a
    kh                    @  s  d dl mZ d dlmZ d dlmZ d dlmZ d dlm	Z	m
Z
mZmZ d dlmZmZmZmZ d dlmZ d dlmZmZmZmZmZ d d	lmZmZ d d
lmZ d dlm Z m!Z! d dl"m#Z# d dl$m%Z%m&Z& d dl'm(Z(m)Z) d dl*m+Z,m-Z-m.Z. d dl/m0Z0m1Z1 d dl2m3Z3 d dl4m5Z5m6Z6m7Z7 d dl8m9Z9 d dl:m;Z;m<Z<m=Z= d dl>m?Z? d dl@mAZA d dlBmCZC d dlDmEZE dd ZFG dd de	ZGedd ZHdd  ZIdId"d#d$d%d&d'ZJG d(d) d)eGZKG d*d+ d+eGZLG d,d- d-eGZMG d.d/ d/eGZNG d0d1 d1eGZOG d2d3 d3eOZPG d4d5 d5eOZQG d6d7 d7e	ZRG d8d9 d9e	ZSG d:d; d;eSZTG d<d= d=eSZUG d>d? d?eSZVG d@dA dAeSZWG dBdC dCeSZXG dDdE dEeSZYG dFdG dGeSZZdHS )J    )annotations)Add)cacheit)Expr)DefinedFunctionArgumentIndexError	PoleError
expand_mul)	fuzzy_notfuzzy_or	FuzzyBool	fuzzy_and)Mod)RationalpiIntegerFloatequal_valued)NeEq)S)SymbolDummy)sympify)	factorialRisingFactorial)	bernoullieuler)argimre)logexp)floor)sqrtMinMax)	Piecewise)	cos_table	ipartfracfermat_coords)And)	factorint)symmetric_poly)numbered_symbolsc                 C  s   t | trdS | tjS dS )z; Helper to extract symbolic coefficient for imaginary unit N)
isinstancer   Zas_coefficientr   ImaginaryUnitr    r2   V/var/www/auris/lib/python3.9/site-packages/sympy/functions/elementary/trigonometric.py_imaginary_unit_as_coefficient!   s    
r4   c                   @  sJ   e Zd ZdZdZejfZdd Zdd Z	dddZ
dd	d
ZdddZdS )TrigonometricFunctionz(Base class for trigonometric functions. Tc                 C  sD   | j | j }|j | j kr:|jd jr@t|jd jr@dS n|jS d S Nr   F)funcargsis_rationalr
   is_zeroselfsr2   r2   r3   _eval_is_rational3   s
    z'TrigonometricFunction._eval_is_rationalc                 C  sd   | j | j }|j | j krZt| jd jr8| jd jr8dS t| jd }|d ur`|jr`dS n|jS d S Nr   FT)r7   r8   r
   r:   Zis_algebraic	_pi_coeffr9   )r<   r=   pi_coeffr2   r2   r3   _eval_is_algebraic;   s    z(TrigonometricFunction._eval_is_algebraicc                 K  s&   | j f d|i|\}}||tj  S )Ndeep)as_real_imagr   r0   )r<   rC   hintsZre_partZim_partr2   r2   r3   _eval_expand_complexF   s    z*TrigonometricFunction._eval_expand_complexc                 K  s   | j d jrF|r6d|d< | j d j|fi |tjfS | j d tjfS |rl| j d j|fi | \}}n| j d  \}}||fS )Nr   Fcomplex)r8   is_extended_realexpandr   ZerorD   )r<   rC   rE   r    r   r2   r2   r3   _as_real_imagJ   s    "z#TrigonometricFunction._as_real_imagNc                 C  s   t | jd }|d u r$t|jd }||s4tjS ||kr@|S ||jv r|jrr||\}}||krr|t	| S |j
r||\}}|j|dd\}}||kr|t	| S tdd S )Nr   F)Zas_Addz%Use the periodicity function instead.)r	   r8   tupleZfree_symbolshasr   rJ   is_MulZas_independentabsis_AddNotImplementedError)r<   Zgeneral_periodsymbolfghar2   r2   r3   _periodW   s$    

zTrigonometricFunction._period)T)T)N)__name__
__module____qualname____doc__Z
unbranchedr   ComplexInfinity_singularitiesr>   rB   rF   rK   rW   r2   r2   r2   r3   r5   -   s   

r5   c                	   C  s   ddddddddd	S )
N)      )r_      )r`      )ra   
   )ra      )rc   rb   )      )(   <   )   rd   re         rf   rg   x   r2   r2   r2   r2   r3   _table2q   s    rl   c                 C  s   t j}g }t| D ],}|t}|r6|jr6||7 }q|| q|t ju rV| t jfS |t j }|| }|j	sd| j	r|j
du rt||t g  |fS | t jfS )a  
    Split ARG into two parts, a "rest" and a multiple of $\pi$.
    This assumes ARG to be an Add.
    The multiple of $\pi$ returned in the second position is always a Rational.

    Examples
    ========

    >>> from sympy.functions.elementary.trigonometric import _peeloff_pi
    >>> from sympy import pi
    >>> from sympy.abc import x, y
    >>> _peeloff_pi(x + pi/2)
    (x, 1/2)
    >>> _peeloff_pi(x + 2*pi/3 + pi*y)
    (x + pi*y + pi/6, 1/2)

       F)r   rJ   r   Z	make_argscoeffr   r9   appendHalf
is_integeris_even)r   rA   Z
rest_termsrV   Km1m2r2   r2   r3   _peeloff_pi   s    





rv      r   intzExpr | None)r   cyclesreturnc                 C  s  | t u rtjS | stjS | j r| t }|r| \}}|jrt|d }|dkrt	t
t|d  }d| }|| }t	|}	t|	|rt|	|}|| }ntt	|}|| }|j r|d }
|
dkr|S |
s|jdurtjS tdS |
| S |S n| jrtjS dS )a6  
    When arg is a Number times $\pi$ (e.g. $3\pi/2$) then return the Number
    normalized to be in the range $[0, 2]$, else `None`.

    When an even multiple of $\pi$ is encountered, if it is multiplying
    something with known parity then the multiple is returned as 0 otherwise
    as 2.

    Examples
    ========

    >>> from sympy.functions.elementary.trigonometric import _pi_coeff
    >>> from sympy import pi, Dummy
    >>> from sympy.abc import x
    >>> _pi_coeff(3*x*pi)
    3*x
    >>> _pi_coeff(11*pi/7)
    11/7
    >>> _pi_coeff(-11*pi/7)
    3/7
    >>> _pi_coeff(4*pi)
    0
    >>> _pi_coeff(5*pi)
    1
    >>> _pi_coeff(5.0*pi)
    1
    >>> _pi_coeff(5.5*pi)
    3/2
    >>> _pi_coeff(2 + pi)

    >>> _pi_coeff(2*Dummy(integer=True)*pi)
    2
    >>> _pi_coeff(2*Dummy(even=True)*pi)
    0

    rw   r   rm   N)r   r   OnerJ   rN   rn   as_coeff_MulZis_FloatrO   rx   roundr!   Zevalfr   r   rq   rr   r   r:   )r   ry   cxcxrS   pmcmic2r2   r2   r3   r@      sB    %




r@   c                      s   e Zd ZdZd8ddZd9ddZedd	 Zee	d
d Z
d: fdd	Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd  Zd!d" Zd#d$ Zd%d& Zd'd( Zd;d*d+Zd,d- Zd.d/ Zd0d1 Zd2d3 Zd4d5 Zd6d7 Z  Z S )<sina  
    The sine function.

    Returns the sine of x (measured in radians).

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

    This function will evaluate automatically in the
    case $x/\pi$ is some rational number [4]_.  For example,
    if $x$ is a multiple of $\pi$, $\pi/2$, $\pi/3$, $\pi/4$, and $\pi/6$.

    Examples
    ========

    >>> from sympy import sin, pi
    >>> from sympy.abc import x
    >>> sin(x**2).diff(x)
    2*x*cos(x**2)
    >>> sin(1).diff(x)
    0
    >>> sin(pi)
    0
    >>> sin(pi/2)
    1
    >>> sin(pi/6)
    1/2
    >>> sin(pi/12)
    -sqrt(2)/4 + sqrt(6)/4


    See Also
    ========

    csc, cos, sec, tan, cot
    asin, acsc, acos, asec, atan, acot, atan2

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
    .. [2] https://dlmf.nist.gov/4.14
    .. [3] https://functions.wolfram.com/ElementaryFunctions/Sin
    .. [4] https://mathworld.wolfram.com/TrigonometryAngles.html

    Nc                 C  s   |  dt |S Nrm   rW   r   r<   rR   r2   r2   r3   period#  s    z
sin.periodrw   c                 C  s$   |dkrt | jd S t| |d S Nrw   r   )cosr8   r   r<   argindexr2   r2   r3   fdiff&  s    z	sin.fdiffc                 C  s  ddl m} ddlm} |jrT|tju r.tjS |jr:tjS |tj	tj
fv rT|ddS |tju rdtjS t||rddlm} |j|j }}t|dt  }|tj
ur||d t  }|tj	ur||d t  }||||td ttdd tjur6||||ttd	d ttd
d tjur6|ddS ||||td ttdd tjurz|tt|t|dS ||||ttd	d ttdd tjur|dtt|t|S |tt|t|tt|t|S nt||r|| S | r| |  S t|}|d urDddlm}	 tj|	| S t|}
|
d urH|
j rdtjS d|
 j r|
j!du rtj"|
tj#  S |
j$s|
t }||kr| |S d S |
j$rH|
d }|dkr| |d t  S d| dkr| d| t S |
td	d d t }t%|}t|t%s*|S |
t |krD| |
t S d S |j&rt'|\}}|r|t }t|t%| t%|t|  S |jrtjS t|t(r|j)d S t|t*r|j)d }|t+d|d   S t|t,r|j)\}}|t+|d |d   S t|t-r,|j)d }t+d|d  S t|t.r^|j)d }dt+dd|d   |  S t|t/r||j)d }d| S t|t0r|j)d }t+dd|d   S d S )Nr   AccumBoundsSetExprrw   	FiniteSetrm   r`   r^      rc   )sinhF)1!sympy.calculus.accumulationboundsr   sympy.sets.setexprr   	is_Numberr   NaNr:   rJ   InfinityNegativeInfinityr\   r/   sympy.sets.setsr   minmaxr#   r   intersectionr   ZEmptySetr%   r   r&   
_eval_funccould_extract_minus_signr4   %sympy.functions.elementary.hyperbolicr   r0   r@   rq   rr   NegativeOnerp   is_Rationalr   rP   rv   asinr8   atanr$   atan2acosacotacscasec)clsr   r   r   r   r   r   di_coeffr   rA   nargr   resultr   yr2   r2   r3   eval,  s    




