rt        j                  S |j!                  «       r
 | | «       S y rß  )rv   r   rw   rM   r\   rN   rx   rz   rX   r   r   rB  rp  rJ   r   Úis_infiniter{   )r�   r]   rø  s      r3   r…   z
acsch.evalM  s  € à�=Š=Ø”a—e‘e‰|Ü—u‘u�ØœŸ
™
Ñ"Ü—v‘v�Øœ×*Ñ*Ñ*Ü—v‘v�Ø—’Ü×(Ñ(Ð(ØœŸ™‘Ü˜1œt A›w™;Ó'Ð'ØœŸ™Ñ%Ü˜Q¤ a£™[Ó)Ð)Ð)à�=Š=Ü$›ˆIà�iÑØ  ‘~¤aÑ'Ð'à”!×#Ñ#Ñ#Ü—6‘6ˆMà�?Š?Ü—6‘6ˆMà�;Š;Ü×$Ñ$Ð$à×'Ñ'Ô)Ù˜˜“I�:Ðð *rC   c                 ó¦  — | dk(  rt        d|z  «      S | dk  s| dz  dk(  rt        j                  S t        |«      }t	        |«      dkD  r+| dkD  r&|d   }| | dz
  | dz
  z  z  |dz  z  d| dz  dz  z  z  S | dz  }t        t        j                  |«      | z  }t        |«      | z  dz  | z  dz  }t        j                  |dz   z  |z  |z  || z  z  dz  S )Nr   r7   r6   rH   r:   )	r   r   r\   r   rˆ   r   rA   r   rB  rÊ  s          r3   rŒ   zacsch.taylor_termo  sò   € ð �Š6Ü�q˜1‘u“:ÐØ�ŠU�a˜!‘e˜q’jÜ—6‘6ˆMä˜“
ˆAÜ�>Ó" QÒ&¨1¨qª5Ø" 2Ñ&�Ø�r˜a !™e a¨¡c™]Ñ+¨a°©dÑ2°A¸¸A¹À¹	±MÑBÐBà˜‘F�Ü#¤A§F¡F¨AÓ.°!Ñ3�Ü˜a“L 1Ñ$¨Ñ)¨AÑ-°Ñ2�Ü—}‘} q¨!¡tÑ,¨qÑ0°1Ñ4°q¸!±tÑ;¸aÑ?Ð?rC   c                 óº  — | j                   d   }|j                  |d«      j                  «       }|t         t        t        j
                  fv r'| j                  t        «      j                  |||¬«      S |t        j                  u r0| j                  |j                  |«      «      }|j                  r|S | S |t        j                  u rd|z  j                  |«      S |j                  rãd|dz  z   j                  rÑ|j!                  ||r|nd«      }t#        |«      j                  r5t%        |«      j                  r‘| j                  |«       t        t&        z  z
  S t#        |«      j(                  r5t%        |«      j(                  rG| j                  |«       t        t&        z  z   S | j                  t        «      j                  |||¬«      S | j                  |«      S rÍ  )ri   rã   rÎ  r   r   r\   r0   r   rè   rw   r}   râ   rå   rz   r/  rõ   rá   r   r   r   ry   rÏ  s           r3   rè   zacsch._eval_as_leading_term�  su  € Ø�i‰i˜‰lˆØ�X‰X�a˜‹^×"Ñ"Ó$ˆà”1�"”aœŸ™�Ñ Ø—<‘<¤Ó$×:Ñ:¸1À4ÈdÐ:ÓSÐSà”—‘‰;Ø—9‘9˜S×0Ñ0°Ó3Ó4ˆDØ�~Š~Ø�à�à”×"Ñ"Ñ"Ø�c‘E×*Ñ*¨1Ó-Ð-à�?Š?  B¨¡E¡	×6Ò6Ø—7‘7˜1¡d™d°Ó2ˆDÜ�$‹x×#Ò#Ü�b“6×%Ò%Ø ŸI™I b›M˜>¬A¬b©DÑ0Ð0Ü�D“×%Ò%Ü�b“6×%Ò%Ø ŸI™I b›M˜>¬A¬b©DÑ0Ð0à—|‘|¤CÓ(×>Ñ>¸qÀtÐRVÐ>ÓWÐWØ�y‰y˜‹}ÐrC   c                 ó"  •— ddl m} | j                  d   }|j                  |d«      }|t        u �r^t        dd¬«      }t        t        |dz  z   «      j                  t        «      j                  |dd|z  «      }	t         | j                  d   z   }
