ó
    ‰*£hÑ ã                   óš  • S SK JrJrJr  S SKJr  S SKJrJr  S SK	J
r
JrJrJr  S SKJrJrJr  S SKJr  S SKJrJrJr  S SKJrJrJr  S S	KJrJrJr  S S
K J!r!J"r"J#r#  S SK$J%r%  S SK&J'r'  S SK(J)r)J*r*J+r+J,r,J-r-J.r.J/r/J0r0J1r1J2r2J3r3  S SK4J5r5  S r6\S 5       r7\S 5       r8\S 5       r9 " S S\5      r:S r; " S S\:5      r< " S S\:5      r= " S S\:5      r> " S S\:5      r? " S S\:5      r@ " S  S!\@5      rA " S" S#\@5      rB " S$ S%\5      rC " S& S'\C5      rD " S( S)\C5      rE " S* S+\C5      rF " S, S-\C5      rG " S. S/\C5      rH " S0 S1\C5      rIg2)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           	      óœ   • U R                  U R                  [        5       Vs0 s H  nXR                  [        5      _M     sn5      $ s  snf ©N)ÚxreplaceÚatomsÚHyperbolicFunctionÚrewriter   )ÚexprÚhs     Úb/home/mande/repo/quber/.venv/lib/python3.13/site-packages/sympy/functions/elementary/hyperbolic.pyÚ_rewrite_hyperbolics_as_expr4      sH   € Ø�=‰=Ø—‘Ô.Ô/ó1Ú/ˆAð ŸY™Y¤s›^Ò+Ù/ñ1ó 2ð 2ùò 1s   £A	c                  ón  • 0 [         [        [         S[        S5      -   -  5      _[         * [        [         * S[        S5      -   -  5      _[        R                  [
        S-  _[        SS5      [
        [        SS5      -  _[        S5      S-  [
        S-  _[        S5      * S-  [
        [        SS5      -  _S[        S5      -  [
        S-  _S[        S5      -  [
        [        SS5      -  _[        S5      S-  [
        S-  _[        S5      * S-  [
        [        SS5      -  _[        S5      S-
  [        S5      -  [
        [        SS	5      -  _[        S5      S-
  * [        S5      -  [
        [        S
S	5      -  _[        S[        S5      -   5      S-  [
        S-  _[        S[        S5      -   5      * S-  [
        [        S
S5      -  _[        S[        S5      -
  5      S-  [
        [        SS5      -  _[        S[        S5      -
  5      * S-  [
        [        SS5      -  _S[        S5      -   S[        S5      -  -  [
        S	-  _S[        S5      -   * S[        S5      -  -  [
        [        SS	5      -  [        S5      S-   S-  [
        S-  [        S5      S-   * S-  [
        [        SS5      -  0E$ )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  • [         [        * S-  [         [        S5      [        S5      -   -  [        * S-  [         S[        S5      -   -  [        * S-  [         S-  [        S[        S5      -
  5      -  [        * S-  [         S-  [        * S-  [         [        SS[        S5      -  -   5      -  [        * S-  [         [        S5      -  [        * S-  [         [        S5      S-
  -  S	[        -  S-  [         S-  [        S
5      -  [        * S
-  [         S-  [        S[        S5      -   5      -  S	[        -  S-  [         [        SS[        S5      -  -
  5      -  S[        -  S-  [         [        S5      [        S5      -
  -  S[        -  S-  [        S5      [         * [	        S[        S5      -   S-  5      -  0$ )Nr7   r;   r>   r6   r<   é
   r=   r:   éýÿÿÿr8   éþÿÿÿéûÿÿÿ)r   r   r   r   r   rB   rC   r3   Ú_acsch_tablerJ   3   sg  € ô ”ˆs�Q‰wÜŒt�A‹wœ˜a›Ñ Ñ!¤B 3¨¡8Üˆq”4˜“7‰{‰Oœb˜S 2™XÜˆa‰C”$�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  • 0 [         [        [         -  S-  * [        S[        S5      -   5      -   _[         * [        [         -  S-  [        S[        S5      -   5      -   _[        S5      [        S5      -
  [        S-  _[        S5      [        S5      -
  S[        -  S-  _[        SS[        S5      -  -
  5      [        S-  _[        SS[        S5      -  -
  5      * S[        -  S-  _S[        S[        S5      -   5      -  [        S	-  _S
[        S[        S5      -   5      -  S[        -  S	-  _S[        S5      -  [        S-  _S
[        S5      -  S[        -  S-  _[        S5      S-
  [        S-  _S[        S5      -
  S[        -  S-  _[        S5      [        S-  _[        S5      * S[        -  S-  _[        SS[        S5      -  -   5      S[        -  S-  _[        SS[        S5      -  -   5      * S[        -  S-  _[	        S5      [        S-  _[	        S5      * S[        -  S-  [        SS[        S5      -   -  5      S[        -  S	-  [        SS[        S5      -   -  5      * S[        -  S	-  S[        S5      -   S[        -  S-  S[        S5      -
  S[        -  S-  [        S5      [        S5      -   S[        -  S-  [        S5      * [        S5      -
  S[        -  S-  [         [        R
                  -  [        * [         -  S-  [         [        R                  -  [        [         -  S-  0	E$ )Nr7   r6   r;   r>   r@   r<   rF   é	   r=   rH   r?   r8   r:   r9   )r   r   r   r   r   ÚInfinityÚNegativeInfinityrB   rC   r3   Ú_asech_tablerO   F   s  € ð
Ü”"”Q‘$˜‘(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   • \ rS rSrSrSrSrg)r/   éj   zQ
Base class for hyperbolic functions.

See Also
========

sinh, cosh, tanh, coth
TrB   N)Ú__name__Ú
__module__Ú__qualname__Ú__firstlineno__Ú__doc__Ú
unbranchedÚ__static_attributes__rB   rC   r3   r/   r/   j   s   † ñð ƒJrC   r/   c                 óf  • [         [        -  n[        R                  " U 5       HY  nX!:X  a  [        R
                  n  OUUR                  (       d  M-  UR                  5       u  p4XA:X  d  MF  UR                  (       d  MY    O   U [        R                  4$ U[        R                  -  nX5-
  nXU-  -
  U4$ )a¥  
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_ipirg   w   s�   € ô" ŒQ‰$€CÜ�]Š]˜3ÖˆØ‹8Ü—‘ˆAÙØ�X�X‰XØ—>‘>Ó#‰DˆAØ�x˜AŸMŸM™MÙñ  ð ”A—F‘Fˆ{Ðà
Œa�f‰f‰*€BØ	
‰€BØ�C‘‰<˜ÐÐrC   c                   óØ   • \ rS rSrSrSS jrSS jr\S 5       r\	\
S 5       5       rS rSS jrSS	 jrSS
 jrSS jrS rS rS rS rS rS rS rS rS rS rS rS rS rS rSrg)Úsinhé™   zî
``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   • US:X  a  [        U R                  S   5      $ [        X5      e)z0
Returns the first derivative of this function.
r6   r   )ÚcoshÚargsr   ©ÚselfÚargindexs     r3   ÚfdiffÚ
sinh.fdiff­   s)   € ð �q‹=Ü˜Ÿ	™	 !™Ó%Ð%ä$ TÓ4Ð4rC   c                 ó   • [         $ ©z'
Returns the inverse of this function.
©Úasinhrn   s     r3   ÚinverseÚsinh.inverse¶   ó	   € ô ˆrC   c                 óú  • UR                   (       a¦  U[        R                  L a  [        R                  $ U[        R                  L a  [        R                  $ U[        R                  L a  [        R                  $ UR
                  (       a  [        R                  $ UR                  (       a
  U " U* 5      * $ g U[        R                  L a  [        R                  $ [        U5      nUb  [        [        U5      -  $ UR                  5       (       a
  U " U* 5      * $ UR                  (       aS  [        U5      u  p4U(       a?  U[        -  [        -  n[!        U5      [#        U5      -  [#        U5      [!        U5      -  -   $ UR
                  (       a  [        R                  $ UR$                  [&        :X  a  UR(                  S   $ UR$                  [*        :X  a,  UR(                  S   n[-        US-
  5      [-        US-   5      -  $ UR$                  [.        :X  a#  UR(                  S   nU[-        SUS-  -
  5      -  $ UR$                  [0        :X  a/  UR(                  S   nS[-        US-
  5      [-        US-   5      -  -  $ g ©Nr   r6   r7   )Ú	is_Numberr   ÚNaNrM   rN   Úis_zeror_   Úis_negativeÚComplexInfinityr)   r   r'   Úcould_extract_minus_signÚis_Addrg   r   ri   rl   Úfuncrv   rm   Úacoshr   ÚatanhÚacoth)Úclsr`   Úi_coeffÚxÚms        r3   ÚevalÚ	sinh.eval¼   sÞ  € à�=�=Ø”a—e‘eŠ|Ü—u‘u�ØœŸ
™
Ò"Ü—z‘zÐ!Øœ×*Ñ*Ò*Ü×)Ñ)Ð)Ø——Ü—v‘v�Ø——Ù˜S˜D›	�zÐ!ð !ð ”a×'Ñ'Ò'Ü—u‘u�ä4°SÓ9ˆGàÑ"Üœ3˜w›<Ñ'Ð'à×/Ñ/×1Ñ1Ù  ›I˜:Ð%à�z�zÜ# CÓ(‘�ÞØœ"™œQ™�AÜ ›7¤4¨£7™?¬T°!«W´T¸!³W©_Ñ<Ð<à�{�{Ü—v‘v�à�x‰xœ5Ó Ø—x‘x ‘{Ð"à�x‰xœ5Ó Ø—H‘H˜Q‘K�Ü˜A ™E“{¤T¨!¨a©%£[Ñ0Ð0à�x‰xœ5Ó Ø—H‘H˜Q‘K�Øœ˜a ! Q¡$™h›Ñ'Ð'à�x‰xœ5Ó Ø—H‘H˜Q‘K�Øœ$˜q 1™u›+¬¨Q°©U«Ñ3Ñ4Ð4ð !rC   c                 ó¼   • U S:  d	  U S-  S:X  a  [         R                  $ [        U5      n[        U5      S:”  a  US   nX1S-  -  X S-
  -  -  $ X-  [	        U 5      -  $ )z7
Returns the next term in the Taylor series expansion.
r   r7   rH   r6   ©r   r_   r   Úlenr   ©Únr‰   Úprevious_termsrd   s       r3   Útaylor_termÚsinh.taylor_termí   sf   € ð ˆq‹5�A˜‘E˜Q“JÜ—6‘6ˆMä˜“
ˆAä�>Ó" QÓ&Ø" 2Ñ&�Ø˜a™4‘x 1¨!¡e¡9Ñ-Ð-à‘v¤	¨!£Ñ,Ð,rC   c                 óZ   • U R                  U R                  S   R                  5       5      $ ©Nr   ©rƒ   rm   Ú	conjugate©ro   s    r3   Ú_eval_conjugateÚsinh._eval_conjugateþ   ó"   € Ø�y‰y˜Ÿ™ 1™×/Ñ/Ó1Ó2Ð2rC   c                 óÊ  • U R                   S   R                  (       aA  U(       a(  SUS'   U R                  " U40 UD6[        R                  4$ U [        R                  4$ U(       a1  U R                   S   R                  " U40 UD6R                  5       u  p4OU R                   S   R                  5       u  p4[        U5      [        U5      -  [        U5      [        U5      -  4$ )z0
Returns this function as a complex coordinate.