"
"(





 






zsin.evalc                 G  sr   | dk s| d dkrt jS t|}t|dkrP|d }| |d  | | d   S t j| d  ||   t|  S d S Nr   rm   rw   r   rJ   r   lenr   r   nr   previous_termsr   r2   r2   r3   taylor_term  s    zsin.taylor_termr   c                   sZ   | j d }|d ur"|t||}||dtjtjrFtd|  t j	||||dS Nr   zCannot expand %s around 0)r   logxcdir
r8   subsr!   rM   r   r   r\   r   super_eval_nseriesr<   r   r   r   r   r   	__class__r2   r3   r     s    
zsin._eval_nseriesc                 K  sX   ddl m} tj}t|t|fr6||jd t	}t	|| t	| |  d|  S Nr   HyperbolicFunctionrm   
r   r   r   r0   r/   r5   r7   r8   rewriter"   )r<   r   kwargsr   Ir2   r2   r3   _eval_rewrite_as_exp  s
    zsin._eval_rewrite_as_expc                 K  s@   t |tr<tj}|jd }|||   d |||  d  S d S Nr   rm   r/   r!   r   r0   r8   r<   r   r   r   r   r2   r2   r3   _eval_rewrite_as_Pow  s    

zsin._eval_rewrite_as_Powc                 K  s   t |td  ddS Nrm   Fevaluater   r   r<   r   r   r2   r2   r3   _eval_rewrite_as_cos  s    zsin._eval_rewrite_as_cosc                 K  s"   t tj| }d| d|d   S Nrm   rw   tanr   rp   r<   r   r   Ztan_halfr2   r2   r3   _eval_rewrite_as_tan  s    zsin._eval_rewrite_as_tanc                 K  s   t |t| t| S Nr   r   r   r2   r2   r3   _eval_rewrite_as_sincos  s    zsin._eval_rewrite_as_sincosc                 K  sL   t tj| }tdttt|dtt|tdfd| d|d   dfS )Nr   rm   rw   T	cotr   rp   r'   r+   r   r   r   r   r<   r   r   Zcot_halfr2   r2   r3   _eval_rewrite_as_cot  s    $zsin._eval_rewrite_as_cotc                 K  s    | j tfi |j tfi |S r   )r   r   powr   r2   r2   r3   _eval_rewrite_as_pow  s    zsin._eval_rewrite_as_powc                 K  s    | j tfi |j tfi |S r   )r   r   r$   r   r2   r2   r3   _eval_rewrite_as_sqrt  s    zsin._eval_rewrite_as_sqrtc                 K  s   dt | S Nrw   cscr   r2   r2   r3   _eval_rewrite_as_csc  s    zsin._eval_rewrite_as_cscc                 K  s   dt |td  dd S )Nrw   rm   Fr   secr   r   r2   r2   r3   _eval_rewrite_as_sec  s    zsin._eval_rewrite_as_secc                 K  s   |t | S r   )sincr   r2   r2   r3   _eval_rewrite_as_sinc  s    zsin._eval_rewrite_as_sincc                 K  s(   ddl m} tt| d |tj| S )Nr   besseljrm   sympy.functions.special.besselr   r$   r   r   rp   r<   r   r   r   r2   r2   r3   _eval_rewrite_as_besselj  s    zsin._eval_rewrite_as_besseljc                 C  s   |  | jd  S Nr   r7   r8   	conjugater<   r2   r2   r3   _eval_conjugate  s    zsin._eval_conjugateTc                 K  sH   ddl m}m} | jf d|i|\}}t||| t||| fS Nr   coshr   rC   )r   r   r   rK   r   r   r<   rC   rE   r   r   r    r   r2   r2   r3   rD     s    zsin.as_real_imagc                 K  s   ddl m}m} | jd }d }|jr| \}}t|dd }t|dd }t|dd }	t|dd }
||
 ||	  S |j	r|j
dd\}}|jr|jrtj|d d  ||t| S ttj|d d  t| ||d t| dd	S t|S )
Nr   )
chebyshevt
chebyshevuFr   TZrationalrw   rm   )rC   )#sympy.functions.special.polynomialsr  r  r8   rP   as_two_termsr   _eval_expand_trigr   rN   r|   
is_IntegerZis_oddr   r   r	   )r<   rE   r  r  r   r   r   sxsyr~   cyr   r2   r2   r3   r    s*    
 zsin._eval_expand_trigc           	      C  s   ddl m} | jd }||d }|t }|jrT||t  |}tj	| | S |tj
u r||j|dt|jrtdndd}|tjtjfv r|ddS |jr| |S | S )Nr   r   -+dirr   rw   r   r   r8   r   cancelr   rq   as_leading_termr   r   r\   limitr    is_negativer   r   	is_finiter7   	r<   r   r   r   r   r   x0r   ltr2   r2   r3   _eval_as_leading_term  s    


zsin._eval_as_leading_termc                 C  s   | j d jrdS d S Nr   Tr8   rH   r   r2   r2   r3   _eval_is_extended_real  s    zsin._eval_is_extended_realc                 C  s   | j d }|jrdS d S r  r  r<   r   r2   r2   r3   _eval_is_finite  s    
zsin._eval_is_finitec                 C  s"   t | jd \}}|jr|jS d S r   rv   r8   r:   rq   r<   restZpi_multr2   r2   r3   _eval_is_zero  s    zsin._eval_is_zeroc                 C  s    | j d js| j d jrdS d S r  r8   rH   
is_complexr   r2   r2   r3   _eval_is_complex#  s    
zsin._eval_is_complex)N)rw   )r   )T)!rX   rY   rZ   r[   r   r   classmethodr   staticmethodr   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   rD   r  r  r  r  r!  r$  __classcell__r2   r2   r   r3   r      s:   /


t
r   c                      s   e Zd ZdZd8ddZd9ddZedd	 Zee	d
d Z
d: fdd	Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdddd Zd!d" Zd#d$ Zd%d& Zd'd( Zd;d*d+Zd,d- Zd.d/ Zd0d1 Zd2d3 Zd4d5 Zd6d7 Z  ZS )<r   a  
    The cosine function.

    Returns the cosine of x (measured in radians).

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

    See :func:`sin` for notes about automatic evaluation.

    Examples
    ========

    >>> from sympy import cos, pi
    >>> from sympy.abc import x
    >>> cos(x**2).diff(x)
    -2*x*sin(x**2)
    >>> cos(1).diff(x)
    0
    >>> cos(pi)
    -1
    >>> cos(pi/2)
    0
    >>> cos(2*pi/3)
    -1/2
    >>> cos(pi/12)
    sqrt(2)/4 + sqrt(6)/4

    See Also
    ========

    sin, csc, sec, tan, cot
    asin, acsc, acos, asec, atan, acot, atan2

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
    .. [2] https://dlmf.nist.gov/4.14
    .. [3] https://functions.wolfram.com/ElementaryFunctions/Cos

    Nc                 C  s   |  dt |S r   r   r   r2   r2   r3   r   U  s    z
cos.periodrw   c                 C  s&   |dkrt | jd  S t| |d S r   )r   r8   r   r   r2   r2   r3   r   X  s    z	cos.fdiffc                 C  sZ  ddl m} ddlm} ddlm} |jr`|tju r:tjS |j	rFtj
S |tjtjfv r`|ddS |tju rptjS t||rt|td  S t||r|| S |jr|jdu r|ddS | r| | S t|}|d urdd	lm} ||S t|}|d ur|jrtj| S d| jr0|jdu r0tjS |jsV|t }||krR| |S d S |jr|j}	|jd|	  }
|
|	kr|d t }| | S d|
 |	krd| t }| | S t  }|	|v r8||	 \}}|
t | |
t |  }}| || | }}d ||fv rd S || | td | | td |   S |	d
krFd S tj!t"dd d d}|	|v r||j }||j|# S d|	d kr|d t }| |}d |krd S d| d d }d|dk rdndt$t%|  }|t"d| d  S d S |j&r>t'|\}}|r>|t }t(|t(| t|t|  S |j	rLtj
S t|t)rb|j*d S t|t+r|j*d }dt"d|d   S t|t,r|j*\}}|t"|d |d   S t|t-r|j*d }t"d|d  S t|t.r|j*d }dt"dd|d    S t|t/r8|j*d }t"dd|d   S t|t0rV|j*d }d| S d S )Nr   r  r   r   r   rw   rm   F)r   rh   r`   r_   )r^   r`   )1r  r  r   r   r   r   r   r   r   r:   r{   r   r   r\   r/   r   r   r   rH   r  r   r4   r   r   r@   rq   r   rr   rJ   r   qr   rl   rp   r$   rI   rx   rO   rP   rv   r   r   r8   r   r   r   r   r   r   )r   r   r  r   r   r   r   rA   r   r)  r   table2rV   bnvalanvalbcst_table_someZctsnvalr   sign_cosr   r   r2   r2   r3   r   ^  s    










	