|
j                  |«      }|
|z
  |z  }|j                  |d«      s0|dk(  r |d«      S t         t        z  dz   |t        |«      «      z   S t        t        j                  |z   «      j!                  |||¬«      }|j#                  «       t        |«      z  j%                  «       }|	j#                  «       j                  ||«      j%                  «       j'                  «        |||z  |«      z   }|S |t        j(                  t        z  k(  �r[t        dd¬«      }t        t         |dz  z   «      j                  t        «      j                  |dd|z  «      }	t        | j                  d   z   }
|
j                  |«      }|
|z
  |z  }|j                  |d«      s/|dk(  r |d«      S t        t        z  dz   |t        |«      «      z   S t        t        j                  |z   «      j!                  |||¬«      }|j#                  «       t        |«      z  j%                  «       }|	j#                  «       j                  ||«      j%                  «       j'                  «        |||z  |«      z   S t*        ‰| �A  |||¬«      }|t        j,                  u r|S |j.                  r×d|dz  z   j0                  rÅ| j                  d   j3                  ||r|nd«      }t5        |«      j0                  r(t7        |«      j0                  r| t        t        z  z
  S |S t5        |«      j8                  r(t7        |«      j8                  r| t        t        z  z   S |S | j                  t        «      j!                  ||||¬	«      S |S r;  )rg  r<  ri   rã   r   r   rN  r0   r   r>  râ   r?  r   r   r   rX   rÕ  r@  r˜   rA  rB  rÖ  rz   r/  rõ   rá   r   r   ry   rB  s                   €r3   rÕ  zacsch._eval_nseriesž  sr  ø€ Ý(Ø�i‰i˜‰lˆØ�x‰x˜˜1‹~ˆð ”1Š9Ü�c DÔ)ˆAÜœ˜A˜q™D™“/×)Ñ)¬#Ó.×6Ñ6°q¸!¸Q¸q¹SÓAˆCÜ�2˜Ÿ	™	 !™Ñ$ˆDØ×$Ñ$ QÓ'ˆAØ˜‘˜A‘ˆAØ×#Ñ# A qÔ)Ø  Ašv‘q˜“tÐ?¬A¨2¬b©5°©7±Q´t¸A³w³ZÑ+?Ð?ÜœŸ™ ™	“?×0Ñ0°°a¸dÐ0ÓCˆDØ—<‘<“>¤$ q£'Ñ)×1Ñ1Ó3ˆCØ—+‘+“-×$Ñ$ Q¨Ó,×3Ñ3Ó5×=Ñ=Ó?Á!ÀAÀqÁDÈ!Ã*ÑLˆCØˆJà”1—=‘=¤‘?Ó"Ü�c DÔ)ˆAÜœ˜˜Q ™T™	Ó"×*Ñ*¬3Ó/×7Ñ7¸¸1¸aÀ¹cÓBˆCÜ�t—y‘y ‘|Ñ#ˆDØ×$Ñ$ QÓ'ˆAØ˜‘˜A‘ˆAØ×#Ñ# A qÔ)Ø  Ašv‘q˜“tÐ>¬1¬R©4°©6±A´d¸1³g³JÑ+>Ð>ÜœŸ™ ™	“?×0Ñ0°°a¸dÐ0ÓCˆDØ—<‘<“>¤$ q£'Ñ)×1Ñ1Ó3ˆCØ—;‘;“=×%Ñ% a¨Ó-×4Ñ4Ó6×>Ñ>Ó@Á1ÀQÈÁTÈ1Ã:ÑMÐMä‰gÑ# A¨°Ð#Ó6ˆØ”1×$Ñ$Ñ$ØˆJð ×Ò ! d¨A¡g¡+×!:Ò!:Ø—9‘9˜Q‘<×#Ñ# A©t¡t¸Ó;ˆDÜ�$‹x×#Ò#Ü�d“8×'Ò'Ø˜4¤!¤B¡$™;Ð&ð ˆ
ô �D“×%Ò%Ü�d“8×'Ò'Ø˜4¤!¤B¡$™;Ð&ð ˆ
ð —|‘|¤CÓ(×6Ñ6°q¸!À$ÈTÐ6ÓRÐRØˆ