r   FÚcomplex©
rm   Úis_extended_realÚexpandr   r_   Úas_real_imagri   r#   rl   r'   ©ro   ÚdeepÚhintsr   r   s        r3   r¢   Úsinh.as_real_imag  s·   € ð �9‰9�Q‰<×(×(ÞØ#(��iÑ ØŸš DÑ2¨EÑ2´A·F±FÐ;Ð;àœaŸf™f�~Ð%ÞØ—Y‘Y˜q‘\×(Ò(¨Ñ7°Ñ7×DÑDÓF‰FˆB�à—Y‘Y˜q‘\×.Ñ.Ó0‰FˆBÜ�R“œ˜R›Ñ ¤$ r£(¬3¨r«7Ñ"2Ð3Ð3rC   c                 óD   • U R                   " SSU0UD6u  p4X4[        -  -   $ ©Nr¤   rB   ©r¢   r   ©ro   r¤   r¥   Úre_partÚim_parts        r3   Ú_eval_expand_complexÚsinh._eval_expand_complex  ó*   € Ø×,Ò,Ñ@°$Ð@¸%Ñ@ÑˆØ¤™Ñ"Ð"rC   c                 óô  • U(       a!  U R                   S   R                  " U40 UD6nOU R                   S   nS nUR                  (       a  UR                  5       u  pEORUR	                  SS9u  pgU[
        R                  La.  UR                  (       a  U[
        R                  La
  UnUS-
  U-  nUb<  [        U5      [        W5      -  [        U5      [        U5      -  -   R                  SS9$ [        U5      $ ©Nr   T©Úrationalr6   )Útrig)
rm   r¡   r‚   r]   Úas_coeff_Mulr   r[   Ú
is_Integerri   rl   ©ro   r¤   r¥   r`   r‰   ÚyÚcoeffÚtermss           r3   Ú_eval_expand_trigÚsinh._eval_expand_trig  óÓ   € ÞØ—)‘)˜A‘,×%Ò% dÑ4¨eÑ4‰Cà—)‘)˜A‘,ˆCØˆØ�:�:Ø×#Ñ#Ó%‰DˆAˆqà×+Ñ+°TÐ+Ð:‰LˆEØœAŸE™EÒ! e×&6×&6¸5ÌÏÉÒ;MØ�Ø˜Q‘Y ‘M�Ø‰=Ü˜“GœD ›G‘O¤d¨1£g¬d°1«g¡oÑ5×=Ñ=À4Ð=ÐHÐHÜ�C‹yÐrC   Nc                 ó8   • [        U5      [        U* 5      -
  S-  $ ©Nr7   ©r   ©ro   r`   ÚlimitvarÚkwargss       r3   Ú_eval_rewrite_as_tractableÚsinh._eval_rewrite_as_tractable&  ó   € Ü�C“œ3 ˜t›9Ñ$¨Ñ)Ð)rC   c                 ó8   • [        U5      [        U* 5      -
  S-  $ r¿   rÀ   ©ro   r`   rÃ   s      r3   Ú_eval_rewrite_as_expÚsinh._eval_rewrite_as_exp)  rÆ   rC   c                 ó6   • [         * [        [         U-  5      -  $ r,   ©r   r'   rÈ   s      r3   Ú_eval_rewrite_as_sinÚsinh._eval_rewrite_as_sin,  ó   € Üˆr”Cœ˜C™“LÑ Ð rC   c                 ó6   • [         * [        [         U-  5      -  $ r,   ©r   r%   rÈ   s      r3   Ú_eval_rewrite_as_cscÚsinh._eval_rewrite_as_csc/  rÏ   rC   c                 óJ   • [         * [        U[        [         -  S-  -   5      -  $ r¿   ©r   rl   r   rÈ   s      r3   Ú_eval_rewrite_as_coshÚsinh._eval_rewrite_as_cosh2  s    € Üˆr”$�sœR¤™T !™V‘|Ó$Ñ$Ð$rC   c                 óV   • [        [        R                  U-  5      nSU-  SUS-  -
  -  $ ©Nr7   r6   ©Útanhr   rA   ©ro   r`   rÃ   Ú	tanh_halfs       r3   Ú_eval_rewrite_as_tanhÚsinh._eval_rewrite_as_tanh5  s,   € ÜœŸ™ ™Ó$ˆ	Ø�‰{˜A 	¨1¡Ñ,Ñ-Ð-rC   c                 óV   • [        [        R                  U-  5      nSU-  US-  S-
  -  $ rÙ   ©Úcothr   rA   ©ro   r`   rÃ   Ú	coth_halfs       r3   Ú_eval_rewrite_as_cothÚsinh._eval_rewrite_as_coth9  s,   € ÜœŸ™ ™Ó$ˆ	Ø�‰{˜I q™L¨1Ñ,Ñ-Ð-rC   c                 ó   • S[        U5      -  $ ©Nr6   ©ÚcschrÈ   s      r3   Ú_eval_rewrite_as_cschÚsinh._eval_rewrite_as_csch=  ó   € Ø”4˜“9‰}ÐrC   c                 ó<  • U R                   S   R                  XUS9nUR                  US5      nU[        R                  L a$  UR                  USUR                  (       a  SOSS9nUR                  (       a  U$ UR                  (       a  U R                  U5      $ U $ ©Nr   ©ÚlogxÚcdirÚ-Ú+)Údir)
rm   Úas_leading_termÚsubsr   r}   Úlimitr   r~   Ú	is_finiterƒ   ©ro   r‰   rñ   rò   r`   Úarg0s         r3   Ú_eval_as_leading_termÚsinh._eval_as_leading_term@  s}   € Ø�i‰i˜‰l×*Ñ*¨1¸dÐ*ÐCˆØ�x‰x˜˜1‹~ˆà”1—5‘5Š=Ø—9‘9˜Q ¨d×.>×.>¡sÀC�9ÐHˆDØ�<�<ØˆJØ�^�^Ø—9‘9˜T“?Ð"àˆKrC   c                 óŽ   • U R                   S   nUR                  (       a  gUR                  5       u  p#U[        -  R                  $ ©Nr   T©rm   Úis_realr¢   r   r~   ©ro   r`   r   r   s       r3   Ú_eval_is_realÚsinh._eval_is_realM  s9   € Ø�i‰i˜‰lˆØ�;�;Øð ×!Ñ!Ó#‰ˆØ”2‘�‰ÐrC   c                 óB   • U R                   S   R                  (       a  gg rÿ   ©rm   r    r™   s    r3   Ú_eval_is_extended_realÚsinh._eval_is_extended_realW  ó   € Ø�9‰9�Q‰<×(×(Øð )rC   c                 ór   • U R                   S   R                  (       a  U R                   S   R                  $ g r–   ©rm   r    Úis_positiver™   s    r3   Ú_eval_is_positiveÚsinh._eval_is_positive[  ó,   € Ø�9‰9�Q‰<×(×(Ø—9‘9˜Q‘<×+Ñ+Ð+ð )rC   c                 ór   • U R                   S   R                  (       a  U R                   S   R                  $ g r–   ©rm   r    r   r™   s    r3   Ú_eval_is_negativeÚsinh._eval_is_negative_  r  rC   c                 ó8   • U R                   S   nUR                  $ r–   ©rm   rù   ©ro   r`   s     r3   Ú_eval_is_finiteÚsinh._eval_is_finitec  ó   € Ø�i‰i˜‰lˆØ�}‰}ÐrC   c                 ór   • [        U R                  S   5      u  pUR                  (       a  UR                  $ g r–   )rg   rm   r~   Ú
is_integer©ro   ÚrestÚipi_mults      r3   Ú_eval_is_zeroÚsinh._eval_is_zerog  s.   € Ü% d§i¡i°¡lÓ3‰ˆØ�<�<Ø×&Ñ&Ð&ð rC   rB   ©r6   ©Tr,   ) rR   rS   rT   rU   rV   rq   rw   Úclassmethodr‹   Ústaticmethodr   r“   rš   r¢   r­   r»   rÄ   rÉ   rÍ   rÒ   rÖ   rÞ   rå   rë   rü   r  r  r  r  r  r  rX   rB   rC   r3   ri   ri   ™   s¡   † ñô&5ôð ñ.5ó ð.5ð` Øñ-ó ó ð-ò3ô4ô #ôô"*ò*ò!ò!ò%ò.ò.òòòòò,ò,òõ'rC   ri   c                   óÈ   • \ rS rSrSrSS jr\S 5       r\\	S 5       5       r
S rSS jrSS jrSS	 jrSS jrS rS rS rS rS rS rS rS rS rS rS rS rS rSrg
)rl   im  zò
``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   • US:X  a  [        U R                  S   5      $ [        X5      e©Nr6   r   ©ri   rm   r   rn   s     r3   rq   Ú
cosh.fdiff�  s'   € Ø�q‹=Ü˜Ÿ	™	 !™Ó%Ð%ä$ TÓ4Ð4rC   c                 óÎ  • SSK Jn  UR                  (       a¥  U[        R                  L a  [        R                  $ U[        R
                  L a  [        R
                  $ U[        R                  L a  [        R
                  $ UR                  (       a  [        R                  $ UR                  (       a	  U " U* 5      $ g U[        R                  L a  [        R                  $ [        U5      nUb  U" U5      $ UR                  5       (       a	  U " U* 5      $ UR                  (       aS  [        U5      u  pEU(       a?  U[        -  [         -  n[#        U5      [#        U5      -  [%        U5      [%        U5      -  -   $ UR                  (       a  [        R                  $ UR&                  [(        :X  a  [+        SUR,                  S   S-  -   5      $ UR&                  [.        :X  a  UR,                  S   $ UR&                  [0        :X  a!  S[+        SUR,                  S   S-  -
  5      -  $ UR&                  [2        :X  a/  UR,                  S   nU[+        US-
  5      [+        US-   5      -  -  $ g )Nr   )r#   r6   r7   )Ú(sympy.functions.elementary.trigonometricr#   r|   r   r}   rM   rN   r~   r[   r   r€   r)   r�   r‚   rg   r   r   rl   ri   rƒ   rv   r   rm   r„   r…   r†   )r‡   r`   r#   rˆ   r‰   rŠ   s         r3   r‹   Ú	cosh.eval‡  sÇ  € å@Ø�=�=Ø”a—e‘eŠ|Ü—u‘u�ØœŸ
™
Ò"Ü—z‘zÐ!Øœ×*Ñ*Ò*Ü—z‘zÐ!Ø——Ü—u‘u�Ø——Ù˜C˜4“yÐ ð !ð ”a×'Ñ'Ò'Ü—u‘u�ä4°SÓ9ˆGàÑ"Ù˜7“|Ð#à×/Ñ/×1Ñ1Ù ˜t›9Ð$à�z�zÜ# CÓ(‘�ÞØœ"™œQ™�AÜ ›7¤4¨£7™?¬T°!«W´T¸!³W©_Ñ<Ð<à�{�{Ü—u‘u�à�x‰xœ5Ó Ü˜A §¡¨¡¨Q¡Ñ.Ó/Ð/à�x‰xœ5Ó Ø—x‘x ‘{Ð"à�x‰xœ5Ó Øœ˜a #§(¡(¨1¡+¨q¡.Ñ0Ó1Ñ1Ð1à�x‰xœ5Ó Ø—H‘H˜Q‘K�Øœ$˜q 1™u›+¬¨Q°©U«Ñ3Ñ4Ð4ð !rC   c                 ó¼   • U S:  d	  U S-  S:X  a  [         R                  $ [        U5      n[        U5      S:”  a  US   nX1S-  -  X S-
  -  -  $ X-  [	        U 5      -  $ )Nr   r7   r6   rH   rŽ   r�   s       r3   r“   Úcosh.taylor_term·  sf   € ð ˆq‹5�A˜‘E˜Q“JÜ—6‘6ˆMä˜“
ˆAä�>Ó" QÓ&Ø" 2Ñ&�Ø˜a™4‘x 1¨!¡e¡9Ñ-Ð-à‘vœi¨›lÑ*Ð*rC   c                 óZ   • U R                  U R                  S   R                  5       5      $ r–   r—   r™   s    r3   rš   Úcosh._eval_conjugateÅ  rœ   rC   c                 óÊ  • U R                   S   R                  (       aA  U(       a(  SUS'   U R                  " U40 UD6[        R                  4$ U [        R                  4$ U(       a1  U R                   S   R                  " U40 UD6R                  5       u  p4OU R                   S   R                  5       u  p4[        U5      [        U5      -  [        U5      [        U5      -  4$ )Nr   Frž   )
rm   r    r¡   r   r_   r¢   rl   r#   ri   r'   r£   s        r3   r¢   Úcosh.as_real_imagÈ  sµ   € Ø�9‰9�Q‰<×(×(ÞØ#(��iÑ ØŸš DÑ2¨EÑ2´A·F±FÐ;Ð;àœaŸf™f�~Ð%ÞØ—Y‘Y˜q‘\×(Ò(¨Ñ7°Ñ7×DÑDÓF‰FˆB�à—Y‘Y˜q‘\×.Ñ.Ó0‰FˆBä�R“œ˜R›Ñ ¤$ r£(¬3¨r«7Ñ"2Ð3Ð3rC   c                 óD   • U R                   " SSU0UD6u  p4X4[        -  -   $ r¨   r©   rª   s        r3   r­   Úcosh._eval_expand_complexÖ  r¯   rC   c                 óô  • U(       a!  U R                   S   R                  " U40 UD6nOU R                   S   nS nUR                  (       a  UR                  5       u  pEORUR	                  SS9u  pgU[
        R                  La.  UR                  (       a  U[
        R                  La
  UnUS-
  U-  nUb<  [        U5      [        W5      -  [        U5      [        U5      -  -   R                  SS9$ [        U5      $ r±   )
rm   r¡   r‚   r]   rµ   r   r[   r¶   rl   ri   r·   s           r3   r»   Úcosh._eval_expand_trigÚ  r½   rC   Nc                 ó8   • [        U5      [        U* 5      -   S-  $ r¿   rÀ   rÁ   s       r3   rÄ   Úcosh._eval_rewrite_as_tractableë  rÆ   rC   c                 ó8   • [        U5      [        U* 5      -   S-  $ r¿   rÀ   rÈ   s      r3   rÉ   Úcosh._eval_rewrite_as_expî  rÆ   rC   c                 ó$   • [        [        U-  SS9$ ©NF©Úevaluate©r#   r   rÈ   s      r3   Ú_eval_rewrite_as_cosÚcosh._eval_rewrite_as_cosñ  ó   € Ü”1�s‘7 UÑ+Ð+rC   c                 ó*   • S[        [        U-  SS9-  $ ©Nr6   Fr=  ©r&   r   rÈ   s      r3   Ú_eval_rewrite_as_secÚcosh._eval_rewrite_as_secô  ó   € Ø”3”q˜3‘w¨Ñ/Ñ/Ð/rC   c                 óH   • [         * [        U[        [         -  S-  -   SS9-  $ ©Nr7   Fr=  ©r   ri   r   rÈ   s      r3   Ú_eval_rewrite_as_sinhÚcosh._eval_rewrite_as_sinh÷  s"   € Üˆr”$�sœR¤™T !™V‘|¨eÑ4Ñ4Ð4rC   c                 óV   • [        [        R                  U-  5      S-  nSU-   SU-
  -  $ rÙ   rÚ   rÜ   s       r3   rÞ   Úcosh._eval_rewrite_as_tanhú  s,   € ÜœŸ™ ™Ó$ aÑ'ˆ	Ø�I‘  I¡Ñ.Ð.rC   c                 óV   • [        [        R                  U-  5      S-  nUS-   US-
  -  $ rÙ   rá   rã   s       r3   rå   Úcosh._eval_rewrite_as_cothþ  s,   € ÜœŸ™ ™Ó$ aÑ'ˆ	Ø˜A‘ 	¨A¡Ñ.Ð.rC   c                 ó   • S[        U5      -  $ rè   ©ÚsechrÈ   s      r3   Ú_eval_rewrite_as_sechÚcosh._eval_rewrite_as_sech  rí   rC   c                 óX  • U R                   S   R                  XUS9nUR                  US5      nU[        R                  L a$  UR                  USUR                  (       a  SOSS9nUR                  (       a  [        R                  $ UR                  (       a  U R                  U5      $ U $ rï   )rm   rö   r÷   r   r}   rø   r   r~   r[   rù   rƒ   rú   s         r3   rü   Úcosh._eval_as_leading_term  s�   € Ø�i‰i˜‰l×*Ñ*¨1¸dÐ*ÐCˆØ�x‰x˜˜1‹~ˆà”1—5‘5Š=Ø—9‘9˜Q ¨d×.>×.>¡sÀC�9ÐHˆDØ�<�<Ü—5‘5ˆLØ�^�^Ø—9‘9˜T“?Ð"àˆKrC   c                 ó°   • U R                   S   nUR                  (       d  UR                  (       a  gUR                  5       u  p#U[        -  R