(



" 






zcos.evalc                 G  sr   | dk s| d dkrt jS t|}t|dkrP|d }| |d  | | d   S t j| d  ||   t|  S d S )Nr   rm   rw   r   r   r   r2   r2   r3   r     s    zcos.taylor_termr   c                   sZ   | j d }|d ur"|t||}||dtjtjrFtd|  t j	||||dS r   r   r   r   r2   r3   r     s    
zcos._eval_nseriesc                 K  s\   t j}ddlm} t|t|fr>||jd jt	fi |}t	|| t	| |  d S r   
r   r0   r   r   r/   r5   r7   r8   r   r"   )r<   r   r   r   r   r2   r2   r3   r      s
    zcos._eval_rewrite_as_expc                 K  s8   t |tr4tj}|jd }|| d ||  d  S d S r   r   r   r2   r2   r3   r     s    

zcos._eval_rewrite_as_Powc                 K  s   t |td  ddS r   )r   r   r   r2   r2   r3   _eval_rewrite_as_sin  s    zcos._eval_rewrite_as_sinc                 K  s"   t tj| d }d| d|  S r   r   r   r2   r2   r3   r     s    zcos._eval_rewrite_as_tanc                 K  s   t |t| t | S r   r   r   r2   r2   r3   r     s    zcos._eval_rewrite_as_sincosc              	   K  sP   t tj| d }tdttt|dtt|dt df|d |d  dfS )Nrm   rw   r   Tr   r   r2   r2   r3   r     s    (zcos._eval_rewrite_as_cotc                 K  s   | j |fi |S r   )r   r   r2   r2   r3   r     s    zcos._eval_rewrite_as_powr   r1   c                   s  ddl m} t|  d u r d S t tr.d S t ts<d S t } j|v rv| j| j  } jdk rr|	 }|S  jd sֈ d }t
|t jtfi |}|d d }t|d rdnd}	|	td| d  S t j}
|
r|
}ndd t j D }t| } fd	d
t||D }dd t|tdD }t
tdd
 |D  |}|
rnt|
dkrr|S |jtfi |S )Nr   r(  i  rm   rw   r   c                 S  s   g | ]\}}|| qS r2   r2   ).0r+  er2   r2   r3   
<listcomp>?      z-cos._eval_rewrite_as_sqrt.<locals>.<listcomp>c                 3  s"   | ]\}} j t|| V  qd S r   )r   r   )r3  r   r   rA   r2   r3   	<genexpr>B  r6  z,cos._eval_rewrite_as_sqrt.<locals>.<genexpr>c                 S  s    g | ]}|d  |d t  fqS )rw   r   )r   r3  r   r2   r2   r3   r5  C  r6  zc                 s  s   | ]}|d  V  qdS )r   Nr2   r9  r2   r2   r3   r8  D  r6  )r  r  r@   r/   r   r   r(   r)  r   rI   r   r   r   r$   rx   r*   r,   itemsr)   zipr.   sumr  r   r   )r<   r   r   r  r.  rvZpico2r/  r   r0  ZFCZdenomsZapartdecompXZpclsr2   r7  r3   r     s>    





 zcos._eval_rewrite_as_sqrtc                 K  s   dt | S r   r   r   r2   r2   r3   r   J  s    zcos._eval_rewrite_as_secc                 K  s   dt |jtfi | S r   )r   r   r   r   r2   r2   r3   r   M  s    zcos._eval_rewrite_as_cscc                 K  s:   ddl m} ttt| d |tj | t|dfdS )Nr   r   rm   rw   Tr   r   r'   r$   r   r   rp   r   r   r2   r2   r3   r   P  s
    &zcos._eval_rewrite_as_besseljc                 C  s   |  | jd  S r   r   r   r2   r2   r3   r   W  s    zcos._eval_conjugateTc                 K  sJ   ddl m}m} | jf d|i|\}}t||| t| || fS r   )r   r   r   rK   r   r   r   r2   r2   r3   rD   Z  s    zcos.as_real_imagc                 K  s   ddl m} | jd }d }|jr|| \}}t|dd }t|dd }t|dd }t|dd }	||	 ||  S |jr|j	dd\}
}|
j
r||
t|S t|S )Nr   r(  Fr   Tr  )r  r  r8   rP   r  r   r  r   rN   r|   r  )r<   rE   r  r   r   r   r  r	  r~   r
  rn   termsr2   r2   r3   r  _  s    
zcos._eval_expand_trigc           	      C  s   ddl m} | jd }||d }|td  t }|jrd||t  td  |}tj	| | S |tj
u r|j|dt|jrdndd}|tjtjfv r|ddS |jr| |S | S )	Nr   r   rm   r  r  r  r   rw   r  r  r2   r2   r3   r  p  s    


zcos._eval_as_leading_termc                 C  s   | j d jrdS d S r  r  r   r2   r2   r3   r  ~  s    zcos._eval_is_extended_realc                 C  s   | j d }|jrdS d S r  r  r  r2   r2   r3   r    s    
zcos._eval_is_finitec                 C  s    | j d js| j d jrdS d S r  r"  r   r2   r2   r3   r$    s    
zcos._eval_is_complexc                 C  s,   t | jd \}}|jr(|r(|tj jS d S r   rv   r8   r:   r   rp   rq   r  r2   r2   r3   r!    s    
zcos._eval_is_zero)N)rw   )r   )T) rX   rY   rZ   r[   r   r   r%  r   r&  r   r   r   r   r   r2  r   r   r   r   r   r   r   r   r   rD   r  r  r  r  r$  r!  r'  r2   r2   r   r3   r   )  s:   +


 +
r   c                      s   e Zd ZdZd:ddZd;ddZd<dd	Zed
d Ze	e
dd Zd= fdd	Zdd Zdd Zd>ddZdd Zdd Zdd Zdd Zd d! Zd"d# Zd$d% Zd&d' Zd(d) Zd*d+ Zd,d- Zd.d/ Zd0d1 Zd2d3 Zd4d5 Zd6d7 Zd8d9 Z   Z!S )?r   a  
    The tangent function.

    Returns the tangent of x (measured in radians).

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

    See :class:`sin` for notes about automatic evaluation.

    Examples
    ========

    >>> from sympy import tan, pi
    >>> from sympy.abc import x
    >>> tan(x**2).diff(x)
    2*x*(tan(x**2)**2 + 1)
    >>> tan(1).diff(x)
    0
    >>> tan(pi/8).expand()
    -1 + sqrt(2)

    See Also
    ========

    sin, csc, cos, sec, cot
    asin, acsc, acos, asec, atan, acot, atan2

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
    .. [2] https://dlmf.nist.gov/4.14
    .. [3] https://functions.wolfram.com/ElementaryFunctions/Tan

    Nc                 C  s   |  t|S r   r   r   r2   r2   r3   r     s    z
tan.periodrw   c                 C  s$   |dkrt j| d  S t| |d S Nrw   rm   )r   r{   r   r   r2   r2   r3   r     s    z	tan.fdiffc                 C  s   t S z7
        Returns the inverse of this function.
        r   r   r2   r2   r3   inverse  s    ztan.inversec                 C  s  ddl m} |jrL|tju r"tjS |jr.tjS |tjtjfv rL|tjtjS |tj	u r\tjS t
|| r|j|j }}t|t }|tjur||t  }|tjur||t  }ddlm} ||||td ttdd r|tjtjS |t|t|S | r| |  S t|}|d ur@ddlm} tj|| S t|d}	|	d urr|	jrbtjS |	js|	t }
|
|kr| |
S d S |	jrr|	j}|	j| }tddtd d  tddtd  tddtd d  tddtd  d	}|d
v r0d| | }|dkr(d| }||  S || S |	jd s|	t d }
t|
t|
td   }}t
|tst
|ts|dkrtj	S d| ||  S t  }||v r|| \}}| |t | | |t |  }}d ||fv rd S || d||   S |	tj! d tj! t }
t|
t|
td   }}t
|ts`t
|ts`|dkrXtj	S || S |
|krr| |
S |j"rt#|\}}|rt|t }|tj	u rt$| S t|S |jrtjS t
|t%r|j&d S t
|t'r|j&\}}|| S t
|t(r"|j&d }|td|d   S t
|t)rL|j&d }td|d  | S t
|t*rj|j&d }d| S t
|t+r|j&d }dtdd|d   |  S t
|t,r|j&d }tdd|d   | S d S )Nr   r   r   rm   r^   )tanhrw   r`   )rw   rm   r^   r_   r`   rb   rb   )-r   r   r   r   r   r:   rJ   r   r   r\   r/   r   r   r#   r   r   r   r   r   r   r   r4   r   rJ  r0   r@   rq   r   r)  r   r$   r   rl   rp   rP   rv   r   r   r8   r   r   r   r   r   r   )r   r   r   r   r   r   r   r   rJ  rA   r   r)  r   Ztable10r   cresultsresultr*  rV   r+  r,  r-  r   r   Ztanmr   r2   r2   r3   r     s    



$










"









ztan.evalc                 G  s~   | dk s| d dkrt jS t|}| d d d| d   }}t| d }t| d }t j| | |d  | | ||   S d S Nr   rm   rw   )r   rJ   r   r   r   r   )r   r   r   rV   r+  BFr2   r2   r3   r   J  s    ztan.taylor_termr   c                   sL   | j d |dd t }|r:|jr:| tj|||dS t j|||dS )Nr   rm   r   r   )r8   r  r   r  r   r   r   r   r<   r   r   r   r   r   r   r2   r3   r   Y  s    
ztan._eval_nseriesc                 K  sF   t |trBtj}|jd }|||  ||   ||  ||   S d S r   r   r   r2   r2   r3   r   _  s    

ztan._eval_rewrite_as_Powc                 C  s   |  | jd  S r   r   r   r2   r2   r3   r   e  s    ztan._eval_conjugateTc                 K  sx   | j f d|i|\}}|rdddlm}m} td| |d|  }td| | |d| | fS | |tjfS d S NrC   r   r   rm   	rK   r   r   r   r   r   r7   r   rJ   r<   rC   rE   r    r   r   r   Zdenomr2   r2   r3   rD   h  s     ztan.as_real_imagc                   sF  | j d }d }|jrt|j }g }|j D ]}t|dd }|| q(td  fddt|D }ddg}t|d D ]2}	|d|	d    t|	|d	|	d
 d   7  < qz|d |d  	t
t||S |jr>|jdd\}
}|
jr>|
dkr>tj}tddd}d||  |
  }t|t| 	|t|fgS t|S )Nr   Fr   Yc                   s   g | ]}t  qS r2   nextr3  r   ZYgr2   r3   r5  |  r6  z)tan._eval_expand_trig.<locals>.<listcomp>rw   rm   r   r_   Tr  dummyreal)r8   rP   r   r   r  ro   r.   ranger-   r   listr<  rN   r|   r  r   r0   r   rI   r   r    )r<   rE   r   r   r   ZTXZtxrV  r   r   rn   rD  r   r:  Pr2   rZ  r3   r  q  s,    