rC   c                 ó   — t         S ro   rÖ   rj   s     r3   rr   zacsch.inverseÎ  ré  rC   c                 óH   — t        d|z  t        d|dz  z  dz   «      z   «      S rß  rÛ  r»   s      r3   rÝ  zacsch._eval_rewrite_as_logÔ  s'   € Ü�1�S‘5œ4  # q¡&¡¨1¡Ó-Ñ-Ó.Ð.rC   c                 ó   — t        d|z  «      S rÕ   rp   r»   s      r3   r  zacsch._eval_rewrite_as_asinhÙ  rJ  rC   c                 óÀ   — t         t        dt         |z  z
  «      t        t         |z  dz
  «      z  t        t         |z  d¬«      z  t        t        j
                  z  z
  z  S r  )r   r   r~   r   r   rA   r»   s      r3   rã  zacsch._eval_rewrite_as_acoshÜ  sT   € Ü”$�qœ1˜S™5‘y“/¤$¤q¨¡u¨q¡y£/Ñ1Ü %¤a¨¡e°eÔ <ñ=Ü?AÄ!Ç&Á&¹yñIñ Jð 	JrC   c                 ó´   — |dz  }|dz   }t        | «      |z  t        t        j                  z  t        |dz   «      |z  t	        t        |«      «      z  z
  z  S rÈ   )r   r   r   rA   r   )rk   r]   r·   Úarg2Úarg2p1s        r3   rà  zacsch._eval_rewrite_as_atanhà  s^   € Ø�A‰vˆØ˜‘ˆÜ�T�E‹{˜3‰¤¤1§6¡6¡	Ü $ f¨a¡i ZÓ 0°Ñ 7¼¼dÀ6»lÓ8KÑ Kñ!Lñ Mð 	MrC   c                 ó4   — | j                   d   j                  S rŽ   )ri   rW  r‘   s    r3   r  zacsch._eval_is_zeroæ  s   € Ø�y‰y˜‰|×'Ñ'Ð'rC   c                 ó4   — | j                   d   j                  S rŽ   rð   r‘   s    r3   rñ   zacsch._eval_is_extended_realé  rì  rC   c                 óF   — t        | j                  d   j                  «      S rŽ   rR  r‘   s    r3   rþ   zacsch._eval_is_finiteì  rS  rC   r  rï  )rQ   rR   rS   rT   rm   r  r…   r	  r   rŒ   rè   rÕ  rr   rÝ  r¸   r  rã  rà  r  rñ   rþ   rð  rñ  s   @r3   rN  rN     s~   ø„ ñ#óJ5ð ñó ððB Øñ@ó ó ð@ò õ:.ó`ò/ð "6ÐòòJòMò(ò-ö/rC   rN  N)JÚ
sympy.corer   r   r   Úsympy.core.addr   Úsympy.core.functionr   r   Úsympy.core.logicr	   r
   r   r   Úsympy.core.numbersr   r   r   Úsympy.core.symbolr   Ú(sympy.functions.combinatorial.factorialsr   r   r   Ú%sympy.functions.combinatorial.numbersr   r   r   Ú$sympy.functions.elementary.complexesr   r   r   Ú&sympy.functions.elementary.exponentialr   r   r   Ú#sympy.functions.elementary.integersr   Ú(sympy.functions.elementary.miscellaneousr   r  r   r    r!   r"   r#   r$   r%   r&   r'   r(   r)   Úsympy.polys.specialpolysr*   r4   rD   rJ   rO   r/   rd   rf   rh   rÊ   rÐ   r‹  r×   r+  r¿  rq   r~   r   r€   r3  rN  rB   rC   r3   ú<module>rr     sŒ  ðß *Ñ *Ý ß Cß FÓ Fß .Ñ .Ý #÷Gñ Gç FÑ Fß <Ñ <ß LÑ LÝ 5Ý 9÷$÷ $÷ $ñ $õ 4ò2ð
 	ñó 	ðð2 	ñ
ó 	ð
ð$ 	ñ
ó 	ð
ôF
˜ô 
òôDQ'Ðô Q'ôhs2Ðô s2ôlRÐô RôjiÐô iôXG?Ð#5ô G?ôTD,Ð'ô D,ôN4Ð'ô 4ôv	 ô 	ôo&Ð%ô o&ôdy&Ð%ô y&ôxbÐ%ô bôJP]Ð%ô P]ôfA/Ð%ô A/ôHM/Ð%õ M/rC   