                  $ rÿ   )rm   r  Úis_imaginaryr¢   r   r~   r  s       r3   r  Úcosh._eval_is_real  sC   € Ø�i‰i˜‰lˆð �;�;˜#×*×*Øð
 ×!Ñ!Ó#‰ˆØ”2‘�‰ÐrC   c                 ó  • U R                   S   nUR                  5       u  p#US[        -  -  nUR                  nU(       a  gUR                  nUSL a  U$ [	        U[        U[	        U[        S-  :  US[        -  S-  :„  /5      /5      /5      $ ©Nr   r7   TFr8   ©rm   r¢   r   r~   r	   r
   ©ro   Úzr‰   r¸   ÚymodÚyzeroÚxzeros          r3   r  Úcosh._eval_is_positive  s–   € ð �I‰I�a‰Lˆà�~‰~Ó‰ˆØ�A”b‘D‰zˆà—‘ˆæØà—	‘	ˆà�EŠ>ØˆLäàäØÜ˜d¤R¨¡T™k¨4°!´B±$°q±&©=Ð9Ó:ðó ð	ó ð 	rC   c                 ó  • U R                   S   nUR                  5       u  p#US[        -  -  nUR                  nU(       a  gUR                  nUSL a  U$ [	        U[        U[	        U[        S-  :*  US[        -  S-  :¬  /5      /5      /5      $ r]  r^  r_  s          r3   Ú_eval_is_nonnegativeÚcosh._eval_is_nonnegative?  s”   € Ø�I‰I�a‰Lˆà�~‰~Ó‰ˆØ�A”b‘D‰zˆà—‘ˆæØà—	‘	ˆà�EŠ>ØˆLäàäØÜ˜d¤b¨¡d™l¨D°A´b±D¸±F©NÐ;Ó<ðó ð	ó ð 	rC   c                 ó8   • U R                   S   nUR                  $ r–   r  r  s     r3   r  Úcosh._eval_is_finiteY  r  rC   c                 ó¤   • [        U R                  S   5      u  pU(       a/  UR                  (       a  U[        R                  -
  R
                  $ g g r–   )rg   rm   r~   r   rA   r  r  s      r3   r  Úcosh._eval_is_zero]  s;   € Ü% d§i¡i°¡lÓ3‰ˆÞ˜ŸŸØœqŸv™vÑ%×1Ñ1Ð1ð %ˆ8rC   rB   r!  r"  r,   )rR   rS   rT   rU   rV   rq   r#  r‹   r$  r   r“   rš   r¢   r­   r»   rÄ   rÉ   r@  rF  rL  rÞ   rå   rU  rü   r  r  rf  r  r  rX   rB   rC   r3   rl   rl   m  s˜   † ñô&5ð ñ-5ó ð-5ð^ Øñ
+ó ó ð
+ò3ô4ô#ôô"*ò*ò,ò0ò5ò/ò/òòòòò@ò4õ2rC   rl   c                   óÄ   • \ rS rSrSrSS jrSS jr\S 5       r\	\
S 5       5       rS rSS jrS	 rSS jrS rS rS rS rS rS rS rS rS rS rS rS rS rSrg
)rÛ   ic  z÷
``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|   • US:X  a,  [         R                  [        U R                  S   5      S-  -
  $ [	        X5      e©Nr6   r   r7   )r   r[   rÛ   rm   r   rn   s     r3   rq   Ú
tanh.fdiffw  s5   € Ø�q‹=Ü—5‘5œ4 §	¡	¨!¡Ó-¨qÑ0Ñ0Ð0ä$ TÓ4Ð4rC   c                 ó   • [         $ rt   ©r…   rn   s     r3   rw   Útanh.inverse}  ry   rC   c                 ó  • UR                   (       a¦  U[        R                  L a  [        R                  $ U[        R                  L a  [        R                  $ U[        R
                  L a  [        R                  $ UR                  (       a  [        R                  $ UR                  (       a
  U " U* 5      * $ g U[        R                  L a  [        R                  $ [        U5      nUb;  UR                  5       (       a  [        * [        U* 5      -  $ [        [        U5      -  $ UR                  5       (       a
  U " U* 5      * $ UR                  (       aV  [!        U5      u  p4U(       aB  [#        U[$        -  [        -  5      nU[        R                  L a  ['        U5      $ [#        U5      $ UR                  (       a  [        R                  $ UR(                  [*        :X  a#  UR,                  S   nU[/        SUS-  -   5      -  $ UR(                  [0        :X  a/  UR,                  S   n[/        US-
  5      [/        US-   5      -  U-  $ UR(                  [2        :X  a  UR,                  S   $ UR(                  [4        :X  a  SUR,                  S   -  $ g r{   )r|   r   r}   rM   r[   rN   ÚNegativeOner~   r_   r   r€   r)   r�   r   r(   r‚   rg   rÛ   r   râ   rƒ   rv   rm   r   r„   r…   r†   )r‡   r`   rˆ   r‰   rŠ   Útanhms         r3   r‹   Ú	tanh.evalƒ  sè  € à�=�=Ø”a—e‘eŠ|Ü—u‘u�ØœŸ
™
Ò"Ü—u‘u�Øœ×*Ñ*Ò*Ü—}‘}Ð$Ø——Ü—v‘v�Ø——Ù˜S˜D›	�zÐ!ð !ð ”a×'Ñ'Ò'Ü—u‘u�ä4°SÓ9ˆGàÑ"Ø×3Ñ3×5Ñ5Ü˜2¤ W H£Ñ-Ð-Üœ3˜w›<Ñ'Ð'à×/Ñ/×1Ñ1Ù  ›I˜:Ð%à�z�zÜ# CÓ(‘�ÞÜ  ¤2¡¤a¡›L�EØ¤× 1Ñ 1Ò1Ü# A›w˜ä# A›w˜à�{�{Ü—v‘v�à�x‰xœ5Ó Ø—H‘H˜Q‘K�Øœ˜a ! Q¡$™h›Ñ'Ð'à�x‰xœ5Ó Ø—H‘H˜Q‘K�Ü˜A ™E“{¤T¨!¨a©%£[Ñ0°1Ñ4Ð4à�x‰xœ5Ó Ø—x‘x ‘{Ð"à�x‰xœ5Ó Ø˜Ÿ™ !™‘}Ð$ð !rC   c                 óÂ   • U S:  d	  U S-  S:X  a  [         R                  $ [        U5      nSU S-   -  n[        U S-   5      n[	        U S-   5      nX3S-
  -  U-  U-  X-  -  $ ©Nr   r7   r6   )r   r_   r   r   r   )r‘   r‰   r’   rb   ÚBÚFs         r3   r“   Útanh.taylor_term¸  sk   € ð ˆq‹5�A˜‘E˜Q“JÜ—6‘6ˆMä˜“
ˆAà�A˜‘E‘
ˆAä˜!˜a™%Ó ˆAÜ˜!˜a™%Ó ˆAà˜!‘e‘9˜q‘= ‘? Q¡TÑ)Ð)rC   c                 óZ   • U R                  U R                  S   R                  5       5      $ r–   r—   r™   s    r3   rš   Útanh._eval_conjugateÇ  rœ   rC   c                 ó  • U R                   S   R                  (       aA  U(       a(  SUS'   U R                  " U40 UD6[        R                  4$ U [        R                  4$ U(       a1  U R                   S   R                  " U40 UD6R                  5       u  p4OU R                   S   R                  5       u  p4[        U5      S-  [        U5      S-  -   n[        U5      [        U5      -  U-  [        U5      [        U5      -  U-  4$ )Nr   Frž   r7   rŸ   )ro   r¤   r¥   r   r   Údenoms         r3   r¢   Útanh.as_real_imagÊ  sØ   € Ø�9‰9�Q‰<×(×(ÞØ#(��iÑ ØŸš DÑ2¨EÑ2´A·F±FÐ;Ð;àœaŸf™f�~Ð%ÞØ—Y‘Y˜q‘\×(Ò(¨Ñ7°Ñ7×DÑDÓF‰FˆB�à—Y‘Y˜q‘\×.Ñ.Ó0‰FˆBÜ�R“˜!‘œc "›g q™jÑ(ˆÜ�R“œ˜b›Ñ! %Ñ'¬¨R«´°R³©¸Ñ)>Ð?Ð?rC   c                 ó  • U R                   S   nUR                  (       aƒ  [        UR                   5      nUR                    Vs/ s H  n[        USS9R	                  5       PM     nnSS/n[        US-   5       H  nXgS-  ==   [        Xu5      -  ss'   M     US   US   -  $ UR                  (       aµ  UR                  5       u  p‰UR                  (       a’  US:”  aŒ  [        U	5      n
[        SUS-   S5       Vs/ s H  n[        [        U5      U5      X«-  -  PM     nn[        SUS-   S5       Vs/ s H  n[        [        U5      U5      X«-  -  PM     nn[        U6 [        U6 -  $ [        U5      $ s  snf s  snf s  snf )Nr   Fr=  r6   r7   )rm   r‚   r�   rÛ   r»   Úranger*   r\   rµ   r¶   r   r   )ro   r¥   r`   r‘   r‰   ÚTXrd   Úir¹   rº   ÚTÚkÚds                r3   r»   Útanh._eval_expand_trigØ  s\  € Ø�i‰i˜‰lˆØ�:�:Ü�C—H‘H“ˆAàŸšó#Ú!�Aô �q 5Ñ)×;Ñ;Ö=Ù!ð ð #à�A�ˆAÜ˜1˜q™5–\�Ø�a‘%“œN¨1Ó1Ñ1•ñ "à�Q‘4˜˜!™‘9ÐØ�Z�ZØ×+Ñ+Ó-‰LˆEØ×× E¨A£IÜ˜“K�Ü7<¸QÀÈÁ	È1Ô7MÓNÒ7M°!”Rœ˜e› aÓ(¨©Ô-Ñ7M�ÐNÜ7<¸QÀÈÁ	È1Ô7MÓNÒ7M°!”Rœ˜e› aÓ(¨©Ô-Ñ7M�ÐNÜ˜A�wœs A˜w‘Ð&Ü�C‹yÐùò#ùò OùÚNs   Á"E5Ã;$E:Ä3$E?Nc                 ó@   • [        U* 5      [        U5      pTXT-
  XT-   -  $ r,   rÀ   ©ro   r`   rÂ   rÃ   Úneg_expÚpos_exps         r3   rÄ   Útanh._eval_rewrite_as_tractableë  ó$   € Ü ˜t›9¤c¨#£h�ØÑ! GÑ$5Ñ6Ð6rC   c                 ó@   • [        U* 5      [        U5      pCXC-
  XC-   -  $ r,   rÀ   ©ro   r`   rÃ   r‹  rŒ  s        r3   rÉ   Útanh._eval_rewrite_as_expï  rŽ  rC   c                 ó4   • [         * [        [         U-  SS9-  $ r<  )r   r(   rÈ   s      r3   Ú_eval_rewrite_as_tanÚtanh._eval_rewrite_as_tanó  ó   € Üˆr”Cœ˜C™¨%Ñ0Ñ0Ð0rC   c                 ó4   • [         * [        [         U-  SS9-  $ r<  )r   r$   rÈ   s      r3   Ú_eval_rewrite_as_cotÚtanh._eval_rewrite_as_cotö  r•  rC   c                 ó^   • [         [        U5      -  [        [        [         -  S-  U-
  SS9-  $ rJ  rK  rÈ   s      r3   rL  Útanh._eval_rewrite_as_sinhù  s(   € Ü”�c“‰{œ4¤¤1¡ Q¡¨¡°uÑ=Ñ=Ð=rC   c                 ó^   • [         [        [        [         -  S-  U-
  SS9-  [        U5      -  $ rJ  rÕ   rÈ   s      r3   rÖ   Útanh._eval_rewrite_as_coshü  s)   € Ü””bœ‘d˜1‘f˜s‘l¨UÑ3Ñ3´D¸³IÑ=Ð=rC   c                 ó   • S[        U5      -  $ rè   ©râ   rÈ   s      r3   rå   Útanh._eval_rewrite_as_cothÿ  ó   € Ø”�c“‰{ÐrC   c                 óÈ   • SSK Jn  U R                  S   R                  U5      nXR                  ;   a  U" SU5      R                  U5      (       a  U$ U R                  U5      $ ©Nr   )ÚOrderr6   ©Úsympy.series.orderr£  rm   rö   Úfree_symbolsÚcontainsrƒ   ©ro   r‰   rñ   rò   r£  r`   s         r3   rü   Útanh._eval_as_leading_term  sR   € Ý,Ø�i‰i˜‰l×*Ñ*¨1Ó-ˆà× Ñ Ó ¡U¨1¨a£[×%9Ñ%9¸#×%>Ñ%>ØˆJà—9‘9˜S“>Ð!rC   c                 óÊ   • U R                   S   nUR                  (       a  gUR                  5       u  p#US:X  a  U[        -  [        S-  :X  a  g U[        S-  -  R                  $ )Nr   Tr7   r   r  s       r3   r  Útanh._eval_is_real  sY   € Ø�i‰i˜‰lˆØ�;�;Øà×!Ñ!Ó#‰ˆð �‹7�rœB‘w¤" Q¡$“Øð ”b˜‘d‘×$Ñ$Ð$rC   c                 óB   • U R                   S   R                  (       a  gg rÿ   r  r™   s    r3   r  Útanh._eval_is_extended_real  r	  rC   c                 ór   • U R                   S   R                  (       a  U R                   S   R                  $ g r–   r  r™   s    r3   r  Útanh._eval_is_positive  r  rC   c                 ór   • U R                   S   R                  (       a  U R                   S   R                  $ g r–   r  r™   s    r3   r  Útanh._eval_is_negative"  r  rC   c                 óÖ   • U R                   S   nUR                  5       u  p#[        U5      S-  [        U5      S-  -   nUS:X  a  gUR                  (       a  gUR
                  (       a  gg )Nr   r7   FT)rm   r¢   r#   ri   Ú	is_numberr    )ro   r`   r   r   r  s        r3   r  Útanh._eval_is_finite&  s^   € Ø�i‰i˜‰lˆà×!Ñ!Ó#‰ˆÜ�B“˜‘
œT "›X q™[Ñ(ˆØ�A‹:ØØ�_�_ØØ××Øð  rC   c                 óF   • U R                   S   nUR                  (       a  gg rÿ   ©rm   r~   r  s     r3   r  Útanh._eval_is_zero2  s   € Ø�i‰i˜‰lˆØ�;�;Øð rC   rB   r!  r"  r,   )rR   rS   rT   rU   rV   rq   rw   r#  r‹   r$  r   r“   rš   r¢   r»   rÄ   rÉ   r“  r—  rL  rÖ   rå   rü   r  r  r  r  r  r  rX   rB   rC   r3   rÛ   rÛ   c  s˜   † ñô&5ôð ñ2%ó ð2%ðh Øñ*ó ó ð*ò3ô@òô&7ò7ò1ò1ò>ò>òò"ò%òò,ò,ò
õrC   rÛ   c                   ó    • \ rS rSrSrSS jrSS jr\S 5       r\	\
S 5       5       rS rSS jrSS
 jrS rS rS rS rS rS rS rS rSrg	)râ   i8  zû
``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`   • US:X  a  S[        U R                  S   5      S-  -  $ [        X5      e)Nr6   r9   r   r7   r(  rn   s     r3   rq   Ú
coth.fdiffL  s1   € Ø�q‹=Ø”d˜4Ÿ9™9 Q™<Ó(¨!Ñ+Ñ+Ð+ä$ TÓ4Ð4rC   c                 ó   • [         $ rt   )r†   rn   s     r3   rw   Úcoth.inverseR  ry   rC   c                 ó  • UR                   (       a¦  U[        R                  L a  [        R                  $ U[        R                  L a  [        R                  $ U[        R
                  L a  [        R                  $ UR                  (       a  [        R                  $ UR                  (       a
  U " U* 5      * $ g U[        R                  L a  [        R                  $ [        U5      nUb;  UR                  5       (       a  [        [        U* 5      -  $ [        * [        U5      -  $ UR                  5       (       a