0  ztan._eval_expand_trigc                 K  sf   t j}ddlm} t|t|fr6||jd t	}t	| | t	||  }}|||  ||  S Nr   r   r1  )r<   r   r   r   r   neg_exppos_expr2   r2   r3   r     s    ztan._eval_rewrite_as_expc                 K  s   dt |d  t d|  S r   r   r<   r   r   r2   r2   r3   r2    s    ztan._eval_rewrite_as_sinc                 K  s   t |td  ddt | S r   r   re  r2   r2   r3   r     s    ztan._eval_rewrite_as_cosc                 K  s   t |t| S r   r   r   r2   r2   r3   r     s    ztan._eval_rewrite_as_sincosc                 K  s   dt | S r   r   r   r2   r2   r3   r     s    ztan._eval_rewrite_as_cotc                 K  s4   t |jtfi |}t|jtfi |}|| S r   )r   r   r   r   )r<   r   r   sin_in_sec_formcos_in_sec_formr2   r2   r3   r     s    ztan._eval_rewrite_as_secc                 K  s4   t |jtfi |}t|jtfi |}|| S r   )r   r   r   r   )r<   r   r   sin_in_csc_formcos_in_csc_formr2   r2   r3   r     s    ztan._eval_rewrite_as_cscc                 K  s2   | j tfi |j tfi |}|tr.d S |S r   r   r   r   rM   r<   r   r   r   r2   r2   r3   r     s     
ztan._eval_rewrite_as_powc                 K  s2   | j tfi |j tfi |}|tr.d S |S r   r   r   r$   rM   rl  r2   r2   r3   r     s     
ztan._eval_rewrite_as_sqrtc                 K  s&   ddl m} |tj||tj | S Nr   r   r   r   r   rp   r   r2   r2   r3   r     s    ztan._eval_rewrite_as_besseljc           
      C  s   ddl m} ddlm} | jd }||d }d| t }|jrl||t d  	|}	|j
rd|	S d|	 S |tju r|j|d||jrdndd}|tjtjfv r|tjtjS |jr| |S | S )	Nr   r   r    rm   r   r  r  r  r   r   $sympy.functions.elementary.complexesr    r8   r   r  r   rq   r  rr   r   r\   r  r  r   r   r  r7   
r<   r   r   r   r   r    r   r  r   r  r2   r2   r3   r    s    

ztan._eval_as_leading_termc                 C  s   | j d jS r   r  r   r2   r2   r3   r    s    ztan._eval_is_extended_realc                 C  s,   | j d }|jr(|t tj jdu r(dS d S r?   r8   is_realr   r   rp   rq   r  r2   r2   r3   _eval_is_real  s    
ztan._eval_is_realc                 C  s6   | j d }|jr(|t tj jdu r(dS |jr2dS d S r?   )r8   ru  r   r   rp   rq   is_imaginaryr  r2   r2   r3   r    s
    
ztan._eval_is_finitec                 C  s"   t | jd \}}|jr|jS d S r   r  r  r2   r2   r3   r!    s    ztan._eval_is_zeroc                 C  s,   | j d }|jr(|t tj jdu r(dS d S r?   rt  r  r2   r2   r3   r$    s    
ztan._eval_is_complex)N)rw   )rw   )r   )T)"rX   rY   rZ   r[   r   r   rI  r%  r   r&  r   r   r   r   r   rD   r  r   r2  r   r   r   r   r   r   r   r   r  r  rv  r  r!  r$  r'  r2   r2   r   r3   r     s>   %



 
		r   c                   @  s   e Zd ZdZd<ddZd=ddZd>dd	Zed
d Ze	e
dd Zd?ddZdd Zd@ddZdd Zdd Zdd Zdd Zdd Zd d! Zd"d# Zd$d% Zd&d' Zd(d) Zd*d+ Zd,d- Zd.d/ Zd0d1 Zd2d3 Zd4d5 Zd6d7 Zd8d9 Z d:d; Z!dS )Ar   a  
    The cotangent function.

    Returns the cotangent of x (measured in radians).

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

    See :class:`sin` for notes about automatic evaluation.

    Examples
    ========

    >>> from sympy import cot, pi
    >>> from sympy.abc import x
    >>> cot(x**2).diff(x)
    2*x*(-cot(x**2)**2 - 1)
    >>> cot(1).diff(x)
    0
    >>> cot(pi/12)
    sqrt(3) + 2

    See Also
    ========

    sin, csc, cos, sec, tan
    asin, acsc, acos, asec, atan, acot, atan2

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
    .. [2] https://dlmf.nist.gov/4.14
    .. [3] https://functions.wolfram.com/ElementaryFunctions/Cot

    Nc                 C  s   |  t|S r   r   r   r2   r2   r3   r     s    z
cot.periodrw   c                 C  s$   |dkrt j| d  S t| |d S rF  )r   r   r   r   r2   r2   r3   r     s    z	cot.fdiffc                 C  s   t S rG  r   r   r2   r2   r3   rI    s    zcot.inversec                 C  s  ddl m} |jrL|tju r"tjS |jr.tjS |tjtjfv rL|tjtjS |tju r\tjS t	||rxt
|td   S | r| |  S t|}|d urddlm} tj || S t|d}|d url|jrtjS |js|t }||kr| |S d S |jrl|jdv rt
td | S |jdkr|jd s|t d }t|t|td   }}t	|tst	|tsd| ||  S |j}	|j|	 }
t }|	|v r||	 \}}| |
t | | |
t |  }}d ||fv rd S d||  ||  S |tj d tj t }t|t|td   }}t	|tsZt	|tsZ|dkrRtjS || S ||krl| |S |jrt|\}}|rt|t }|tju rt|S t
| S |jrtjS t	|tr|jd S t	|tr|jd }d| S t	|tr|j\}}|| S t	|t r:|jd }t!d|d  | S t	|t"rd|jd }|t!d|d   S t	|t#r|jd }t!dd|d   | S t	|t$r|jd }dt!dd|d   |  S d S )Nr   r   rm   )cothrK  rw   )%r   r   r   r   r   r:   r\   r   r   r/   r   r   r   r4   r   ry  r0   r@   rq   r   r)  r   r   rl   rp   rP   rv   r   r   r8   r   r   r   r$   r   r   r   )r   r   r   r   ry  rA   r   rL  rM  r)  r   r*  rV   r+  r,  r-  r   r   Zcotmr   r2   r2   r3   r     s    






"









zcot.evalc                 G  s   | dkrdt | S | dk s(| d dkr.tjS t |}t| d }t| d }tj| d d  d| d   | | ||   S d S Nr   rw   rm   )r   r   rJ   r   r   r   )r   r   r   rO  rP  r2   r2   r3   r     s    zcot.taylor_termr   c                 C  sL   | j d |dt }|r6|jr6| tj|||dS | tj|||dS )Nr   rQ  )r8   r  r   r  r   r   r   r   rR  r2   r2   r3   r     s    
zcot._eval_nseriesc                 C  s   |  | jd  S r   r   r   r2   r2   r3   r     s    zcot._eval_conjugateTc                 K  sz   | j f d|i|\}}|rfddlm}m} td| |d|  }td|  | |d| | fS | |tjfS d S rS  rT  rU  r2   r2   r3   rD     s    "zcot.as_real_imagc                 K  sn   ddl m} tj}t|t|fr>||jd jt	fi |}t	| | t	||  }}|||  ||  S ra  r   )r<   r   r   r   r   rb  rc  r2   r2   r3   r     s    zcot._eval_rewrite_as_expc                 K  sH   t |trDtj}|jd }| ||  ||   ||  ||   S d S r   r   r   r2   r2   r3   r     s    

zcot._eval_rewrite_as_Powc                 K  s   t d| dt |d   S r   rd  re  r2   r2   r3   r2    s    zcot._eval_rewrite_as_sinc                 K  s   t |t |td  dd S r   r   re  r2   r2   r3   r     s    zcot._eval_rewrite_as_cosc                 K  s   t |t| S r   r   r   r   r2   r2   r3   r     s    zcot._eval_rewrite_as_sincosc                 K  s   dt | S r   r   r   r2   r2   r3   r     s    zcot._eval_rewrite_as_tanc                 K  s4   t |jtfi |}t|jtfi |}|| S r   )r   r   r   r   )r<   r   r   rh  rg  r2   r2   r3   r     s    zcot._eval_rewrite_as_secc                 K  s4   t |jtfi |}t|jtfi |}|| S r   )r   r   r   r   )r<   r   r   rj  ri  r2   r2   r3   r     s    zcot._eval_rewrite_as_cscc                 K  s2   | j tfi |j tfi |}|tr.d S |S r   rk  rl  r2   r2   r3   r     s     
zcot._eval_rewrite_as_powc                 K  s2   | j tfi |j tfi |}|tr.d S |S r   rm  rl  r2   r2   r3   r     s     
zcot._eval_rewrite_as_sqrtc                 K  s&   ddl m} |tj ||tj| S rn  ro  r   r2   r2   r3   r     s    zcot._eval_rewrite_as_besseljc           
      C  s   ddl m} ddlm} | jd }||d }d| t }|jrn||t d  	|}	|j
rhd|	 S |	 S |tju r|j|d||jrdndd}|tjtjfv r|tjtjS |jr| |S | S )	Nr   r   rp  rm   rw   r  r  r  rq  rs  r2   r2   r3   r    s    

zcot._eval_as_leading_termc                 C  s   | j d jS r   r  r   r2   r2   r3   r    s    zcot._eval_is_extended_realc                   sF  | j d }d }|jrt|j }g }|j D ]}t|dd }|| q(td  fddt|D }ddg}t|ddD ]6}	|||	 d   t|	|d||	 d	 d   7  < qz|d |d
  	t
t||S |jr>|jdd\}
}|
jr>|
d
kr>tj}tddd}|| |
  }t|t| 	|t|fgS t|S )Nr   Fr   rV  c                   s   g | ]}t  qS r2   rW  rY  rZ  r2   r3   r5    r6  z)cot._eval_expand_trig.<locals>.<listcomp>r   rm   r_   rw   Tr  r[  r\  )r8   rP   r   r   r  ro   r.   r^  r-   r   r_  r<  rN   r|   r  r   r0   r   rI   r    r   )r<   rE   r   r   r   ZCXr~   rV  r   r   rn   rD  r   r:  r`  r2   rZ  r3   r    s,    