  U " U* 5      * $ UR                  (       aV  [        U5      u  p4U(       aB  [!        U["        -  [        -  5      nU[        R                  L a  [!        U5      $ [%        U5      $ UR                  (       a  [        R                  $ UR&                  [(        :X  a#  UR*                  S   n[-        SUS-  -   5      U-  $ UR&                  [.        :X  a/  UR*                  S   nU[-        US-
  5      [-        US-   5      -  -  $ UR&                  [0        :X  a  SUR*                  S   -  $ UR&                  [2        :X  a  UR*                  S   $ g r{   )r|   r   r}   rM   r[   rN   rt  r~   r€   r   r)   r�   r   r$   r‚   rg   râ   r   rÛ   rƒ   rv   rm   r   r„   r…   r†   )r‡   r`   rˆ   r‰   rŠ   Úcothms         r3   r‹   Ú	coth.evalX  sî  € à�=�=Ø”a—e‘eŠ|Ü—u‘u�ØœŸ
™
Ò"Ü—u‘u�Øœ×*Ñ*Ò*Ü—}‘}Ð$Ø——Ü×(Ñ(Ð(Ø——Ù˜S˜D›	�zÐ!ð !ð ”a×'Ñ'Ò'Ü—u‘u�ä4°SÓ9ˆGàÑ"Ø×3Ñ3×5Ñ5Üœs G 8›}Ñ,Ð,Ü�rœC ›LÑ(Ð(à×/Ñ/×1Ñ1Ù  ›I˜:Ð%à�z�zÜ# CÓ(‘�ÞÜ  ¤2¡¤a¡›L�EØ¤× 1Ñ 1Ò1Ü# A›w˜ä# A›w˜à�{�{Ü×(Ñ(Ð(à�x‰xœ5Ó Ø—H‘H˜Q‘K�Ü˜A  1¡™H“~ aÑ'Ð'à�x‰xœ5Ó Ø—H‘H˜Q‘K�Øœ$˜q 1™u›+¬¨Q°©U«Ñ3Ñ4Ð4à�x‰xœ5Ó Ø˜Ÿ™ !™‘}Ð$à�x‰xœ5Ó Ø—x‘x ‘{Ð"ð !rC   c                 óÜ   • U S:X  a  S[        U5      -  $ U S:  d	  U S-  S:X  a  [        R                  $ [        U5      n[        U S-   5      n[	        U S-   5      nSU S-   -  U-  U-  X-  -  $ r{   ©r   r   r_   r   r   ©r‘   r‰   r’   ry  rz  s        r3   r“   Úcoth.taylor_term�  sv   € ð �‹6Ø”w˜q“z‘>Ð!Ø�‹U�a˜!‘e˜q“jÜ—6‘6ˆMä˜“
ˆAä˜!˜a™%Ó ˆAÜ˜!˜a™%Ó ˆAà�q˜1‘u‘: ‘> !Ñ# a¡dÑ*Ð*rC   c                 óZ   • U R                  U R                  S   R                  5       5      $ r–   r—   r™   s    r3   rš   Úcoth._eval_conjugateœ  rœ   rC   c                 ó  • SSK JnJn  U R                  S   R                  (       aA  U(       a(  SUS'   U R
                  " U40 UD6[        R                  4$ U [        R                  4$ U(       a1  U R                  S   R
                  " U40 UD6R                  5       u  pVOU R                  S   R                  5       u  pV[        U5      S-  U" U5      S-  -   n[        U5      [        U5      -  U-  U" U5      * U" U5      -  U-  4$ )Nr   )r#   r'   Frž   r7   )r+  r#   r'   rm   r    r¡   r   r_   r¢   ri   rl   )ro   r¤   r¥   r#   r'   r   r   r  s           r3   r¢   Úcoth.as_real_imagŸ  sÞ   € ßGØ�9‰9�Q‰<×(×(ÞØ#(��iÑ ØŸš DÑ2¨EÑ2´A·F±FÐ;Ð;àœaŸf™f�~Ð%ÞØ—Y‘Y˜q‘\×(Ò(¨Ñ7°Ñ7×DÑDÓF‰FˆB�à—Y‘Y˜q‘\×.Ñ.Ó0‰FˆBÜ�R“˜!‘™c "›g q™jÑ(ˆÜ�R“œ˜b›Ñ! %Ñ'©#¨b«'¨±#°b³'Ñ)9¸%Ñ)?Ð@Ð@rC   Nc                 ó@   • [        U* 5      [        U5      pTXT-   XT-
  -  $ r,   rÀ   rŠ  s         r3   rÄ   Úcoth._eval_rewrite_as_tractable®  rŽ  rC   c                 ó@   • [        U* 5      [        U5      pCXC-   XC-
  -  $ r,   rÀ   r�  s        r3   rÉ   Úcoth._eval_rewrite_as_exp²  rŽ  rC   c                 ó`   • [         * [        [        [         -  S-  U-
  SS9-  [        U5      -  $ rJ  rK  rÈ   s      r3   rL  Úcoth._eval_rewrite_as_sinh¶  s+   € Üˆr”$”rœ!‘t˜A‘v ‘|¨eÑ4Ñ4´T¸#³YÑ>Ð>rC   c                 ó`   • [         * [        U5      -  [        [        [         -  S-  U-
  SS9-  $ rJ  rÕ   rÈ   s      r3   rÖ   Úcoth._eval_rewrite_as_cosh¹  s*   € Üˆr”$�s“)‰|œD¤¤A¡ a¡¨#¡¸Ñ>Ñ>Ð>rC   c                 ó   • S[        U5      -  $ rè   ©rÛ   rÈ   s      r3   rÞ   Úcoth._eval_rewrite_as_tanh¼  r   rC   c                 ór   • U R                   S   R                  (       a  U R                   S   R                  $ g r–   r  r™   s    r3   r  Úcoth._eval_is_positive¿  r  rC   c                 ór   • U R                   S   R                  (       a  U R                   S   R                  $ g r–   r  r™   s    r3   r  Úcoth._eval_is_negativeÃ  r  rC   c                 óÎ   • SSK Jn  U R                  S   R                  U5      nXR                  ;   a"  U" SU5      R                  U5      (       a  SU-  $ U R                  U5      $ r¢  r¤  r¨  s         r3   rü   Úcoth._eval_as_leading_termÇ  sV   € Ý,Ø�i‰i˜‰l×*Ñ*¨1Ó-ˆà× Ñ Ó ¡U¨1¨a£[×%9Ñ%9¸#×%>Ñ%>Ø�S‘5ˆLà—9‘9˜S“>Ð!rC   c                 óÄ  • U R                   S   nUR                  (       a˜  UR                    Vs/ s H  n[        USS9R                  5       PM     nn/ / /n[	        UR                   5      n[        USS5       H%  nXVU-
  S-     R                  [        Xt5      5        M'     [        US   6 [        US   6 -  $ UR                  (       aˆ  UR                  SS9u  pƒUR                  (       af  US:”  a`  [        USS9n	/ / /n[        USS5       H*  nXXU-
  S-     R                  [        X‡5      X—-  -  5        M,     [        US   6 [        US   6 -  $ [        U5      $ s  snf )	Nr   Fr=  r9   r7   r6   Tr²   )rm   r‚   râ   r»   r�   r‚  Úappendr*   r   r\   rµ   r¶   r   )
ro   r¥   r`   r‰   ÚCXrd   r‘   r„  r¹   Úcs
             r3   r»   Úcoth._eval_expand_trigÐ  sB  € Ø�i‰i˜‰lˆØ�:�:ØGJÇxÂxÓPÂxÀ!”$�q 5Ñ)×;Ñ;Ö=ÁxˆBÐPØ�R�ˆAÜ�C—H‘H“ˆAÜ˜1˜b "Ö%�Ø�q‘5˜A‘+‘×%Ñ%¤n°QÓ&;Ö<ñ &ä˜˜!™�:œc 1 Q¡4˜jÑ(Ð(Ø�Z�ZØ×'Ñ'°Ð'Ð6‰HˆEØ×× E¨A£IÜ˜ UÑ+�Ø˜�H�Ü˜u b¨"Ö-�AØ˜q‘y A‘oÑ&×-Ñ-¬h°uÓ.@ÀÁÑ.EÖFñ .ä˜A˜a™D�z¤# q¨¡t *Ñ,Ð,Ü�C‹yÐùò Qs   ¯"ErB   r!  r"  r,   )rR   rS   rT   rU   rV   rq   rw   r#  r‹   r$  r   r“   rš   r¢   rÄ   rÉ   rL  rÖ   rÞ   r  r  rü   r»   rX   rB   rC   r3   râ   râ   8  sz   † ñô&5ôð ñ2#ó ð2#ðh Øñ+ó ó ð+ò3ôAô7ò7ò?ò?òò,ò,ò"õrC   râ   c                   óª   • \ rS rSr% SrSrSr\\S'   Sr	\\S'   \
S 5       rS rS rS	 rS
 rSS jrS rS rSS jrS rSS jrS rS rS rS rSrg)ÚReciprocalHyperbolicFunctioniä  z=Base class for reciprocal functions of hyperbolic functions. NÚ_is_evenÚ_is_oddc                 óH  • UR                  5       (       a5  U R                  (       a	  U " U* 5      $ U R                  (       a
  U " U* 5      * $ U R                  R	                  U5      n[        US5      (       a#  UR                  5       U :X  a  UR                  S   $ Ub  SU-  $ U$ )Nrw   r   r6   )r�   rà  rá  Ú_reciprocal_ofr‹   Úhasattrrw   rm   )r‡   r`   Úts      r3   r‹   Ú!ReciprocalHyperbolicFunction.evalì  sŠ   € à×'Ñ'×)Ñ)Ø�|�|Ù˜C˜4“yÐ Ø�{�{Ù˜S˜D›	�zÐ!à×Ñ×#Ñ# CÓ(ˆÜ�3˜	×"Ñ" s§{¡{£}¸Ó';Ø—8‘8˜A‘;ÐØ‘mˆq�‰sÐ*¨Ð*rC   c                 ó`   • U R                  U R                  S   5      n[        XA5      " U0 UD6$ r–   )rã  rm   Úgetattr)ro   Úmethod_namerm   rÃ   Úos        r3   Ú_call_reciprocalÚ-ReciprocalHyperbolicFunction._call_reciprocalù  s/   € à×Ñ §	¡	¨!¡Ó-ˆÜ�qÔ&¨Ð7°Ñ7Ð7rC   c                 óB   • U R                   " U/UQ70 UD6nUb  SU-  $ U$ rè   )rë  )ro   ré  rm   rÃ   rå  s        r3   Ú_calculate_reciprocalÚ2ReciprocalHyperbolicFunction._calculate_reciprocalþ  s1   € ð ×!Ò! +Ð?°Ò?¸Ñ?ˆØ‘mˆq�‰sÐ*¨Ð*rC   c                 ó`   • U R                  X5      nUb  X0R                  U5      :w  a  SU-  $ g g rè   )rë  rã  )ro   ré  r`   rå  s       r3   Ú_rewrite_reciprocalÚ0ReciprocalHyperbolicFunction._rewrite_reciprocal  s9   € ð ×!Ñ! +Ó3ˆØ‰=˜Q×"5Ñ"5°cÓ":Ó:Ø�Q‘3ˆJð ;ˆ=rC   c                 ó&   • U R                  SU5      $ )NrÉ   ©rñ  rÈ   s      r3   rÉ   Ú1ReciprocalHyperbolicFunction._eval_rewrite_as_exp  s   € Ø×'Ñ'Ð(>ÀÓDÐDrC   c                 ó&   • U R                  SU5      $ )NrÄ   rô  rÁ   s       r3   rÄ   Ú7ReciprocalHyperbolicFunction._eval_rewrite_as_tractable  s   € Ø×'Ñ'Ð(DÀcÓJÐJrC   c                 ó&   • U R                  SU5      $ )NrÞ   rô  rÈ   s      r3   rÞ   Ú2ReciprocalHyperbolicFunction._eval_rewrite_as_tanh  ó   € Ø×'Ñ'Ð(?ÀÓEÐErC   c                 ó&   • U R                  SU5      $ )Nrå   rô  rÈ   s      r3   rå   Ú2ReciprocalHyperbolicFunction._eval_rewrite_as_coth  rú  rC   c                 óf   • SU R                  U R                  S   5      -  R                  " U40 UD6$ r'  )rã  rm   r¢   )ro   r¤   r¥   s      r3   r¢   Ú)ReciprocalHyperbolicFunction.as_real_imag  s0   € Ø�D×'Ñ'¨¯	©	°!©Ó5Ñ5×CÒCÀDÑRÈEÑRÐRrC   c                 óZ   • U R                  U R                  S   R                  5       5      $ r–   r—   r™   s    r3   rš   Ú,ReciprocalHyperbolicFunction._eval_conjugate  rœ   rC   c                 óF   • U R                   " SSS0UD6u  p4U[        U-  -   $ )Nr¤   TrB   r©   rª   s        r3   r­   Ú1ReciprocalHyperbolicFunction._eval_expand_complex  s,   € Ø×,Ò,Ñ@°$Ð@¸%Ñ@ÑˆØœ˜7™Ñ"Ð"rC   c                 ó&   • U R                   " S0 UD6$ )N)r»   )rî  )ro   r¥   s     r3   r»   Ú.ReciprocalHyperbolicFunction._eval_expand_trig!  s   € Ø×)Ò)ÑGÀÑGÐGrC   c                 ó`   • SU R                  U R                  S   5      -  R                  XUS9$ )Nr6   r   rð   )rã  rm   rü   )ro   r‰   rñ   rò   s       r3   rü   Ú2ReciprocalHyperbolicFunction._eval_as_leading_term$  s1   € Ø�$×%Ñ% d§i¡i°¡lÓ3Ñ3×JÑJÈ1Ð^bÐJÐcÐcrC   c                 óR   • U R                  U R                  S   5      R                  $ r–   )rã  rm   r    r™   s    r3   r  Ú3ReciprocalHyperbolicFunction._eval_is_extended_real'  s!   € Ø×"Ñ" 4§9¡9¨Q¡<Ó0×AÑAÐArC   c                 óX   • SU R                  U R                  S   5      -  R                  $ r'  )rã  rm   rù   r™   s    r3   r  Ú,ReciprocalHyperbolicFunction._eval_is_finite*  s&   € Ø�$×%Ñ% d§i¡i°¡lÓ3Ñ3×>Ñ>Ð>rC   rB   r,   r"  )rR   rS   rT   rU   rV   rã  rà  r   Ú__annotations__rá  r#  r‹   rë  rî  rñ  rÉ   rÄ   rÞ   rå   r¢   rš   r­   r»   rü   r  r  rX   rB   rC   r3   rß  rß  ä  s€   ‡ ÙGð €NØ€HˆiÓØ€GˆYÓàñ
+ó ð
+ò8ò
+òòEôKòFòFôSò3ô#òHòdòBõ?rC   rß  c                   óh   • \ rS rSrSr\rSrSS jr\	\
S 5       5       rS rS rS rS	 rS
 rS rSrg)rê   i.  a  
``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   • US:X  a2  [        U R                  S   5      * [        U R                  S   5      -  $ [        X5      e)z/
Returns the first derivative of this function
r6   r   )râ   rm   rê   r   rn   s     r3   rq   Ú
csch.fdiffE  s>   € ð �q‹=Ü˜Ÿ™ 1™Ó&Ð&¬¨d¯i©i¸©lÓ);Ñ;Ð;ä$ TÓ4Ð4rC   c                 óâ   • U S:X  a  S[        U5      -  $ U S:  d	  U S-  S:X  a  [        R                  $ [        U5      n[        U S-   5      n[	        U S-   5      nSSSU -  -
  -  U-  U-  X-  -  $ )z6
Returns the next term in the Taylor series expansion
r   r6   r7   rÁ  rÂ  s        r3   r“   Úcsch.taylor_termN  s{   € ð �‹6Ø”W˜Q“Z‘<ÐØ�‹U�a˜!‘e˜q“jÜ—6‘6ˆMä˜“
ˆAä˜!˜a™%Ó ˆAÜ˜!˜a™%Ó ˆAà˜˜A˜q™D™‘> AÑ% aÑ'¨!©$Ñ.Ð.rC   c                 ó2   • [         [        [         U-  SS9-  $ r<  rÌ   rÈ   s      r3   rÍ   Úcsch._eval_rewrite_as_sin`  ó   € Ü”3”q˜3‘w¨Ñ/Ñ/Ð/rC   c                 ó2   • [         [        [         U-  SS9-  $ r<  rÑ   rÈ   s      r3   rÒ   Úcsch._eval_rewrite_as_cscc  r  rC   c                 óF   • [         [        U[         [        -  S-  -   SS9-  $ rJ  rÕ   rÈ   s      r3   rÖ   Úcsch._eval_rewrite_as_coshf  s!   € Ü”4˜œa¤"™f q™jÑ(°5Ñ9Ñ9Ð9rC   c                 ó   • S[        U5      -  $ rè   ©ri   rÈ   s      r3   rL  Úcsch._eval_rewrite_as_sinhi  rí   rC   c                 ór   • U R                   S   R                  (       a  U R                   S   R                  $ g r–   r  r™   s    r3   r  Úcsch._eval_is_positivel  r  rC   c                 ór   • U R                   S   R                  (       a  U R                   S   R                  $ g r–   r  r™   s    r3   r  Úcsch._eval_is_negativep  r  rC   rB   Nr!  )rR   rS   rT   rU   rV   ri   rã  rá  rq   r$  r   r“   rÍ   rÒ   rÖ   rL  r  r  rX   rB   rC   r3   rê   rê   .  sR   † ñð& €NØ€Gô5ð Øñ/ó ó ð/ò 0ò0ò:òò,õ,rC   rê   c                   ób   • \ rS rSrSr\rSrSS jr\	\