4  zcot._eval_expand_trigc                 C  s0   | j d }|jr"|t jdu r"dS |jr,dS d S r?   )r8   ru  r   rq   rw  r  r2   r2   r3   r    s
    
zcot._eval_is_finitec                 C  s&   | j d }|jr"|t jdu r"dS d S r?   r8   ru  r   rq   r  r2   r2   r3   rv    s    
zcot._eval_is_realc                 C  s&   | j d }|jr"|t jdu r"dS d S r?   r}  r  r2   r2   r3   r$    s    
zcot._eval_is_complexc                 C  s,   t | jd \}}|r(|jr(|tj jS d S r   rE  )r<   r   Zpimultr2   r2   r3   r!    s    
zcot._eval_is_zeroc                 C  s6   | j d }|||}||kr.|t jr.tjS t|S r   )r8   r   r   rq   r   r\   r   )r<   oldnewr   Zargnewr2   r2   r3   
_eval_subs  s
    
zcot._eval_subs)N)rw   )rw   )r   )T)"rX   rY   rZ   r[   r   r   rI  r%  r   r&  r   r   r   r   rD   r   r   r2  r   r   r   r   r   r   r   r   r  r  r  r  rv  r$  r!  r  r2   r2   r2   r3   r     s>   %



h

	r   c                   @  s   e Zd ZU dZdZejfZdZde	d< dZ
de	d< edd Zdd	 Zd
d Zdd Zdd Zd1ddZdd Zdd Zdd Zdd Zdd Zdd Zdd  Zd!d" Zd2d$d%Zd&d' Zd(d) Zd*d+ Zd,d- Zd3d/d0ZdS )4ReciprocalTrigonometricFunctionz@Base class for reciprocal functions of trigonometric functions. Nr   _is_even_is_oddc                 C  sF  |  r*| jr| | S | jr*| |  S t|}|d urd| js|jr|j}|jd|  }||kr||d t }| | S d| |krd| t }| jr| |S | jr| | S t	|dr|
 | kr|jd S | j|}|d u r|S tdd || fD rd| tS tdd || fD r:d| tS d| S d S )Nrm   rw   rI  r   c                 s  s   | ]}t |tV  qd S r   )r/   r   rY  r2   r2   r3   r8  P  r6  z7ReciprocalTrigonometricFunction.eval.<locals>.<genexpr>c                 s  s   | ]}t |tV  qd S r   )r/   r   rY  r2   r2   r3   r8  R  r6  )r   r  r  r@   rq   r   r)  r   r   hasattrrI  r8   _reciprocal_ofr   anyr   r   r   )r   r   rA   r)  r   r   tr2   r2   r3   r   2  s@    



z$ReciprocalTrigonometricFunction.evalc                 O  s$   |  | jd }t|||i |S r   )r  r8   getattr)r<   method_namer8   r   or2   r2   r3   _call_reciprocalW  s    z0ReciprocalTrigonometricFunction._call_reciprocalc                 O  s,   | j |g|R i |}|d ur(d| S |S r   )r  )r<   r  r8   r   r  r2   r2   r3   _calculate_reciprocal\  s    z5ReciprocalTrigonometricFunction._calculate_reciprocalc                 C  s.   |  ||}|d ur*|| |kr*d| S d S r   )r  r  )r<   r  r   r  r2   r2   r3   _rewrite_reciprocalb  s    z3ReciprocalTrigonometricFunction._rewrite_reciprocalc                 C  s   t | jd }| ||S r   )r	   r8   r  r   )r<   rR   rS   r2   r2   r3   rW   i  s    z'ReciprocalTrigonometricFunction._periodrw   c                 C  s   |  d| | d  S )Nr   rm   r  r   r2   r2   r3   r   m  s    z%ReciprocalTrigonometricFunction.fdiffc                 K  s   |  d|S )Nr   r  r   r2   r2   r3   r   p  s    z4ReciprocalTrigonometricFunction._eval_rewrite_as_expc                 K  s   |  d|S )Nr   r  r   r2   r2   r3   r   s  s    z4ReciprocalTrigonometricFunction._eval_rewrite_as_Powc                 K  s   |  d|S )Nr2  r  r   r2   r2   r3   r2  v  s    z4ReciprocalTrigonometricFunction._eval_rewrite_as_sinc                 K  s   |  d|S )Nr   r  r   r2   r2   r3   r   y  s    z4ReciprocalTrigonometricFunction._eval_rewrite_as_cosc                 K  s   |  d|S )Nr   r  r   r2   r2   r3   r   |  s    z4ReciprocalTrigonometricFunction._eval_rewrite_as_tanc                 K  s   |  d|S )Nr   r  r   r2   r2   r3   r     s    z4ReciprocalTrigonometricFunction._eval_rewrite_as_powc                 K  s   |  d|S )Nr   r  r   r2   r2   r3   r     s    z5ReciprocalTrigonometricFunction._eval_rewrite_as_sqrtc                 C  s   |  | jd  S r   r   r   r2   r2   r3   r     s    z/ReciprocalTrigonometricFunction._eval_conjugateTc                 K  s"   d|  | jd  j|fi |S r   )r  r8   rD   )r<   rC   rE   r2   r2   r3   rD     s    z,ReciprocalTrigonometricFunction.as_real_imagc                 K  s   | j di |S )Nr  )r  r  )r<   rE   r2   r2   r3   r    s    z1ReciprocalTrigonometricFunction._eval_expand_trigc                 C  s   |  | jd  S r   )r  r8   r  r   r2   r2   r3   r    s    z6ReciprocalTrigonometricFunction._eval_is_extended_realc                 C  s    d|  | jd  j|||dS )Nrw   r   r   r   )r  r8   r  )r<   r   r   r   r2   r2   r3   r    s    z5ReciprocalTrigonometricFunction._eval_as_leading_termc                 C  s   d|  | jd  jS r   )r  r8   r  r   r2   r2   r3   r    s    z/ReciprocalTrigonometricFunction._eval_is_finiter   c                 C  s   d|  | jd  |||S r   )r  r8   r   r<   r   r   r   r   r2   r2   r3   r     s    z-ReciprocalTrigonometricFunction._eval_nseries)rw   )T)r   ) rX   rY   rZ   r[   r  r   r\   r]   r  __annotations__r  r%  r   r  r  r  rW   r   r   r   r2  r   r   r   r   r   rD   r  r  r  r  r   r2   r2   r2   r3   r  $  s4   

$

r  c                   @  s   e Zd ZdZeZdZdddZdd Zdd	 Z	d
d Z
dd Zdd Zdd ZdddZdd Zdd Zeedd Zdd ZdS )r   a  
    The secant function.

    Returns the secant of x (measured in radians).

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

    See :class:`sin` for notes about automatic evaluation.

    Examples
    ========

    >>> from sympy import sec
    >>> from sympy.abc import x
    >>> sec(x**2).diff(x)
    2*x*tan(x**2)*sec(x**2)
    >>> sec(1).diff(x)
    0

    See Also
    ========

    sin, csc, cos, tan, cot
    asin, acsc, acos, asec, atan, acot, atan2

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
    .. [2] https://dlmf.nist.gov/4.14
    .. [3] https://functions.wolfram.com/ElementaryFunctions/Sec

    TNc                 C  s
   |  |S r   rW   r   r2   r2   r3   r     s    z
sec.periodc                 K  s    t |d d }|d |d  S r   rf  )r<   r   r   Zcot_half_sqr2   r2   r3   r     s    zsec._eval_rewrite_as_cotc                 K  s   dt | S r   r   r   r2   r2   r3   r     s    zsec._eval_rewrite_as_cosc                 K  s   t |t|t |  S r   r   r   r2   r2   r3   r     s    zsec._eval_rewrite_as_sincosc                 K  s   dt |jtfi | S r   )r   r   r   r   r2   r2   r3   r2    s    zsec._eval_rewrite_as_sinc                 K  s   dt |jtfi | S r   )r   r   r   r   r2   r2   r3   r     s    zsec._eval_rewrite_as_tanc                 K  s   t td | ddS r   )r   r   r   r2   r2   r3   r     s    zsec._eval_rewrite_as_cscrw   c                 C  s2   |dkr$t | jd t| jd  S t| |d S r   )r   r8   r   r   r   r2   r2   r3   r     s    z	sec.fdiffc                 K  sB   ddl m} tdtt| td |tj |  t|dfdS )Nr   r   rw   rm   rB  rC  r   r2   r2   r3   r     s
    .zsec._eval_rewrite_as_besseljc                 C  s,   | j d }|jr(|t tj jdu r(dS d S r?   )r8   r#  r   r   rp   rq   r  r2   r2   r3   r$    s    
zsec._eval_is_complexc                 G  s\   | dk s| d dkrt jS t|}| d }t j| td|  td|  |d|   S d S rN  )r   rJ   r   r   r   r   r   r   r   kr2   r2   r3   r     s
    zsec.taylor_termc           
      C  s   ddl m} ddlm} | jd }||d }|td  t }|jrp||t  td  	|}	t
j| |	 S |t
ju r|j|d||jrdndd}|t
jt
jfv r|t
jt
jS |jr| |S | S )Nr   r   rp  rm   r  r  r  r   r   rr  r    r8   r   r  r   rq   r  r   r   r\   r  r  r   r   r  r7   rs  r2   r2   r3   r    s    

zsec._eval_as_leading_term)N)rw   )rX   rY   rZ   r[   r   r  r  r   r   r   r   r2  r   r   r   r   r$  r&  r   r   r  r2   r2   r2   r3   r     s"   #


r   c                   @  s   e Zd ZdZeZdZdddZdd Zdd	 Z	d
d Z
dd Zdd Zdd Zdd ZdddZdd Zeedd Zdd ZdS )r   a  
    The cosecant function.

    Returns the cosecant of x (measured in radians).

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

    See :func:`sin` for notes about automatic evaluation.

    Examples
    ========

    >>> from sympy import csc
    >>> from sympy.abc import x
    >>> csc(x**2).diff(x)
    -2*x*cot(x**2)*csc(x**2)
    >>> csc(1).diff(x)
    0

    See Also
    ========

    sin, cos, sec, tan, cot
    asin, acsc, acos, asec, atan, acot, atan2

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
    .. [2] https://dlmf.nist.gov/4.14
    .. [3] https://functions.wolfram.com/ElementaryFunctions/Csc

    TNc                 C  s
   |  |S r   r  r   r2   r2   r3   r   /  s    z
csc.periodc                 K  s   dt | S r   rd  r   r2   r2   r3   r2  2  s    zcsc._eval_rewrite_as_sinc                 K  s   t |t|t |  S r   r{  r   r2   r2   r3   r   5  s    zcsc._eval_rewrite_as_sincosc                 K  s    t |d }d|d  d|  S r   rf  r   r2   r2   r3   r   8  s    zcsc._eval_rewrite_as_cotc                 K  s   dt |jtfi | S r   )r   r   r   r   r2   r2   r3   r   <  s    zcsc._eval_rewrite_as_cosc                 K  s   t td | ddS r   r   r   r2   r2   r3   r   ?  s    zcsc._eval_rewrite_as_secc                 K  s   dt |jtfi | S r   )r   r   r   r   r2   r2   r3   r   B  s    zcsc._eval_rewrite_as_tanc                 K  s0   ddl m} tdt dt||tj|   S )Nr   r   rm   rw   r   r   r2   r2   r3   r   E  s    zcsc._eval_rewrite_as_besseljrw   c                 C  s4   |dkr&t | jd  t| jd  S t| |d S r   )r   r8   r   r   r   r2   r2   r3   r   I  s    z	csc.fdiffc                 C  s&   | j d }|jr"|t jdu r"dS d S r?   r}  r  r2   r2   r3   r$  O  s    
zcsc._eval_is_complexc                 G  s   | dkrdt | S | dk s(| d dkr.tjS t |}| d d }tj|d  d dd| d  d  td|  |d| d   td|  S d S rz  )r   r   rJ   r   r   r   r  r2   r2   r3   r   T  s    $

zcsc.taylor_termc           
      C  s   ddl m} ddlm} | jd }||d }|t }|jr`||t  	|}	t
j| |	 S |t
ju r|j|d||jrdndd}|t
jt
jfv r|t
jt
jS |jr| |S | S )Nr   r   rp  r  r  r  r  rs  r2   r2   r3   r  a  s    

zcsc._eval_as_leading_term)N)rw   )rX   rY   rZ   r[   r   r  r  r   r2  r   r   r   r   r   r   r   r$  r&  r   r   r  r2   r2   r2   r3   r     s"   #

r   c                   @  s\   e Zd ZdZejfZdddZedd Z	ddd	Z
d
d Zdd Zdd Zdd ZeZdS )r   a  
    Represents an unnormalized sinc function:

    .. math::

        \operatorname{sinc}(x) =
        \begin{cases}
          \frac{\sin x}{x} & \qquad x \neq 0 \\
          1 & \qquad x = 0
        \end{cases}

    Examples
    ========

    >>> from sympy import sinc, oo, jn
    >>> from sympy.abc import x
    >>> sinc(x)
    sinc(x)

    * Automated Evaluation

    >>> sinc(0)
    1
    >>> sinc(oo)
    0

    * Differentiation

    >>> sinc(x).diff()
    cos(x)/x - sin(x)/x**2

    * Series Expansion

    >>> sinc(x).series()
    1 - x**2/6 + x**4/120 + O(x**6)

    * As zero'th order spherical Bessel Function

    >>> sinc(x).rewrite(jn)
    jn(0, x)

    See also
    ========

    sin

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Sinc_function

    rw   c                 C  s<   | j d }|dkr.t|| t||d   S t| |d S rz  )r8   r   r   r   )r<   r   r   r2   r2   r3   r     s    
z
sinc.fdiffc                 C  s   |j rtjS |jr8|tjtjfv r(tjS |tju r8tjS |tju rHtjS |	 rZ| | S t
|}|d ur|jrt|j rtjS nd| jrtj|tj  | S d S r   )r:   r   r{   r   r   r   rJ   r   r\   r   r@   rq   r
   r   rp   )r   r   rA   r2   r2   r3   r     s$    




z	sinc.evalr   c                 C  s    | j d }t|| |||S r   )r8   r   r   r  r2   r2   r3   r     s    
zsinc._eval_nseriesc                 K  s   ddl m} |d|S )Nr   )jn)r   r  )r<   r   r   r  r2   r2   r3   _eval_rewrite_as_jn  s    zsinc._eval_rewrite_as_jnc                 K  s&   t t|| t|tjftjtjfS r   )r'   r   r   r   rJ   r{   truer   r2   r2   r3   r2    s    zsinc._eval_rewrite_as_sinc                 C  sL   | j d jrdS t| j d \}}|jr8t|j|jgS |jrH|jrHdS d S )Nr   TF)r8   is_infiniterv   r:   r   rq   Z
is_nonzeror   r  r2   r2   r3   r!    s    zsinc._eval_is_zeroc                 C  s    | j d js| j d jrdS d S r  )r8   rH   rw  r   r2   r2   r3   rv    s    zsinc._eval_is_realN)rw   )r   )rX   rY   rZ   r[   r   r\   r]   r   r%  r   r   r  r2  r!  rv  r  r2   r2   r2   r3   r   q  s   4


	r   c                   @  s^   e Zd ZU dZejejejejfZ	de
d< eedd Zeedd Zeedd	 Zd
S )InverseTrigonometricFunctionz/Base class for inverse trigonometric functions.ztuple[Expr, ...]r]   c                -   C  s  t dd td t dd td dt d td t dt d d td t dt dt d  d td t dt d d ttdd t dt dt d  d ttdd tjtd t dt d d td t tjt dd  td t dt d d ttdd t tjt dd  ttdd t dd d td dt d d t d t dd d ttdd t dd t dd  td	 t d d t dd  t d	 t dd t d td	 dt d t d t d	 t dd t dd  ttdd	 dt d t d ttdd	 iS )
Nr^   rm   r_   rw   r`   rc   ra   rb   rh   )r$   r   r   r   rp   r2   r2   r2   r3   _asin_table  s,     &
  "z(InverseTrigonometricFunction._asin_tablec                   C  s  t dd td dt d td t dtd t dd td dt d t d dt d ttdd t ddt d  td t ddt d  ttdd t ddt d d  td t ddt d d  ttdd dt d td d	t d t d dt d ttdd iS )
Nr^   ra   rw   rm   rc   r`   rb   rh   r   r$   r   r   r2   r2   r2   r3   _atan_table  s    "z(InverseTrigonometricFunction._atan_tablec                %   C  s  dt d d td t dtd t ddt d d  td dt tddt dd   td t ddt d d  ttdd dt tddt dd   ttdd dtd t ddt d  td dt dt d  td t ddt d  ttdd dt dt d  ttdd dt d td t dd ttdd t dd  ttd	d t dt d td
 t dt d ttdd
 t dt d  ttdd
 iS )Nrm   r^   r_   r`   rw   rc   ra   rb   rh   r  r2   r2   r2   r3   _acsc_table$  s$    ""(z(InverseTrigonometricFunction._acsc_tableN)rX   rY   rZ   r[   r   r{   r   rJ   r\   r]   r  r&  r   r  r  r  r2   r2   r2   r3   r    s   
r  c                      s   e Zd ZdZd$ddZdd Zdd Zd	d
 Zedd Z	e
edd Zdd Zd% fdd	Zdd Zdd Zdd ZeZdd Zdd Zdd Zd d! Zd&d"d#Z  ZS )'r   ad  
    The inverse sine function.

    Returns the arcsine of x in radians.

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

    ``asin(x)`` will evaluate automatically in the cases
    $x \in \{\infty, -\infty, 0, 1, -1\}$ and for some instances when the
    result is a rational multiple of $\pi$ (see the ``eval`` class method).

    A purely imaginary argument will lead to an asinh expression.

    Examples
    ========

    >>> from sympy import asin, oo
    >>> asin(1)
    pi/2
    >>> asin(-1)
    -pi/2
    >>> asin(-oo)
    oo*I
    >>> asin(oo)
    -oo*I

    See Also
    ========

    sin, csc, cos, sec, tan, cot
    acsc, acos, asec, atan, acot, atan2

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
    .. [2] https://dlmf.nist.gov/4.23
    .. [3] https://functions.wolfram.com/ElementaryFunctions/ArcSin

    rw   c                 C  s0   |dkr"dt d| jd d   S t| |d S Nrw   r   rm   r$   r8   r   r   r2   r2   r3   r   i  s    z
asin.fdiffc                 C  s4   | j | j }|j | j kr*|jd jr0dS n|jS d S r6   r7   r8   r9   r;   r2   r2   r3   r>   o  s
    zasin._eval_is_rationalc                 C  s   |   o| jd jS r   )r  r8   is_positiver   r2   r2   r3   _eval_is_positivew  s    zasin._eval_is_positivec                 C  s   |   o| jd jS r   )r  r8   r  r   r2   r2   r3   _eval_is_negativez  s    zasin._eval_is_negativec                 C  s  |j rt|tju rtjS |tju r,tjtj S |tju rBtjtj S |jrNtjS |tju r`t	d S |tj
u rtt	 d S |tju rtjS | r| |  S |jr|  }||v r|| S t|}|d urddlm} tj|| S |jrtjS t|tr\|jd }|jr\|dt	 ; }|t	kr(t	| }|t	d kr>t	| }|t	 d k rXt	 | }|S t|tr|jd }|jrt	d t| S d S )Nrm   r   )asinh)r   r   r   r   r   r0   r:   rJ   r{   r   r   r\   r   	is_numberr  r4   r   r  r/   r   r8   is_comparabler   r   )r   r   
asin_tabler   r  angr2   r2   r3   r   }  sT    










z	asin.evalc                 G  s   | dk s| d dkrt jS t|}t|dkrb| dkrb|d }|| d d  | | d   |d  S | d d }tt j|}t|}|| ||   |  S d S r   )r   rJ   r   r   r   rp   r   r   r   r   r   r  RrP  r2   r2   r3   r     s    $zasin.taylor_termc                 C  s   | j d }||d }|tju r4| ||S |jrD||S |tj tjtj	fv rt| 
tj|||d S d|d  jr|||r|nd}t|jr|jrt | | S n:t|jr|jrt| | S n| 
tj|||d S | |S Nr   r  rw   rm   )r8   r   r  r   r   r7   r  r:   r{   r\   r   r!   r  rI   r  r  r   r   r  r<   r   r   r   r   r  ndirr2   r2   r3   r    s$    