S 5       5       rS rS rS rS	 rS
 rSrg)rT  iu  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   • US:X  a2  [        U R                  S   5      * [        U R                  S   5      -  $ [        X5      er'  )rÛ   rm   rT  r   rn   s     r3   rq   Ú
sech.fdiffŒ  s<   € Ø�q‹=Ü˜$Ÿ)™) A™,Ó'Ð'¬¨T¯Y©Y°q©\Ó(:Ñ:Ð:ä$ TÓ4Ð4rC   c                 óŽ   • U S:  d	  U S-  S:X  a  [         R                  $ [        U5      n[        U 5      [	        U 5      -  X-  -  $ rx  )r   r_   r   r   r   ©r‘   r‰   r’   s      r3   r“   Úsech.taylor_term’  sA   € ð ˆq‹5�A˜‘E˜Q“JÜ—6‘6ˆMä˜“
ˆAÜ˜“8œi¨›lÑ*¨Q©VÑ3Ð3rC   c                 ó*   • S[        [        U-  SS9-  $ rD  r?  rÈ   s      r3   r@  Úsech._eval_rewrite_as_cos›  rH  rC   c                 ó$   • [        [        U-  SS9$ r<  rE  rÈ   s      r3   rF  Úsech._eval_rewrite_as_secž  rB  rC   c                 óF   • [         [        U[         [        -  S-  -   SS9-  $ rJ  rK  rÈ   s      r3   rL  Úsech._eval_rewrite_as_sinh¡  s    € Ü”4˜œa¤"™f a™i™°%Ñ8Ñ8Ð8rC   c                 ó   • S[        U5      -  $ rè   ©rl   rÈ   s      r3   rÖ   Úsech._eval_rewrite_as_cosh¤  rí   rC   c                 óB   • U R                   S   R                  (       a  gg rÿ   r  r™   s    r3   r  Úsech._eval_is_positive§  r	  rC   rB   Nr!  )rR   rS   rT   rU   rV   rl   rã  rà  rq   r$  r   r“   r@  rF  rL  rÖ   r  rX   rB   rC   r3   rT  rT  u  sM   † ñð& €NØ€Hô5ð Øñ4ó ó ð4ò0ò,ò9òõrC   rT  c                   ó   • \ rS rSrSrSrg)ÚInverseHyperbolicFunctioni°  z,Base class for inverse hyperbolic functions.rB   N)rR   rS   rT   rU   rV   rX   rB   rC   r3   r1  r1  °  s   † Ù6ârC   r1  c                   ó¨   ^ • \ rS rSrSrSS jr\S 5       r\\	S 5       5       r
S rSU 4S jjrS r\rS	 rS
 rS rS rSS jrS rS rS rSrU =r$ )rv   i¶  a  
``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   • US:X  a!  S[        U R                  S   S-  S-   5      -  $ [        X5      ern  )r   rm   r   rn   s     r3   rq   Úasinh.fdiffÌ  s5   € Ø�q‹=Ø”T˜$Ÿ)™) A™,¨™/¨AÑ-Ó.Ñ.Ð.ä$ TÓ4Ð4rC   c                 óŒ  • UR                   (       aú  U[        R                  L a  [        R                  $ U[        R                  L a  [        R                  $ U[        R                  L a  [        R                  $ UR
                  (       a  [        R                  $ U[        R                  L a  [        [        S5      S-   5      $ U[        R                  L a  [        [        S5      S-
  5      $ UR                  (       a
  U " U* 5      * $ OƒU[        R                  L a  [        R                  $ UR
                  (       a  [        R                  $ [        U5      nUb  [        [        U5      -  $ UR!                  5       (       a
  U " U* 5      * $ [#        U[$        5      (       a¡  UR&                  S   R(                  (       a‚  UR&                  S   nUR*                  (       a  U$ [-        U5      u  pEUbO  UbK  [/        U[0        S-  -   [0        -  5      nU[        [0        -  U-  -
  nUR2                  nUSL a  U$ USL a  U* $ g g g g g )Nr7   r6   r   TF)r|   r   r}   rM   rN   r~   r_   r[   r   r   rt  r   r€   r)   r   r!   r�   Ú
isinstanceri   rm   r³  r  r   r   r   Úis_even)	r‡   r`   rˆ   r`  Úrr„  ÚfrŠ   Úevens	            r3   r‹   Ú
asinh.evalÒ  sÅ  € à�=�=Ø”a—e‘eŠ|Ü—u‘u�ØœŸ
™
Ò"Ü—z‘zÐ!Øœ×*Ñ*Ò*Ü×)Ñ)Ð)Ø——Ü—v‘v�ØœŸ™’Üœ4 ›7 Q™;Ó'Ð'ØœŸ™Ò%Üœ4 ›7 Q™;Ó'Ð'Ø——Ù˜S˜D›	�zÐ!ð !ð ”a×'Ñ'Ò'Ü×(Ñ(Ð(à�{�{Ü—v‘v�ä4°SÓ9ˆGàÑ"Üœ4 ›=Ñ(Ð(à×/Ñ/×1Ñ1Ù  ›I˜:Ð%ä�cœ4× Ñ  S§X¡X¨a¡[×%:×%:Ø—‘˜‘ˆAØ�y�yØ�Ü" 1Ó%‰DˆAØ‰} ¡Ü˜1œr !™t™8¤R™-Ó(�Øœœ"™˜Q™‘J�Ø—y‘y�Ø˜4’<Ø�HØ˜U’]Ø˜2�Ið #ð "/ˆ}ð &;Ð rC   c                 óZ  • U S:  d	  U S-  S:X  a  [         R                  $ [        U5      n[        U5      S:¼  a%  U S:”  a  US   nU* U S-
  S-  -  X S-
  -  -  US-  -  $ U S-
  S-  n[	        [         R
                  U5      n[        U5      n[         R                  U-  U-  U-  X-  -  U -  $ ©Nr   r7   rH   r6   )r   r_   r   r�   r   rA   r   rt  ©r‘   r‰   r’   rd   r†  ÚRrz  s          r3   r“   Úasinh.taylor_term   s·   € ð ˆq‹5�A˜‘E˜Q“JÜ—6‘6ˆMä˜“
ˆAÜ�>Ó" aÓ'¨A°«EØ" 2Ñ&�Ø�r˜Q ™U Q™J‘¨¨q©5©	Ñ2°Q¸±TÑ9Ð9à˜‘U˜q‘L�Ü#¤A§F¡F¨AÓ.�Ü˜a“L�Ü—}‘} aÑ'¨!Ñ+¨aÑ/°!±$Ñ6¸Ñ:Ð:rC   c                 óØ  • U R                   S   nUR                  US5      R                  5       nUR                  (       a  UR	                  U5      $ U[
        R                  L a5  U R                  UR	                  U5      5      nUR                  (       a  U$ U $ U[        * [        [
        R                  4;   a#  U R                  [        5      R                  XUS9$ SUS-  -   R                  (       aç  UR                  X(       a  UOS5      n[!        U5      R"                  (       a;  [%        U5      R                  (       a   U R                  U5      * [        [&        -  -
  $ Ox[!        U5      R                  (       a;  [%        U5      R"                  (       a   U R                  U5      * [        [&        -  -   $ O#U R                  [        5      R                  XUS9$ U R                  U5      $ ©Nr   rð   r6   r7   )rm   r÷   Úcancelr~   rö   r   r}   rƒ   rù   r   r€   r0   r   rü   r   rõ   r   r  r   r   ©ro   r‰   rñ   rò   r`   Úx0r1   Úndirs           r3   rü   Úasinh._eval_as_leading_term  sq  € Ø�i‰i˜‰lˆØ�X‰X�a˜‹^×"Ñ"Ó$ˆØ�:�:Ø×&Ñ& qÓ)Ð)à”—‘Š;Ø—9‘9˜S×0Ñ0°Ó3Ó4ˆDØ�~�~Ø�à�ð ”1�"”aœ×*Ñ*Ð+Ó+Ø—<‘<¤Ó$×:Ñ:¸1ÈdÐ:ÐSÐSà��A‘‰I×"×"Ø—7‘7˜1¥d™d°Ó2ˆDÜ�$‹x×#×#Ü�b“6×%×%Ø ŸI™I b›M˜>¬A¬b©DÑ0Ð0ð &ä�D“×%×%Ü�b“6×%×%Ø ŸI™I b›M˜>¬A¬b©DÑ0Ð0ð &ð —|‘|¤CÓ(×>Ñ>¸qÐRVÐ>ÐWÐWØ�y‰y˜‹}ÐrC   c                 óÀ  >• U R                   S   nUR                  US5      nU[        [        * 4;   a#  U R                  [        5      R                  XX4S9$ [        T	U ]  XUS9nU[        R                  L a  U$ SUS-  -   R                  (       aÍ  UR                  X(       a  UOS5      n[        U5      R                  (       a.  [        U5      R                  (       a  U* [        [        -  -
  $  U$ [        U5      R                  (       a.  [        U5      R                  (       a  U* [        [        -  -   $  U$ U R                  [        5      R                  XX4S9$ U$ ©Nr   rð   ©r‘   rñ   r6   r7   )rm   r÷   r   r0   r   Ú_eval_nseriesÚsuperr   r€   r   rõ   r   r  r   r   ©
ro   r‰   r‘   rñ   rò   r`   rû   ÚresrF  Ú	__class__s
            €r3   rK  Úasinh._eval_nseries-  s/  ø€ Ø�i‰i˜‰lˆØ�x‰x˜˜1‹~ˆð ”Aœ�r�7‹?Ø—<‘<¤Ó$×2Ñ2°1¸dÐ2ÐNÐNä‰gÑ# A°Ð#Ð6ˆØ”1×$Ñ$Ò$ØˆJð ��a‘‰K×$×$Ø—7‘7˜1¥d™d°Ó2ˆDÜ�$‹x×#×#Ü�d“8×'×'Ø˜4¤!¤B¡$™;Ð&ð (ð ˆ
ô �D“×%×%Ü�d“8×'×'Ø˜4¤!¤B¡$™;Ð&ð (ð ˆ
ð —|‘|¤CÓ(×6Ñ6°qÀ$Ð6ÐRÐRØˆ
rC   c                 ó<   • [        U[        US-  S-   5      -   5      $ rÙ   ©r   r   ©ro   r‰   rÃ   s      r3   Ú_eval_rewrite_as_logÚasinh._eval_rewrite_as_logF  s   € Ü�1”t˜A˜q™D 1™H“~Ñ%Ó&Ð&rC   c                 ó<   • [        U[        SUS-  -   5      -  5      $ ©Nr6   r7   )r…   r   rS  s      r3   Ú_eval_rewrite_as_atanhÚasinh._eval_rewrite_as_atanhK  s   € Ü�Q”t˜A  1¡™H“~Ñ%Ó&Ð&rC   c                 óˆ   • [         U-  n[         [        SU-
  5      [        US-
  5      -  [        U5      -  [        S-  -
  -  $ rW  )r   r   r„   r   )ro   r‰   rÃ   Úixs       r3   Ú_eval_rewrite_as_acoshÚ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                 ó4   • [         * [        [         U-  SS9-  $ r<  )r   r!   rS  s      r3   Ú_eval_rewrite_as_asinÚasinh._eval_rewrite_as_asinR  s   € Üˆr”Dœ˜Q™¨Ñ/Ñ/Ð/rC   c                 óT   • [         [        [         U-  SS9-  [         [        -  S-  -
  $ )NFr=  r7   )r   r   r   rS  s      r3   Ú_eval_rewrite_as_acosÚasinh._eval_rewrite_as_acosU  s%   € Ü”4œ˜A™¨Ñ.Ñ.´´2±°a±Ñ7Ð7rC   c                 ó   • [         $ rt   r  rn   s     r3   rw   Úasinh.inverseX  ó	   € ô ˆrC   c                 ó4   • U R                   S   R                  $ r–   r¶  r™   s    r3   r  Úasinh._eval_is_zero^  s   € Ø�y‰y˜‰|×#Ñ#Ð#rC   c                 ó4   • U R                   S   R                  $ r–   r  r™   s    r3   r  Úasinh._eval_is_extended_reala  ó   € Ø�y‰y˜‰|×,Ñ,Ð,rC   c                 ó4   • U R                   S   R                  $ r–   r  r™   s    r3   r  Úasinh._eval_is_finited  ó   € Ø�y‰y˜‰|×%Ñ%Ð%rC   rB   r!  ©r   )rR   rS   rT   rU   rV   rq   r#  r‹   r$  r   r“   rü   rK  rT  rÄ   rX  r\  r_  rb  rw   r  r  r  rX   Ú__classcell__©rO  s   @r3   rv   rv   ¶  sƒ   ø† ñô*5ð ñ+ó ð+ðZ Øñ;ó ó ð;ò÷:ò2'ð "6Ðò'ò@ò0ò8ôò$ò-÷&ð &rC   rv   c                   ó¨   ^ • \ rS rSrSrSS jr\S 5       r\\	S 5       5       r
S rSU 4S jjrS r\rS	 rS
 rS rS rSS jrS rS rS rSrU =r$ )r„   ih  a  
``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‚   • US:X  a/  U R                   S   nS[        US-
  5      [        US-   5      -  -  $ [        X5      er'  ©rm   r   r   )ro   rp   r`   s      r3   rq   Úacosh.fdiff~  sA   € Ø�q‹=Ø—)‘)˜A‘,ˆCØ”d˜3 ™7“m¤D¨¨q©£MÑ1Ñ2Ð2ä$ TÓ4Ð4rC   c                 óö  • UR                   (       aÍ  U[        R                  L a  [        R                  $ U[        R                  L a  [        R                  $ U[        R                  L a  [        R                  $ UR
                  (       a  [        [        -  S-  $ U[        R                  L a  [        R                  $ U[        R                  L a  [        [        -  $ UR                  (       a/  [        5       nX;   a   UR                  (       a  X!   [        -  $ X!   $ U[        R                  L a  [        R                  $ U[        [        R                  -  :X  a!  [        R                  [        [        -  S-  -   $ U[        * [        R                  -  :X  a!  [        R                  [        [        -  S-  -
  $ UR
                  (       a  [        [        -  [        R                  -  $ [!        U["        5      (       aû  UR$                  S   R                  (       aÜ  UR$                  S   nUR&                  (       a  [)        U5      $ [+        U5      u  pEUb   Ubœ  [-        U[        -  5      nU[        [        -  U-  -
  nUR.                  nUSL a(  UR0                  (       a  U$ UR2                  (       a  U* $ g USL a8  U[        [        -  -  nUR4                  (       a  U* $ UR6                  (       a  U$ g g g g g g )Nr7   r   TF)r|   r   r}   rM   rN   r~   r   r   r[   r_   rt  r³  rD   r    r€   rA   r6  rl   rm   r  r   r   r   r7  Úis_nonnegativer   Úis_nonpositiver  )	r‡   r`   Ú	cst_tabler`  r8  r„  r9  rŠ   r:  s	            r3   r‹   Ú
acosh.eval…  s$  € à�=�=Ø”a—e‘eŠ|Ü—u‘u�ØœŸ
™