zasin._eval_as_leading_termr   c                   s  ddl m} | jd |d}|tju rtddd}ttj|d  t	
|dd| }tj| jd  }	|	|}
|	|
 |
 }||ds|dkr|dS td |t| S ttj| j|||d}| t|
  }| ||  ||| | S |tju rtddd}ttj|d  t	
|dd| }tj| jd  }	|	|}
|	|
 |
 }||ds|dkr|dS t d |t| S ttj| j|||d}| t|
  }| ||  ||| | S t j|||d}|tju r|S d|d  jr| jd ||r<|nd}t|jrb|jrt | S n6t|jr|jrt| S n| t	j||||d	S |S 
Nr   Or  TZpositiverm   rw   rQ  r  )sympy.series.orderr  r8   r   r   r{   r   r   r   r!   nseriesr  is_meromorphicr   r$   r   removeOrI   powsimpr   r   r\   r  r  r   r  r<   r   r   r   r   r  arg0r  ZserZarg1rS   rT   Zres1resr  r   r2   r3   r     sJ    &
$&&
(&
zasin._eval_nseriesc                 K  s   t d t| S r   r   r   re  r2   r2   r3   _eval_rewrite_as_acos	  s    zasin._eval_rewrite_as_acosc                 K  s    dt |dtd|d     S r   )r   r$   re  r2   r2   r3   _eval_rewrite_as_atan	  s    zasin._eval_rewrite_as_atanc                 K  s&   t j tt j| td|d    S rF  r   r0   r!   r$   re  r2   r2   r3   _eval_rewrite_as_log	  s    zasin._eval_rewrite_as_logc                 K  s    dt dtd|d   |  S r   )r   r$   r   r2   r2   r3   _eval_rewrite_as_acot	  s    zasin._eval_rewrite_as_acotc                 K  s   t d td|  S r   r   r   r   r2   r2   r3   _eval_rewrite_as_asec	  s    zasin._eval_rewrite_as_asecc                 K  s   t d| S r   )r   r   r2   r2   r3   _eval_rewrite_as_acsc	  s    zasin._eval_rewrite_as_acscc                 C  s   | j d }|jodt| jS Nr   rw   r8   rH   rO   is_nonnegativer<   r   r2   r2   r3   r  	  s    
zasin._eval_is_extended_realc                 C  s   t S rG  rd  r   r2   r2   r3   rI   	  s    zasin.inverse)rw   )r   )rw   )rX   rY   rZ   r[   r   r>   r  r  r%  r   r&  r   r   r  r   r  r  r  _eval_rewrite_as_tractabler  r  r  r  rI  r'  r2   r2   r   r3   r   >  s*   *

6,r   c                      s   e Zd ZdZd$ddZdd Zedd Zee	d	d
 Z
dd Zdd Zdd Zd% fdd	Zdd ZeZdd Zdd Zd&ddZdd Zdd Zd d! Zd"d# Z  ZS )'r   a  
    The inverse cosine function.

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

    Returns the arc cosine of x (measured in radians).

    ``acos(x)`` will evaluate automatically in the cases
    $x \in \{\infty, -\infty, 0, 1, -1\}$ and for some instances when
    the result is a rational multiple of $\pi$ (see the eval class method).

    ``acos(zoo)`` evaluates to ``zoo``
    (see note in :class:`sympy.functions.elementary.trigonometric.asec`)

    A purely imaginary argument will be rewritten to asinh.

    Examples
    ========

    >>> from sympy import acos, oo
    >>> acos(1)
    0
    >>> acos(0)
    pi/2
    >>> acos(oo)
    oo*I

    See Also
    ========

    sin, csc, cos, sec, tan, cot
    asin, acsc, asec, atan, acot, atan2

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
    .. [2] https://dlmf.nist.gov/4.23
    .. [3] https://functions.wolfram.com/ElementaryFunctions/ArcCos

    rw   c                 C  s0   |dkr"dt d| jd d   S t| |d S Nrw   r   r   rm   r  r   r2   r2   r3   r   S	  s    z
acos.fdiffc                 C  s4   | j | j }|j | j kr*|jd jr0dS n|jS d S r6   r  r;   r2   r2   r3   r>   Y	  s
    zacos._eval_is_rationalc                 C  s  |j rn|tju rtjS |tju r,tjtj S |tju rBtjtj S |jrPtd S |tju r`tj	S |tj
u rntS |tju r~tjS |jr|  }||v rtd ||  S | |v rtd ||   S t|}|d urtd t| S |jrt|jdkr|jd dkr|jd }d}n|}d}t|trr|jd }|jrr|rLt| }|dt ; }|tkrndt | }|S t|tr|jd }|jr|rtd t| S td t| S d S Nrm   r   r   rw   TF)r   r   r   r   r0   r   r:   r   r{   rJ   r   r\   r  r  r4   r   rN   r   r8   r/   r   r  r   )r   r   r  r   r   minusr  r2   r2   r3   r   a	  sX    






(



z	acos.evalc                 G  s   | dkrt d S | dk s$| d dkr*tjS t|}t|dkrr| dkrr|d }|| d d  | | d   |d  S | d d }ttj|}t|}| | ||   |  S d S r   )r   r   rJ   r   r   r   rp   r   r  r2   r2   r3   r   	  s    $zacos.taylor_termc                 C  s  | j d }||d }|tju r4| ||S |dkrXtdttj| | S |tj tj	fv r| 
tj|||dS d|d  jr|||r|nd}t|jr|jrdt | | S n8t|jr|jr| | S n| 
tj|||d S | |S Nr   rw   rm   r  )r8   r   r  r   r   r7   r  r$   r{   r\   r   r!   r  r  r  r   r   r  rI   r  r2   r2   r3   r  	  s$    



zacos._eval_as_leading_termc                 C  s   | j d }|jodt| jS r  r  r  r2   r2   r3   r  	  s    
zacos._eval_is_extended_realc                 C  s   |   S r   )r  r   r2   r2   r3   _eval_is_nonnegative	  s    zacos._eval_is_nonnegativer   c                   s  ddl m} | jd |d}|tju rtddd}ttj|d  t	
|dd| }tj| jd  }	|	|}
|	|
 |
 }||ds|dkr|dS |t|S ttj| j|||d}| t|
  }| ||  ||| | S |tju rtddd}ttj|d  t	
|dd| }tj| jd  }	|	|}
|	|
 |
 }||ds|dkr|dS t|t| S ttj| j|||d}| t|
  }| ||  ||| | S t j|||d}|tju r|S d|d  jr| jd ||r,|nd}t|jrT|jrdt | S n4t|jrp|jr| S n| t	j||||d	S |S r  )r  r  r8   r   r   r{   r   r   r   r!   r  r  r  r$   r   r  rI   r  r   r   r   r\   r  r  r   r  r  r   r2   r3   r   	  sJ    
&
&&
"&zacos._eval_nseriesc                 K  s,   t d tjttj| td|d     S r   r   r   r0   r!   r$   re  r2   r2   r3   r  	  s    
zacos._eval_rewrite_as_logc                 K  s   t d t| S r   r   r   re  r2   r2   r3   _eval_rewrite_as_asin	  s    zacos._eval_rewrite_as_asinc                 K  s8   t td|d  | td d|td|d      S rF  )r   r$   r   re  r2   r2   r3   r  	  s    zacos._eval_rewrite_as_atanc                 C  s   t S rG  r  r   r2   r2   r3   rI   
  s    zacos.inversec                 K  s(   t d dtdtd|d   |   S r   )r   r   r$   r   r2   r2   r3   r  
  s    zacos._eval_rewrite_as_acotc                 K  s   t d| S r   )r   r   r2   r2   r3   r  	
  s    zacos._eval_rewrite_as_asecc                 K  s   t d td|  S r   r   r   r   r2   r2   r3   r  
  s    zacos._eval_rewrite_as_acscc                 C  sN   | j d }| | j d  }|jdu r,|S |jrJ|d jrJ|d jrJ|S d S Nr   Frw   )r8   r7   r   rH   r  Zis_nonpositive)r<   r:  rr2   r2   r3   r   
  s    

zacos._eval_conjugate)rw   )r   )rw   )rX   rY   rZ   r[   r   r>   r%  r   r&  r   r   r  r  r  r   r  r  r  r  rI  r  r  r  r   r'  r2   r2   r   r3   r   '	  s*   +

6,
r   c                      s   e Zd ZU dZded< ejej fZd*ddZdd Z	d	d
 Z
dd Zdd Zdd Zedd Zeedd Zdd Zd+ fdd	Zdd ZeZ fddZd,ddZd d! Zd"d# Zd$d% Zd&d' Zd(d) Z  ZS )-r   a  
    The inverse tangent function.

    Returns the arc tangent of x (measured in radians).

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

    ``atan(x)`` will evaluate automatically in the cases
    $x \in \{\infty, -\infty, 0, 1, -1\}$ and for some instances when the
    result is a rational multiple of $\pi$ (see the eval class method).

    Examples
    ========

    >>> from sympy import atan, oo
    >>> atan(0)
    0
    >>> atan(1)
    pi/4
    >>> atan(oo)
    pi/2

    See Also
    ========

    sin, csc, cos, sec, tan, cot
    asin, acsc, acos, asec, acot, atan2

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
    .. [2] https://dlmf.nist.gov/4.23
    .. [3] https://functions.wolfram.com/ElementaryFunctions/ArcTan

    ztuple[Expr]r8   rw   c                 C  s,   |dkrdd| j d d   S t| |d S r  r8   r   r   r2   r2   r3   r   C
  s    z
atan.fdiffc                 C  s4   | j | j }|j | j kr*|jd jr0dS n|jS d S r6   r  r;   r2   r2   r3   r>   I
  s
    zatan._eval_is_rationalc                 C  s   | j d jS r   )r8   Zis_extended_positiver   r2   r2   r3   r  Q
  s    zatan._eval_is_positivec                 C  s   | j d jS r   )r8   Zis_extended_nonnegativer   r2   r2   r3   r  T
  s    zatan._eval_is_nonnegativec                 C  s   | j d jS r   )r8   r:   r   r2   r2   r3   r!  W
  s    zatan._eval_is_zeroc                 C  s   | j d jS r   r  r   r2   r2   r3   rv  Z
  s    zatan._eval_is_realc                 C  s  |j rn|tju rtjS |tju r(td S |tju r<t d S |jrHtjS |tju rZtd S |tj	u rnt d S |tj
u rddlm} |t d td S | r| |  S |jr|  }||v r|| S t|}|d urddlm} tj|| S |jrtjS t|trB|jd }|jrB|t; }|td kr>|t8 }|S t|tr|jd }|jrtd t| }|td kr|t8 }|S d S )Nrm   r_   r   r   )atanh)r   r   r   r   r   r   r:   rJ   r{   r   r\   r   r   r   r  r  r4   r   r  r0   r/   r   r8   r  r   r   )r   r   r   
atan_tabler   r  r  r2   r2   r3   r   ]
  sT    