Ò"Ü—z‘zÐ!Øœ×*Ñ*Ò*Ü—z‘zÐ!Ø——Üœ!‘t˜a‘x�ØœŸ™’Ü—v‘v�ØœŸ™Ò%Üœ!‘t�à�=�=Ü$›ˆIàÓØ×'×'Ø$™>¬!Ñ+Ð+Ø ‘~Ð%à”!×#Ñ#Ò#Ü×$Ñ$Ð$Ø”!”A—J‘J‘,ÓÜ—:‘:¤¤"¡ Q¡Ñ&Ð&Ø”1�"”Q—Z‘Z‘-ÓÜ—:‘:¤¤"¡ Q¡Ñ&Ð&à�;�;Ü”a‘4œŸ™‘;Ðä�cœ4× Ñ  S§X¡X¨a¡[×%:×%:Ø—‘˜‘ˆAØ�y�yÜ˜1“v�Ü" 1Ó%‰DˆAØ‰} ¡Ü˜!œB™$“K�Øœœ"™˜Q™‘J�Ø—y‘y�Ø˜4’<Ø×'×'Ø ˜ØŸŸØ !˜r˜	ð 'à˜U’]Øœœ2™‘I�AØ×'×'Ø !˜r˜	ØŸŸØ ˜ð 'ð	 #ð "/ˆ}ð &;Ð rC   c                 ój  • U S:X  a  [         [        -  S-  $ U S:  d	  U S-  S:X  a  [        R                  $ [	        U5      n[        U5      S:¼  a#  U S:”  a  US   nX0S-
  S-  -  X S-
  -  -  US-  -  $ U S-
  S-  n[        [        R                  U5      n[        U5      nU* U-  [         -  X-  -  U -  $ r=  )	r   r   r   r_   r   r�   r   rA   r   r>  s          r3   r“   Úacosh.taylor_term¼  s¾   € ð �‹6Ü”R‘4˜‘6ˆMØ�‹U�a˜!‘e˜q“jÜ—6‘6ˆMä˜“
ˆAÜ�>Ó" aÓ'¨A°«EØ" 2Ñ&�Ø ™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¡DÑ(¨1Ñ,Ð,rC   c                 óx  • U R                   S   nUR                  US5      R                  5       nU[        R                  * [        R
                  [        R                  [        R                  4;   a#  U R                  [        5      R                  XUS9$ U[        R                  L a5  U R                  UR                  U5      5      nUR                  (       a  U$ U $ US-
  R                  (       a¹  UR                  X(       a  UOS5      n[!        U5      R                  (       aH  US-   R                  (       a"  U R                  U5      S["        -  [$        -  -
  $ U R                  U5      * $ [!        U5      R&                  (       d#  U R                  [        5      R                  XUS9$ U R                  U5      $ rB  )rm   r÷   rC  r   r[   r_   r€   r0   r   rü   r}   rƒ   rö   rù   r   rõ   r   r   r   r  rD  s           r3   rü   Úacosh._eval_as_leading_termÎ  sC  € Ø�i‰i˜‰lˆØ�X‰X�a˜‹^×"Ñ"Ó$ˆà”1—5‘5�&œ!Ÿ&™&¤!§%¡%¬×):Ñ):Ð;Ó;Ø—<‘<¤Ó$×:Ñ:¸1ÈdÐ:ÐSÐSà”—‘Š;Ø—9‘9˜S×0Ñ0°Ó3Ó4ˆDØ�~�~Ø�à�ð �‰F××Ø—7‘7˜1¥d™d°Ó2ˆDÜ�$‹x×#×#Ø˜‘F×'×'ØŸ9™9 R›=¨1¬Q©3¬r©6Ñ1Ð1ØŸ	™	 "›�~Ð%Ü˜“X×)×)Ø—|‘|¤CÓ(×>Ñ>¸qÐRVÐ>ÐWÐWØ�y‰y˜‹}ÐrC   c                 ó|  >• U R                   S   nUR                  US5      nU[        R                  [        R                  4;   a#  U R                  [        5      R                  XX4S9$ [        T	U ]  XUS9nU[        R                  L a  U$ US-
  R                  (       a›  UR                  X(       a  UOS5      n[        U5      R                  (       a*  US-   R                  (       a  US[        -  [        -  -
  $ U* $ [        U5      R                  (       d#  U R                  [        5      R                  XX4S9$ U$ rI  ©rm   r÷   r   r[   rt  r0   r   rK  rL  r€   r   rõ   r   r   r   r  rM  s
            €r3   rK  Úacosh._eval_nseriesç  s  ø€ Ø�i‰i˜‰lˆØ�x‰x˜˜1‹~ˆð ”A—E‘Eœ1Ÿ=™=Ð)Ó)Ø—<‘<¤Ó$×2Ñ2°1¸dÐ2ÐNÐNä‰gÑ# A°Ð#Ð6ˆØ”1×$Ñ$Ò$ØˆJð �1‰H×!×!Ø—7‘7˜1¥d™d°Ó2ˆDÜ�$‹x×#×#Ø˜1‘H×)×)Ø ¤1¡¤R¡™<Ð'Ø�t�Ü˜“X×)×)Ø—|‘|¤CÓ(×6Ñ6°qÀ$Ð6ÐRÐRØˆ
rC   c                 óT   • [        U[        US-   5      [        US-
  5      -  -   5      $ rè   rR  rS  s      r3   rT  Úacosh._eval_rewrite_as_logþ  s'   € Ü�1”t˜A ™E“{¤T¨!¨a©%£[Ñ0Ñ0Ó1Ð1rC   c                 óT   • [        US-
  5      [        SU-
  5      -  [        U5      -  $ rè   )r   r   rS  s      r3   rb  Úacosh._eval_rewrite_as_acos  s&   € Ü�A˜‘E‹{œ4  A¡›;Ñ&¬¨a«Ñ0Ð0rC   c                 óh   • [        US-
  5      [        SU-
  5      -  [        S-  [        U5      -
  -  $ rW  )r   r   r!   rS  s      r3   r_  Úacosh._eval_rewrite_as_asin  s.   € Ü�A˜‘E‹{œ4  A¡›;Ñ&¬"¨Q©$´°a³©.Ñ9Ð9rC   c                 ó‚   • [        US-
  5      [        SU-
  5      -  [        S-  [        [        [        U-  SS9-  -   -  $ ©Nr6   r7   Fr=  )r   r   r   rv   rS  s      r3   Ú_eval_rewrite_as_asinhÚacosh._eval_rewrite_as_asinh	  s;   € Ü�A˜‘E‹{œ4  A¡›;Ñ&¬"¨Q©$´´5¼¸1¹ÀuÑ3MÑ1MÑ*MÑNÐNrC   c                 óð   • [        US-
  5      n[        SU-
  5      n[        US-  S-
  5      n[        S-  U-  U-  SU[        SUS-  -  5      -  -
  -  U[        US-   5      -  U-  [        XQ-  5      -  -   $ rW  )r   r   r…   )ro   r‰   rÃ   Úsxm1Ús1mxÚsx2m1s         r3   rX  Úacosh._eval_rewrite_as_atanh  s�   € Ü�A˜‘E‹{ˆÜ�A˜‘E‹{ˆÜ�Q˜‘T˜A‘X“ˆÜ�1‘�T‘	˜$‘  A¬¨Q¨q°!©t©V«Ñ$4Ñ 4Ñ5Ø”T˜!˜a™%“[Ñ  Ñ&¬¨u©w«Ñ7ñ8ð 	9rC   c                 ó   • [         $ rt   r,  rn   s     r3   rw   Úacosh.inverse  rf  rC   c                 óH   • U R                   S   S-
  R                  (       a  gg )Nr   r6   Tr¶  r™   s    r3   r  Úacosh._eval_is_zero  s    € Ø�I‰I�a‰L˜1Ñ×%×%Øð &rC   c                 ó~   • [        U R                  S   R                  U R                  S   S-
  R                  /5      $ ©Nr   r6   )r
   rm   r    Úis_extended_nonnegativer™   s    r3   r  Úacosh._eval_is_extended_real  s3   € Ü˜$Ÿ)™) A™,×7Ñ7¸$¿)¹)ÀA¹,ÈÑ:J×9cÑ9cÐdÓeÐerC   c                 ó4   • U R                   S   R                  $ r–   r  r™   s    r3   r  Úacosh._eval_is_finite   rn  rC   rB   r!  ro  )rR   rS   rT   rU   rV   rq   r#  r‹   r$  r   r“   rü   rK  rT  rÄ   rb  r_  rŠ  rX  rw   r  r  r  rX   rp  rq  s   @r3   r„   r„   h  s„   ø† ñô*5ð ñ4!ó ð4!ðl Øñ-ó ó ð-ò ÷2ò.2ð "6Ðò1ò:òOò9ôòòf÷&ð &rC   r„   c                   óœ   ^ • \ rS rSrSrSS jr\S 5       r\\	S 5       5       r
S rSU 4S jjrS r\rS	 rS
 rS rS rS rSS jrSrU =r$ )r…   i$  zù
``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   • US:X  a  SSU R                   S   S-  -
  -  $ [        X5      ern  ©rm   r   rn   s     r3   rq   Úatanh.fdiff8  ó0   € Ø�q‹=Ø�a˜$Ÿ)™) A™,¨™/Ñ)Ñ*Ð*ä$ TÓ4Ð4rC   c                 ó¾  • UR                   (       aò  U[        R                  L a  [        R                  $ UR                  (       a  [        R                  $ U[        R
                  L a  [        R                  $ U[        R                  L a  [        R                  $ U[        R                  L a  [        * [        U5      -  $ U[        R                  L a  [        [        U* 5      -  $ UR                  (       a
  U " U* 5      * $ OwU[        R                  L a%  SSKJn  [        U" [        * S-  [        S-  5      -  $ [!        U5      nUb  [        [        U5      -  $ UR#                  5       (       a
  U " U* 5      * $ UR                  (       a  [        R                  $ [%        U[&        5      (       a­  UR(                  S   R*                  (       aŽ  UR(                  S   nUR,                  (       a  U$ [/        U5      u  pVUb[  UbW  [1        SU-  [        -  5      nUR2                  nU[        U-  [        -  S-  -
  n	USL a  U	$ USL a  U	[        [        -  S-  -
  $ g g g g g )Nr   ©ÚAccumBoundsr7   TF)r|   r   r}   r~   r_   r[   rM   rt  rN   r   r"   r   r€   Ú!sympy.calculus.accumulationboundsr¢  r   r)   r�   r6  rÛ   rm   r³  r  r   r   r7  )
r‡   r`   r¢  rˆ   r`  r8  r„  r9  r:  rŠ   s
             r3   r‹   Ú
atanh.eval>  sÜ  € à�=�=Ø”a—e‘eŠ|Ü—u‘u�Ø——Ü—v‘v�ØœŸ™’Ü—z‘zÐ!ØœŸ™Ò%Ü×)Ñ)Ð)ØœŸ
™
Ò"Ü�rœD ›I‘~Ð%Øœ×*Ñ*Ò*Üœ4  ›:‘~Ð%Ø——Ù˜S˜D›	�zÐ!ð !ð ”a×'Ñ'Ò'ÝIÜ™¤b S¨¡U¬B¨q©DÓ1Ñ1Ð1ä4°SÓ9ˆGàÑ"Üœ4 ›=Ñ(Ð(à×/Ñ/×1Ñ1Ù  ›I˜:Ð%à�;�;Ü—6‘6ˆMä�cœ4× Ñ  S§X¡X¨a¡[×%:×%:Ø—‘˜‘ˆAØ�y�yØ�Ü" 1Ó%‰DˆAØ‰} ¡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                 ód   • U S:  d	  U S-  S:X  a  [         R                  $ [        U5      nX-  U -  $ ©Nr   r7   )r   r_   r   r#  s      r3   r“   Úatanh.taylor_termm  s2   € ð ˆq‹5�A˜‘E˜Q“JÜ—6‘6ˆMä˜“
ˆAØ‘4˜!‘8ˆOrC   c                 óØ  • U R                   S   nUR                  US5      R                  5       nUR                  (       a  UR	                  U5      $ U[
        R                  L a5  U R                  UR	                  U5      5      nUR                  (       a  U$ U $ U[
        R                  * [
        R                  [
        R                  4;   a#  U R                  [        5      R                  XUS9$ SUS-  -
  R                  (       aÓ  UR                  X(       a  UOS5      n[!        U5      R                  (       a1  UR                  (       a  U R                  U5      ["        [$        -  -
  $ On[!        U5      R&                  (       a1  UR&                  (       a  U R                  U5      ["        [$        -  -   $ O#U R                  [        5      R                  XUS9$ U R                  U5      $ rB  )rm   r÷   rC  r~   rö   r   r}   rƒ   rù   r[   r€   r0   r   rü   r   rõ   r   r   r   r  rD  s           r3   rü   Úatanh._eval_as_leading_termv  si  € Ø�i‰i˜‰lˆØ�X‰X�a˜‹^×"Ñ"Ó$ˆØ�:�:Ø×&Ñ& qÓ)Ð)Ø”—‘Š;Ø—9‘9˜S×0Ñ0°Ó3Ó4ˆDØ�~�~Ø�à�ð ”1—5‘5�&œ!Ÿ%™%¤×!2Ñ!2Ð3Ó3Ø—<‘<¤Ó$×:Ñ:¸1ÈdÐ:ÐSÐSà��A‘‰I×"×"Ø—7‘7˜1¥d™d°Ó2ˆDÜ�$‹x×#×#Ø—>—>ØŸ9™9 R›=¬1¬R©4Ñ/Ð/ð "ä�D“×%×%Ø—>—>ØŸ9™9 R›=¬1¬R©4Ñ/Ð/ð "ð —|‘|¤CÓ(×>Ñ>¸qÐRVÐ>ÐWÐWØ�y‰y˜‹}ÐrC   c                 ó¾  >• U R                   S   nUR                  US5      nU[        R                  [        R                  4;   a#  U R                  [        5      R                  XX4S9$ [        T	U ]  XUS9nU[        R                  L a  U$ SUS-  -
  R                  (       a¹  UR                  X(       a  UOS5      n[        U5      R                  (       a$  UR                  (       a  U[        [        -  -
  $  U$ [        U5      R                  (       a$  UR                  (       a  U[        [        -  -   $  U$ U R                  [        5      R                  XX4S9$ U$ rI  r€  rM  s
            €r3   rK  Úatanh._eval_nseries’  s+  ø€ Ø�i‰i˜‰lˆØ�x‰x˜˜1‹~ˆð ”A—E‘Eœ1Ÿ=™=Ð)Ó)Ø—<‘<¤Ó$×2Ñ2°1¸dÐ2ÐNÐNä‰gÑ# A°Ð#Ð6ˆØ”1×$Ñ$Ò$ØˆJð ��a‘‰K×$×$Ø—7‘7˜1¥d™d°Ó2ˆDÜ�$‹x×#×#Ø×#×#Ø¤¤2¡™:Ð%ð $ð ˆ
ô �D“×%×%Ø×#×#Ø¤¤2¡™:Ð%ð $ð ˆ
ð —|‘|¤CÓ(×6Ñ6°qÀ$Ð6ÐRÐRØˆ
rC   c                 óB   • [        SU-   5      [        SU-
  5      -
  S-  $ rW  ©r   rS  s      r3   rT  Úatanh._eval_rewrite_as_log«  s"   € Ü�A˜‘E“
œS  Q¡›ZÑ'¨1Ñ,Ð,rC   c                 óÖ   • [        SUS-  S-
  -  5      n[        U-  S[        US-  * 5      -  -  [        U* 5      [        SUS-  -
  5      -  [        U5      -  U-  [        U5      -  -
  $ rW  )r   r   rv   )ro   r‰   rÃ   r9  s       r3   rŠ  Úatanh._eval_rewrite_as_asinh°  sm   € Ü��A�q‘D˜1‘H‘ÓˆÜ�1‘�aœ˜a ™d˜U›‘mÑ$Ü�a�R“œ˜a ! Q¡$™h›Ñ'¬¨Q«Ñ/°Ñ1´%¸³(Ñ:ñ;ð 	<rC   c                 óB   • U R                   S   R                  (       a  gg rÿ   r¶  r™   s    r3   r  Úatanh._eval_is_zeroµ  s   € Ø�9‰9�Q‰<××Øð  rC   c                 ó´   • [        U R                  S   R                  SU R                  S   -
  R                  U R                  S   S-   R                  /5      $ r–  ©r
   rm   r    rw  r™   s    r3   r  Úatanh._eval_is_extended_real¹  sN   € Ü˜$Ÿ)™) A™,×7Ñ7¸!¸d¿i¹iÈ¹lÑ:J×9ZÑ9ZÐ]a×]fÑ]fÐghÑ]iÐlmÑ]m×\}Ñ\}Ð~ÓÐrC   c                 ó–   • [        [        U R                  S   S-
  R                  U R                  S   S-   R                  /5      5      $ r–  ©r   r	   rm   r~   r™   s    r3   r  Úatanh._eval_is_finite¼  ó=   € Üœ D§I¡I¨a¡L°1Ñ$4×#=Ñ#=ÀÇ	Á	È!ÁÈqÑ@P×?YÑ?YÐ"ZÓ[Ó\Ð\rC   c                 ó4   • U R                   S   R                  $ r–   )rm   rZ  r™   s    r3   Ú_eval_is_imaginaryÚatanh._eval_is_imaginary¿  s   € Ø�y‰y˜‰|×(Ñ(Ð(rC   c                 ó   • [         $ rt   rÑ  rn   s     r3   rw   Úatanh.inverseÂ  rf  rC   rB   r!  ro  )rR   rS   rT   rU   rV   rq   r#  r‹   r$  r   r“   rü   rK  rT  rÄ   rŠ  r  r  r  r»  rw   rX   rp  rq  s   @r3   r…   r…   $  sz   ø† ñô&5ð ñ,&ó ð,&ð\ Øñó ó ðò÷8ò2-ð "6Ðò<ò
ò@ò]ò)÷ò rC   r…   c                   ó–   ^ • \ rS rSrSrSS jr\S 5       r\\	S 5       5       r
S rSU 4S jjrS r\rS	 rS
 rSS jrS rS rSrU =r$ )r†   iÉ  zý
``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   • US:X  a  SSU R                   S   S-  -
  -  $ [        X5      ern  r�  rn   s     r3   rq   Úacoth.fdiffÝ  rŸ  rC   c                 ó"  • UR                   (       aì  U[        R                  L a  [        R                  $ U[        R                  L a  [        R                  $ U[        R
                  L a  [        R                  $ UR                  (       a  [        [        -  S-  $ U[        R                  L a  [        R                  $ U[        R                  L a  [        R
                  $ UR                  (       a
  U " U* 5      * $ OcU[        R                  L a  [        R                  $ [        U5      nUb  [        * [        U5      -  $ UR                  5       (       a
  U " U* 5      * $ UR                  (       a  [        [        -  [        R                   -  $ g r¿   )r|   r   r}   rM   r_   rN   r~   r   r   r[   rt  r   r€   r)   r    r�   rA   )r‡   r`   rˆ   s      r3   r‹   Ú
acoth.evalã  s  € à�=�=Ø”a—e‘eŠ|Ü—u‘u�ØœŸ
™
Ò"Ü—v‘v�Øœ×*Ñ*Ò*Ü—v‘v�Ø——Üœ!‘t˜a‘x�ØœŸ™’Ü—z‘zÐ!ØœŸ™Ò%Ü×)Ñ)Ð)Ø——Ù˜S˜D›	�zÐ!ð !ð ”a×'Ñ'Ò'Ü—v‘v�ä4°SÓ9ˆGàÑ"Ü�rœD ›MÑ)Ð)à×/Ñ/×1Ñ1Ù  ›I˜:Ð%à�;�;Ü”a‘4œŸ™‘;Ðð rC   c                 ó’   • U S:X  a  [         * [        -  S-  $ U S:  d	  U S-  S:X  a  [        R                  $ [	        U5      nX-  U -  $ r¦  )r   r   r   r_   r   r#  s      r3   r“   Úacoth.taylor_term  sH   € ð �‹6Ü�2”b‘5˜‘7ˆNØ�‹U�a˜!‘e˜q“jÜ—6‘6ˆMä˜“
ˆAØ‘4˜!‘8ˆOrC   c                 ó  • U R                   S   nUR                  US5      R                  5       nU[        R                  L a  SU-  R                  U5      $ U[        R                  L a5  U R                  UR                  U5      5      nUR                  (       a  U$ U $ U[        R                  * [        R                  [        R                  4;   a#  U R                  [        5      R                  XUS9$ UR                  (       aê  SUS-  -
  R                  (       aÓ  UR!                  X(       a  UOS5      n[#        U5      R$                  (       a1  UR                  (       a  U R                  U5      [&        [(        -  -   $ On[#        U5      R                  (       a1  UR$                  (       a  U R                  U5      [&        [(        -  -
  $ O#U R                  [        5      R                  XUS9$ U R                  U5      $ )Nr   r6   rð   r7   )rm   r÷   rC  r   r€   rö   r}   rƒ   rù   r[   r_   r0   r   rü   r  r  rõ   r   r   r   r   rD  s           r3   rü   Úacoth._eval_as_leading_term  sx  € Ø�i‰i˜‰lˆØ�X‰X�a˜‹^×"Ñ"Ó$ˆØ”×"Ñ"Ò"Ø�c‘E×*Ñ*¨1Ó-Ð-Ø”—‘Š;Ø—9‘9˜S×0Ñ0°Ó3Ó4ˆDØ�~�~Ø�à�ð ”1—5‘5�&œ!Ÿ%™%¤§¡Ð(Ó(Ø—<‘<¤Ó$×:Ñ:¸1ÈdÐ:ÐSÐSà�:�:˜1˜r 1™u™9×1×1Ø—7‘7˜1¥d™d°Ó2ˆDÜ�$‹x×#×#Ø—>—>ØŸ9™9 R›=¬1¬R©4Ñ/Ð/ð "ä�D“×%×%Ø—>—>ØŸ9™9 R›=¬1¬R©4Ñ/Ð/ð "ð —|‘|¤CÓ(×>Ñ>¸qÐRVÐ>ÐWÐWØ�y‰y˜‹}ÐrC   c                 óà  >• U R                   S   nUR                  US5      nU[        R                  [        R                  4;   a#  U R                  [        5      R                  XX4S9$ [        T	U ]  XUS9nU[        R                  L a  U$ UR                  (       aÐ  SUS-  -
  R                  (       a¹  UR                  X(       a  UOS5      n[        U5      R                  (       a$  UR                  (       a  U[        [         -  -   $  U$ [        U5      R                  (       a$  UR                  (       a  U[        [         -  -
  $  U$ U R                  [        5      R                  XX4S9$ U$ rI  )rm   r÷   r   r[   rt  r0   r   rK  rL  r€   r  r  rõ   r   r   r   r   rM  s
            €r3   rK  Úacoth._eval_nseries*  s1  ø€ Ø�i‰i˜‰lˆØ�x‰x˜˜1‹~ˆð ”A—E‘Eœ1Ÿ=™=Ð)Ó)Ø—<‘<¤Ó$×2Ñ2°1¸dÐ2ÐNÐNä‰gÑ# A°Ð#Ð6ˆØ”1×$Ñ$Ò$ØˆJð �<�<˜Q  q¡™[×5×5Ø—7‘7˜1¥d™d°Ó2ˆDÜ�$‹x×#×#Ø×#×#Ø¤¤2¡™:Ð%ð $ð ˆ
ô �D“×%×%Ø×#×#Ø¤¤2¡™:Ð%ð $ð ˆ
ð —|‘|¤CÓ(×6Ñ6°qÀ$Ð6ÐRÐRØˆ
rC   c                 óN   • [        SSU-  -   5      [        SSU-  -
  5      -
  S-  $ rW  r­  rS  s      r3   rT  Úacoth._eval_rewrite_as_logC  s*   € Ü�A˜˜!™‘G“œs 1 q¨¡s¡7›|Ñ+¨qÑ0Ð0rC   c                 ó   • [        SU-  5      $ rè   rq  rS  s      r3   rX  Úacoth._eval_rewrite_as_atanhH  s   € Ü�Q�q‘S‹zÐrC   c           	      ó  • [         [        -  S-  [        US-
  U-  5      [        XS-
  -  5      -  [        SSU-  -   5      [        XS-   -  5      -  -
  -  U[        SUS-  -  5      -  [        [        SUS-  S-
  -  5      5      -  -   $ rÙ   )r   r   r   rv   rS  s      r3   rŠ  Úacoth._eval_rewrite_as_asinhK  s‡   € Ü”1‘�Q‘œ˜a !™e Q™Y›¬¨Q°A±©Y«Ñ7¼$¸qÀ1ÀQÁ3¹w»-ÌÈQÐTUÑPUÉYËÑ:WÑWÑXØ”$�q˜˜A™‘v“,‘œu¤T¨!¨Q°©T°A©X©,Ó%7Ó8Ñ8ñ9ð 	:rC   c                 ó   • [         $ rt   rž  rn   s     r3   rw   Úacoth.inverseO  rf  rC   c                 óÈ   • [        U R                  S   R                  [        U R                  S   S-
  R                  U R                  S   S-   R
                  /5      /5      $ r–  )r
   rm   r    r	   r—  Úis_extended_nonpositiver™   s    r3   r  Úacoth._eval_is_extended_realU  sw   € Ü˜$Ÿ)™) A™,×7Ñ7¼ÀDÇIÁIÈaÁLÐSTÑDT×CmÑCmÐpt×pyÑpyÐz{Ñp|ð  @Añ  qA÷  pZñ  pZð  C[ó  :\ð  ]ó  ^ð  	^rC   c                 ó–   • [        [        U R                  S   S-
  R                  U R                  S   S-   R                  /5      5      $ r–  r·  r™   s    r3   r  Úacoth._eval_is_finiteX  r¹  rC   rB   r!  ro  )rR   rS   rT   rU   rV   rq   r#  r‹   r$  r   r“   rü   rK  rT  rÄ   rX  rŠ  rw   r  r  rX   rp  rq  s   @r3   r†   r†   É  su   ø† ñô&5ð ñó ðð> Øñó ó ðò÷8ò21ð "6Ðòò:ôò^÷]ð ]rC   r†   c                   ó¢   ^ • \ rS rSrSrSS jr\S 5       r\\	S 5       5       r
S rSU 4S jjrSS jrS	 r\rS
 rS rS rS rS rS rSrU =r$ )Úasechi\  aG  
``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   • US:X  a&  U R                   S   nSU[        SUS-  -
  5      -  -  $ [        X5      e©Nr6   r   r9   r7   rt  ©ro   rp   r`  s      r3   rq   Úasech.fdiff‚  s=   € Ø�q‹=Ø—	‘	˜!‘ˆAØ�qœ˜a ! Q¡$™h›Ñ'Ñ(Ð(ä$ TÓ4Ð4rC   c                 óò  • UR                   (       aÍ  U[        R                  L a  [        R                  $ U[        R                  L a  [        [
        -  S-  $ U[        R                  L a  [        [
        -  S-  $ UR                  (       a  [        R                  $ U[        R                  L a  [        R                  $ U[        R                  L a  [        [
        -  $ UR                  (       a/  [        5       nX;   a   UR                  (       a  X!   [
        -  $ X!   $ U[        R                  L a%  SSKJn  [
        U" [        * S-  [        S-  5      -  $ UR                  (       a  [        R                  $ g )Nr7   r   r¡  )r|   r   r}   rM   r   r   rN   r~   r[   r_   rt  r³  rO   r    r€   r£  r¢  )r‡   r`   ry  r¢  s       r3   r‹   Ú
asech.eval‰  s   € à�=�=Ø”a—e‘eŠ|Ü—u‘u�ØœŸ
™
Ò"Üœ!‘t˜a‘x�Øœ×*Ñ*Ò*Üœ!‘t˜a‘x�Ø——Ü—z‘zÐ!ØœŸ™’Ü—v‘v�ØœŸ™Ò%Üœ!‘t�à�=�=Ü$›ˆIàÓØ×'×'Ø$™>¬!Ñ+Ð+Ø ‘~Ð%à”!×#Ñ#Ò#ÝEÜ‘[¤"  Q¡¬¨1©Ó-Ñ-Ð-à�;�;Ü—:‘:Ðð rC   c                 ó‚  • U S:X  a  [        SU-  5      $ U S:  d	  U S-  S:X  a  [        R                  $ [        U5      n[	        U5      S:”  a*  U S:”  a$  US   nX0S-
  U S-
  -  -  US-  -  SU S-  S-  -  -  $ U S-  n[        [        R                  U5      U -  n[        U5      U -  S-  U -  S-  nSU-  U-  X-  -  S-  $ )Nr   r7   r6   rH   r:   r9   )r   r   r_   r   r�   r   rA   r   r>  s          r3   r“   Úasech.taylor_term¨  sÝ   € ð �‹6Ü�q˜1‘u“:ÐØ�‹U�a˜!‘e˜q“jÜ—6‘6ˆMä˜“
ˆAÜ�>Ó" QÓ&¨1¨q«5Ø" 2Ñ&�Ø ™U Q q¡S™MÑ*¨Q°©TÑ1°1¸¸1¹¸q±y±=ÑAÐAà˜‘F�Ü#¤A§F¡F¨AÓ.°Ñ2�Ü˜a“L 1Ñ$¨Ñ)¨AÑ-°Ñ2�Ø˜A‘v ‘z A¡DÑ(¨1Ñ,Ð,rC   c                 ó¼  • U R                   S   nUR                  US5      R                  5       nU[        R                  * [        R
                  [        R                  [        R                  4;   a#  U R                  [        5      R                  XUS9$ U[        R                  L a5  U R                  UR                  U5      5      nUR                  (       a  U$ U $ UR                  (       d  SU-
  R                  (       aÊ  UR                  X(       a  UOS5      n[!        U5      R"                  (       aY  UR"                  (       d  US-   R                  (       a  U R                  U5      * $ U R                  U5      S[$        -  [&        -  -
  $ [!        U5      R                  (       d#  U R                  [        5      R                  XUS9$ U R                  U5      $ rB  )rm   r÷   rC  r   r[   r_   r€   r0   r   rü   r}   rƒ   rö   rù   r   rõ   r   r  r   r   rD  s           r3   rü   Úasech._eval_as_leading_termº  sO  € Ø�i‰i˜‰lˆØ�X‰X�a˜‹^×"Ñ"Ó$ˆà”1—5‘5�&œ!Ÿ&™&¤!§%¡%¬×):Ñ):Ð;Ó;Ø—<‘<¤Ó$×:Ñ:¸1ÈdÐ:ÐSÐSà”—‘Š;Ø—9‘9˜S×0Ñ0°Ó3Ó4ˆDØ�~�~Ø�à�ð �>�>˜a "™f×1×1Ø—7‘7˜1¥d™d°Ó2ˆDÜ�$‹x×#×#Ø—>—> b¨1¡f×%9×%9Ø ŸI™I b›M˜>Ð)Ø—y‘y “} q¬¡s¬2¡vÑ-Ð-Ü˜“X×)×)Ø—|‘|¤CÓ(×>Ñ>¸qÐRVÐ>ÐWÐWØ�y‰y˜‹}ÐrC   c                 ó  >• SSK Jn  U R                  S   nUR                  US5      nU[        R
                  L GaY  [        SSS9n[        [        R
                  US-  -
  5      R                  [        5      R                  USSU-  5      n	[        R
                  U R                  S   -
  n
U
R                  U5      nX«-
  U-  nUR                  US5      (       d  US:X  a  U" S5      $ U" [        U5      5      $ [        [        R
                  U-   5      R                  XUS9nUR                  5       [        U5      -  R!                  5       nU	R                  5       R                  XŽ5      R!                  5       R#                  5       U" X-  U5      -   $ U[        R$                  L Gag  [        SSS9n[        [        R$                  US-  -   5      R                  [        5      R                  USSU-  5      n	[        R
                  U R                  S   -   n
U
R                  U5      nX«-
  U-  nUR                  US5      (       d-  US:X  a  U" S5      $ [&        [(        -  U" [        U5      5      -   $ [        [        R
                  U-   5      R                  XUS9nUR                  5       [        U5      -  R!                  5       nU	R                  5       R                  XŽ5      R!                  5       R#                  5       U" X-  U5      -   $ [*        TU ]9  XUS9nU[        R,                  L a  U$ UR.                  (       d  SU-
  R.                  (       a¬  UR1                  X(       a  UOS5      n[3        U5      R4                  (       a;  UR4                  (       d  US-   R.                  (       a  U* $ US[&        -  [(        -  -
  $ [3        U5      R.                  (       d#  U R                  [        5      R                  XX4S	9$ U$ ©
Nr   )ÚOrå  T)Úpositiver7   r6   rJ  rð   )r¥  rå  rm   r÷   r   r[   r   rØ  r0   r   Únseriesrö   Úis_meromorphicr   rK  ÚremoveOr¡   Úpowsimprt  r   r   rL  r€   r   rõ   r   r  ©ro   r‰   r‘   rñ   rò   rå  r`   rû   rå  ÚserÚarg1r9  ÚgÚres1rN  rF  rO  s                   €r3   rK  Úasech._eval_nseriesÓ  s-  ø€ Ý(Ø�i‰i˜‰lˆØ�x‰x˜˜1‹~ˆð ”1—5‘5‹=Ü�c DÑ)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 q×)Ñ)Ø  A›v‘q˜“tÐ5©1¬T°!«W«:Ð5ÜœŸ™ ™	“?×0Ñ0°¸dÐ0ÐCˆDØ—<‘<“>¤$ q£'Ñ)×1Ñ1Ó3ˆCØ—;‘;“=×%Ñ% aÓ-×4Ñ4Ó6×>Ñ>Ó@Á1ÀQÁTÈ1Ã:ÑMÐMà”1—=‘=Ó Ü�c DÑ)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 q×)Ñ)Ø  A›v‘q˜“tÐ<¬1¬R©4±!´D¸³G³*Ñ+<Ð<ÜœŸ™ ™	“?×0Ñ0°¸dÐ0ÐCˆDØ—<‘<“>¤$ q£'Ñ)×1Ñ1Ó3ˆCØ—;‘;“=×%Ñ% aÓ-×4Ñ4Ó6×>Ñ>Ó@Á1ÀQÁTÈ1Ã:ÑMÐMä‰gÑ# A°Ð#Ð6ˆØ”1×$Ñ$Ò$ØˆJð ××  D¡×5×5Ø—7‘7˜1¥d™d°Ó2ˆDÜ�$‹x×#×#Ø×#×#¨¨q©×'=×'=Ø˜4�KØ˜Qœq™S¤™V‘|Ð#Ü˜“X×)×)Ø—|‘|¤CÓ(×6Ñ6°qÀ$Ð6ÐRÐRØˆ
rC   c                 ó   • [         $ rt   rS  rn   s     r3   rw   Úasech.inverse   rf  rC   c                 óf   • [        SU-  [        SU-  S-
  5      [        SU-  S-   5      -  -   5      $ rè   rR  rÈ   s      r3   rT  Úasech._eval_rewrite_as_log  s3   € Ü�1�S‘5œ4  #¡¨¡	›?¬T°!°C±%¸!±)«_Ñ<Ñ<Ó=Ð=rC   c                 ó   • [        SU-  5      $ rè   )r„   rÈ   s      r3   r\  Úasech._eval_rewrite_as_acosh  ó   € Ü�Q�s‘U‹|ÐrC   c                 óª   • [        SU-  S-
  5      [        SSU-  -
  5      -  [        [        [        U-  SS9-  [        [        R
                  -  -   -  $ rD  )r   r   rv   r   r   rA   rÈ   s      r3   rŠ  Úasech._eval_rewrite_as_asinh  sN   € Ü�A�c‘E˜A‘I‹œt A¨¨#©¡I›Ñ.´´%¼¸#¹ÈÑ2NÑ0NÜ24´Q·V±V±)ñ1<ñ =ð 	=rC   c           	      óh  • [         [        -  S[        U5      [        SU-  5      -  -
  [         S-  [        U* 5      -  [        U5      -  -
  [         S-  [        US-  5      -  [        US-  * 5      -  -
  -  [        SUS-   -  5      [        US-   5      -  [        [        SUS-  -
  5      5      -  -   $ rW  )r   r   r   r…   rS  s      r3   rX  Ú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                 óŽ   • [        SU-  S-
  5      [        SSU-  -
  5      -  [        S-  [        [        [        U-  SS9-  -
  -  $ r‰  )r   r   r   ÚacschrS  s      r3   Ú_eval_rewrite_as_acschÚasech._eval_rewrite_as_acsch  sC   € Ü�A�a‘C˜!‘G‹}œT ! a¨¡c¡'›]Ñ*¬B¨q©D´1´U¼1¸Q¹3ÈÑ5OÑ3OÑ,OÑPÐPrC   c                 ó®   • [        U R                  S   R                  U R                  S   R                  SU R                  S   -
  R                  /5      $ r–  r´  r™   s    r3   r  Úasech._eval_is_extended_real  sI   € Ü˜$Ÿ)™) A™,×7Ñ7¸¿¹À1¹×9TÑ9TÐWXÐ[_×[dÑ[dÐefÑ[gÑWg×VwÑVwÐxÓyÐyrC   c                 óF   • [        U R                  S   R                  5      $ r–   ©r   rm   r~   r™   s    r3   r  Úasech._eval_is_finite  ó   € Ü˜Ÿ™ 1™×-Ñ-Ó.Ð.rC   rB   r!  ro  )rR   rS   rT   rU   rV   rq   r#  r‹   r$  r   r“   rü   rK  rw   rT  rÄ   r\  rŠ  rX  rþ  r  r  rX   rp  rq  s   @r3   rØ  rØ  \  s�   ø† ñ#ôJ5ð ñó ðð< Øñ-ó ó ð-ò ÷2+ôZò>ð "6Ðòò=òEòQòz÷/ð /rC   rØ  c                   ó¢   ^ • \ rS rSrSrSS jr\S 5       r\\	S 5       5       r
S rSU 4S jjrSS jrS	 r\rS
 rS rS rS rS rS rSrU =r$ )rý  i   aI  
``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|   • US:X  a,  U R                   S   nSUS-  [        SSUS-  -  -   5      -  -  $ [        X5      erÚ  rt  rÛ  s      r3   rq   Úacsch.fdiffF  sF   € Ø�q‹=Ø—	‘	˜!‘ˆAØ�q˜!‘tœD  Q q¨!¡t¡V¡Ó,Ñ,Ñ-Ð-ä$ TÓ4Ð4rC   c                 óB  • UR                   (       aß  U[        R                  L a  [        R                  $ U[        R                  L a  [        R                  $ U[        R
                  L a  [        R                  $ UR                  (       a  [        R                  $ U[        R                  L a  [        S[        S5      -   5      $ U[        R                  L a  [        S[        S5      -   5      * $ UR                  (       a  [        5       nX;   a  X!   [        -  $ U[        R                  L a  [        R                  $ UR                  (       a  [        R                  $ UR                  (       a  [        R                  $ UR!                  5       (       a
  U " U* 5      * $ g rW  )r|   r   r}   rM   r_   rN   r~   r€   r[   r   r   rt  r³  rJ   r   Úis_infiniter�   )r‡   r`   ry  s      r3   r‹   Ú
acsch.evalM  s  € à�=�=Ø”a—e‘eŠ|Ü—u‘u�ØœŸ
™
Ò"Ü—v‘v�Øœ×*Ñ*Ò*Ü—v‘v�Ø——Ü×(Ñ(Ð(ØœŸ™’Ü˜1œt A›w™;Ó'Ð'ØœŸ™Ò%Ü˜Q¤ a£™[Ó)Ð)Ð)à�=�=Ü$›ˆIàÓØ ‘~¤aÑ'Ð'à”!×#Ñ#Ò#Ü—6‘6ˆMà�?�?Ü—6‘6ˆMà�;�;Ü×$Ñ$Ð$à×'Ñ'×)Ñ)Ù˜˜“I�:Ðð *rC   c                 ó®  • U S:X  a  [        SU-  5      $ U S:  d	  U S-  S:X  a  [        R                  $ [        U5      n[	        U5      S:”  a,  U S:”  a&  US   nU* U S-
  U S-
  -  -  US-  -  SU S-  S-  -  -  $ U S-  n[        [        R                  U5      U -  n[        U5      U -  S-  U -  S-  n[        R                  US-   -  U-  U-  X-  -  S-  $ )Nr   r7   r6   rH   r:   )	r   r   r_   r   r�   r   rA   r   rt  r>  s          r3   r“   Úacsch.taylor_termo  sð   € ð �‹6Ü�q˜1‘u“:ÐØ�‹U�a˜!‘e˜q“jÜ—6‘6ˆMä˜“
ˆAÜ�>Ó" QÓ&¨1¨q«5Ø" 2Ñ&�Ø�r˜a !™e a¨¡c™]Ñ+¨a°©dÑ2°A¸¸A¹À¹	±MÑBÐBà˜‘F�Ü#¤A§F¡F¨AÓ.°!Ñ3�Ü˜a“L 1Ñ$¨Ñ)¨AÑ-°Ñ2�Ü—}‘} q¨!¡tÑ,¨qÑ0°1Ñ4°q±tÑ;¸aÑ?Ð?rC   c                 ó  • U R                   S   nUR                  US5      R                  5       nU[        * [        [        R
                  4;   a#  U R                  [        5      R                  XUS9$ U[        R                  L a5  U R                  UR                  U5      5      nUR                  (       a  U$ U $ U[        R                  L a  SU-  R                  U5      $ UR                  (       aþ  SUS-  -   R                  (       aç  UR!                  X(       a  UOS5      n[#        U5      R                  (       a;  [%        U5      R                  (       a   U R                  U5      * [        [&        -  -
  $ Ox[#        U5      R(                  (       a;  [%        U5      R(                  (       a   U R                  U5      * [        [&        -  -   $ O#U R                  [        5      R                  XUS9$ U R                  U5      $ rB  )rm   r÷   rC  r   r   r_   r0   r   rü   r}   rƒ   rö   rù   r€   rZ  r  rõ   r   r   r   r   rD  s           r3   rü   Úacsch._eval_as_leading_term�  s}  € Ø�i‰i˜‰lˆØ�X‰X�a˜‹^×"Ñ"Ó$ˆà”1�"”aœŸ™�Ó Ø—<‘<¤Ó$×:Ñ:¸1ÈdÐ:ÐSÐSà”—‘Š;Ø—9‘9˜S×0Ñ0°Ó3Ó4ˆDØ�~�~Ø�à�à”×"Ñ"Ò"Ø�c‘E×*Ñ*¨1Ó-Ð-à�?�?  B¨¡E¡	×6×6Ø—7‘7˜1¥d™d°Ó2ˆDÜ�$‹x×#×#Ü�b“6×%×%Ø ŸI™I b›M˜>¬A¬b©DÑ0Ð0ð &ä�D“×%×%Ü�b“6×%×%Ø ŸI™I b›M˜>¬A¬b©DÑ0Ð0ð &ð —|‘|¤CÓ(×>Ñ>¸qÐRVÐ>ÐWÐWØ�y‰y˜‹}ÐrC   c                 óN  >• SSK Jn  U R                  S   nUR                  US5      nU[        L GaZ  [        SSS9n[        [        US-  -   5      R                  [        5      R                  USSU-  5      n	[        * U R                  S   -   n
U
R                  U5      nX«-
  U-  nUR                  US5      (       d1  US:X  a  U" S5      $ [        * [        -  S-  U" [        U5      5      -   $ [        [        R                  U-   5      R!                  XUS9nUR#                  5       [        U5      -  R%                  5       nU	R#                  5       R                  XŽ5      R%                  5       R'                  5       U" X-  U5      -   nU$ U[        R(                  [        -  :X  GaW  [        SSS9n[        [        * US-  -   5      R                  [        5      R                  USSU-  5      n	[        U R                  S   -   n
U
R                  U5      nX«-
  U-  nUR                  US5      (       d0  US:X  a  U" S5      $ [        [        -  S-  U" [        U5      5      -   $ [        [        R                  U-   5      R!                  XUS9nUR#                  5       [        U5      -  R%                  5       nU	R#                  5       R                  XŽ5      R%                  5       R'                  5       U" X-  U5      -   $ [*        TU ]A  XUS9nU[        R,                  L a  U$ UR.                  (       añ  SUS-  -   R0                  (       aÚ  U R                  S   R3                  X(       a  UOS5      n[5        U5      R0                  (       a.  [7        U5      R0                  (       a  U* [        [        -  -
  $  U$ [5        U5      R8                  (       a.  [7        U5      R8                  (       a  U* [        [        -  -   $  U$ U R                  [        5      R!                  XX4S	9$ U$ rä  )r¥  rå  rm   r÷   r   r   rý  r0   r   rç  rö   rè  r   r   r   r[   rK  ré  r¡   rê  rt  rL  r€   rZ  r  rõ   r   r   r   rë  s                   €r3   rK  Úacsch._eval_nseriesž  sl  ø€ Ý(Ø�i‰i˜‰lˆØ�x‰x˜˜1‹~ˆð ”1‹9Ü�c DÑ)ˆAÜœ˜A˜q™D™“/×)Ñ)¬#Ó.×6Ñ6°q¸!¸Q¸q¹SÓAˆCÜ�2˜Ÿ	™	 !™Ñ$ˆDØ×$Ñ$ QÓ'ˆAØ‘˜A‘ˆAØ×#Ñ# A q×)Ñ)Ø  A›v‘q˜“tÐ?¬A¨2¬b©5°©7±Q´t¸A³w³ZÑ+?Ð?ÜœŸ™ ™	“?×0Ñ0°¸dÐ0ÐCˆDØ—<‘<“>¤$ q£'Ñ)×1Ñ1Ó3ˆCØ—+‘+“-×$Ñ$ QÓ,×3Ñ3Ó5×=Ñ=Ó?Á!ÀAÁDÈ!Ã*ÑLˆCØˆJà”1—=‘=¤‘?Ô"Ü�c DÑ)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 q×)Ñ)Ø  A›v‘q˜“tÐ>¬1¬R©4°©6±A´d¸1³g³JÑ+>Ð>ÜœŸ™ ™	“?×0Ñ0°¸dÐ0ÐCˆDØ—<‘<“>¤$ q£'Ñ)×1Ñ1Ó3ˆCØ—;‘;“=×%Ñ% aÓ-×4Ñ4Ó6×>Ñ>Ó@Á1ÀQÁTÈ1Ã:ÑMÐMä‰gÑ# A°Ð#Ð6ˆØ”1×$Ñ$Ò$ØˆJð ×× ! d¨A¡g¡+×!:×!:Ø—9‘9˜Q‘<×#Ñ# A­t¡t¸Ó;ˆDÜ�$‹x×#×#Ü�d“8×'×'Ø˜4¤!¤B¡$™;Ð&ð (ð ˆ
ô �D“×%×%Ü�d“8×'×'Ø˜4¤!¤B¡$™;Ð&ð (ð ˆ
ð —|‘|¤CÓ(×6Ñ6°qÀ$Ð6ÐRÐRØˆ
rC   c                 ó   • [         $ rt   ré   rn   s     r3   rw   Úacsch.inverseÎ  rf  rC   c                 óH   • [        SU-  [        SUS-  -  S-   5      -   5      $ rW  rR  rÈ   s      r3   rT  Úacsch._eval_rewrite_as_logÔ  s'   € Ü�1�S‘5œ4  # q¡&¡¨1¡Ó-Ñ-Ó.Ð.rC   c                 ó   • [        SU-  5      $ rè   ru   rÈ   s      r3   rŠ  Úacsch._eval_rewrite_as_asinhÙ  r÷  rC   c                 óº   • [         [        S[         U-  -
  5      [        [         U-  S-
  5      -  [        [         U-  SS9-  [        [        R
                  -  -
  -  $ rD  )r   r   r„   r   r   rA   rÈ   s      r3   r\  Úacsch._eval_rewrite_as_acoshÜ  sT   € Ü”$�qœ1˜S™5‘y“/¤$¤q¨¡u¨q¡y£/Ñ1Ü %¤a¨¡e°eÑ <ñ=Ü?AÄ!Ç&Á&¹yñIñ Jð 	JrC   c                 ó´   • US-  nUS-   n[        U* 5      U-  [        [        R                  -  [        US-  * 5      U-  [	        [        U5      5      -  -
  -  $ rÙ   )r   r   r   rA   r…   )ro   r`   rÃ   Úarg2Úarg2p1s        r3   rX  Ú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   • U R                   S   R                  $ r–   )rm   r
  r™   s    r3   r  Úacsch._eval_is_zeroæ  s   € Ø�y‰y˜‰|×'Ñ'Ð'rC   c                 ó4   • U R                   S   R                  $ r–   r  r™   s    r3   r  Úacsch._eval_is_extended_realé  rk  rC   c                 óF   • [        U R                  S   R                  5      $ r–   r  r™   s    r3   r  Úacsch._eval_is_finiteì  r  rC   rB   r!  ro  )rR   rS   rT   rU   rV   rq   r#  r‹   r$  r   r“   rü   rK  rw   rT  rÄ   rŠ  r\  rX  r  r  r  rX   rp  rq  s   @r3   rý  rý     sƒ   ø† ñ#ôJ5ð ñó ððB Øñ@ó ó ð@ò ÷:.ô`ò/ð "6ÐòòJòMò(ò-÷/ð /rC   rý  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/   rg   ri   rl   rÛ   râ   rß  rê   rT  r1  rv   r„   r…   r†   rØ  rý  rB   rC   r3   Ú<module>r1     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   