z	atan.evalc                 G  sD   | dk s| d dkrt jS t|}t j| d d  ||   |  S d S rN  )r   rJ   r   r   r   r   r   r2   r2   r3   r   
  s    zatan.taylor_termc                 C  s  | j d }||d }|tju r4| ||S |jrD||S |tj tjtj	fv rt| 
tj|||d S d|d  jr|||r|nd}t|jrt|jr| |t S n>t|jrt|jr| |t S n| 
tj|||d S | |S r  )r8   r   r  r   r   r7   r  r:   r0   r\   r   r!   r  rI   r  r  r    r   r  r   r  r2   r2   r3   r  
  s$    






zatan._eval_as_leading_termr   c                   s   | j d |d}|tjtjtj fv r@| tj||||dS t j|||d}| j d 	||rf|nd}|tj
u rt|dkr|t S |S d|d  jrt|jrt|jr|t S n6t|jrt|jr|t S n| tj||||dS |S Nr   r  rQ  rw   rm   )r8   r   r   r0   r   r   r!   r   r   r  r\   r    r   r  r   r  r<   r   r   r   r   r  r  r  r   r2   r3   r   
  s$    






zatan._eval_nseriesc                 K  s2   t jd tt jt j|  tt jt j|    S r   )r   r0   r!   r{   re  r2   r2   r3   r  
  s    zatan._eval_rewrite_as_logc                   sN   |d t jt jfv r8td td| jd   |||S t ||||S d S rN  )	r   r   r   r   r   r8   r   r   _eval_aseriesr<   r   Zargs0r   r   r   r2   r3   r  
  s    $zatan._eval_aseriesc                 C  s   t S rG  r|  r   r2   r2   r3   rI  
  s    zatan.inversec                 K  s0   t |d | td tdt d|d     S r   r$   r   r   r   r2   r2   r3   r  
  s    zatan._eval_rewrite_as_asinc                 K  s(   t |d | tdt d|d    S r   r$   r   r   r2   r2   r3   r  
  s    zatan._eval_rewrite_as_acosc                 K  s   t d| S r   rx  r   r2   r2   r3   r  
  s    zatan._eval_rewrite_as_acotc                 K  s$   t |d | tt d|d   S r   r$   r   r   r2   r2   r3   r  
  s    zatan._eval_rewrite_as_asecc                 K  s,   t |d | td tt d|d    S r   r$   r   r   r   r2   r2   r3   r  
  s    zatan._eval_rewrite_as_acsc)rw   )r   )rw   )rX   rY   rZ   r[   r  r   r0   r]   r   r>   r  r  r!  rv  r%  r   r&  r   r   r  r   r  r  r  rI  r  r  r  r  r  r'  r2   r2   r   r3   r   
  s2   
&

4
r   c                      s   e Zd ZdZejej fZd&ddZdd Zdd Z	d	d
 Z
dd Zedd Zeedd Zdd Zd' fdd	Z fddZdd ZeZd(ddZdd Zdd Zd d! Zd"d# Zd$d% Z  ZS ))r   a  
    The inverse cotangent function.

    Returns the arc cotangent of x (measured in radians).

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

    ``acot(x)`` will evaluate automatically in the cases
    $x \in \{\infty, -\infty, \tilde{\infty}, 0, 1, -1\}$
    and for some instances when the result is a rational multiple of $\pi$
    (see the eval class method).

    A purely imaginary argument will lead to an ``acoth`` expression.

    ``acot(x)`` has a branch cut along $(-i, i)$, hence it is discontinuous
    at 0. Its range for real $x$ is $(-\frac{\pi}{2}, \frac{\pi}{2}]$.

    Examples
    ========

    >>> from sympy import acot, sqrt
    >>> acot(0)
    pi/2
    >>> acot(1)
    pi/4
    >>> acot(sqrt(3) - 2)
    -5*pi/12

    See Also
    ========

    sin, csc, cos, sec, tan, cot
    asin, acsc, acos, asec, atan, atan2

    References
    ==========

    .. [1] https://dlmf.nist.gov/4.23
    .. [2] https://functions.wolfram.com/ElementaryFunctions/ArcCot

    rw   c                 C  s,   |dkrdd| j d d   S t| |d S r  r  r   r2   r2   r3   r     s    z
acot.fdiffc                 C  s4   | j | j }|j | j kr*|jd jr0dS n|jS d S r6   r  r;   r2   r2   r3   r>      s
    zacot._eval_is_rationalc                 C  s   | j d jS r   )r8   r  r   r2   r2   r3   r  (  s    zacot._eval_is_positivec                 C  s   | j d jS r   )r8   r  r   r2   r2   r3   r  +  s    zacot._eval_is_negativec                 C  s   | j d jS r   r  r   r2   r2   r3   r  .  s    zacot._eval_is_extended_realc                 C  s  |j rj|tju rtjS |tju r&tjS |tju r6tjS |jrDtd S |tju rVtd S |tj	u rjt d S |tj
u rztjS | r| |  S |jr|  }||v rtd ||  }|td kr|t8 }|S t|}|d u rddlm} tj || S |jrttj S t|trL|jd }|jrL|t; }|td krH|t8 }|S t|tr|jd }|jrtd t| }|td kr|t8 }|S d S )Nrm   r_   r   )acoth)r   r   r   r   rJ   r   r:   r   r{   r   r\   r   r  r  r4   r   r  r0   rp   r/   r   r8   r  r   r   )r   r   r  r  r   r  r2   r2   r3   r   1  sX    










z	acot.evalc                 G  sT   | dkrt d S | dk s$| d dkr*tjS t|}tj| d d  ||   |  S d S rN  )r   r   rJ   r   r   r  r2   r2   r3   r   g  s    zacot.taylor_termc                 C  s  | j d }||d }|tju r4| ||S |tju rLd| |S |tj tjtj	fv r|| 
tj|||d S |jr
d|d  jr
|||r|nd}t|jrt|jr| |t S n>t|jrt|jr| |t S n| 
tj|||d S | |S )Nr   rw   r  rm   )r8   r   r  r   r   r7   r  r\   r0   rJ   r   r!   r  rI   rw  r  r  r    r   r   r  r  r2   r2   r3   r  r  s$    






zacot._eval_as_leading_termr   c                   s  | j d |d}|tjtjtj fv r@| tj||||dS t j|||d}|tj	u r`|S | j d 
||rt|nd}|jrt|dk r|t S |S |jrd|d  jrt|jrt|jr|t S n6t|jrt|jr|t S n| tj||||dS |S r  )r8   r   r   r0   r   r   r!   r   r   r\   r  r:   r    r   rw  r  r   r  r  r   r2   r3   r     s(    






zacot._eval_nseriesc                   sF   |d t jt jfv r0td| jd  |||S t ||||S d S r  )r   r   r   r   r8   r   r   r  r  r   r2   r3   r    s    zacot._eval_aseriesc                 K  s.   t jd tdt j|  tdt j|    S r   )r   r0   r!   re  r2   r2   r3   r    s    zacot._eval_rewrite_as_logc                 C  s   t S rG  rf  r   r2   r2   r3   rI    s    zacot.inversec                 K  s@   |t d|d   td tt |d  t |d  d    S rF  r  r   r2   r2   r3   r    s    *zacot._eval_rewrite_as_asinc                 K  s8   |t d|d   tt |d  t |d  d   S rF  r  r   r2   r2   r3   r    s    zacot._eval_rewrite_as_acosc                 K  s   t d| S r   rH  r   r2   r2   r3   r    s    zacot._eval_rewrite_as_atanc                 K  s0   |t d|d   tt d|d  |d   S rF  r  r   r2   r2   r3   r    s    zacot._eval_rewrite_as_asecc                 K  s8   |t d|d   td tt d|d  |d    S rF  r  r   r2   r2   r3   r    s    zacot._eval_rewrite_as_acsc)rw   )r   )rw   )rX   rY   rZ   r[   r   r0   r]   r   r>   r  r  r  r%  r   r&  r   r   r  r   r  r  r  rI  r  r  r  r  r  r'  r2   r2   r   r3   r   
  s.   *

5	
r   c                      s   e Zd ZdZedd ZdddZdddZee	d	d
 Z
dd Zd  fdd	Zdd Zdd ZeZdd Zdd Zdd Zdd Zdd Z  ZS )!r   a  
    The inverse secant function.

    Returns the arc secant of x (measured in radians).

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

    ``asec(x)`` will evaluate automatically in the cases
    $x \in \{\infty, -\infty, 0, 1, -1\}$ and for some instances when the
    result is a rational multiple of $\pi$ (see the eval class method).

    ``asec(x)`` has branch cut in the interval $[-1, 1]$. For complex arguments,
    it can be defined [4]_ as

    .. math::
        \operatorname{sec^{-1}}(z) = -i\frac{\log\left(\sqrt{1 - z^2} + 1\right)}{z}

    At ``x = 0``, for positive branch cut, the limit evaluates to ``zoo``. For
    negative branch cut, the limit

    .. math::
        \lim_{z \to 0}-i\frac{\log\left(-\sqrt{1 - z^2} + 1\right)}{z}

    simplifies to :math:`-i\log\left(z/2 + O\left(z^3\right)\right)` which
    ultimately evaluates to ``zoo``.

    As ``acos(x) = asec(1/x)``, a similar argument can be given for
    ``acos(x)``.

    Examples
    ========

    >>> from sympy import asec, oo
    >>> asec(1)
    0
    >>> asec(-1)
    pi
    >>> asec(0)
    zoo
    >>> asec(-oo)
    pi/2

    See Also
    ========

    sin, csc, cos, sec, tan, cot
    asin, acsc, acos, atan, acot, atan2

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
    .. [2] https://dlmf.nist.gov/4.23
    .. [3] https://functions.wolfram.com/ElementaryFunctions/ArcSec
    .. [4] https://reference.wolfram.com/language/ref/ArcSec.html

    c                 C  s  |j rtjS |jr@|tju r"tjS |tju r2tjS |tju r@tS |tj	tj
tjfv r\td S |jr|  }||v rtd ||  S | |v rtd ||   S |jrtd S |jrt|jdkr|jd dkr|jd }d}n|}d}t|tr8|jd }|jr8|rt| }|dt ; }|tkr4dt | }|S t|tr||jd }|jr||rltd t|  td t| S d S r  )r:   r   r\   r   r   r{   rJ   r   r   r   r   r  r  r  rN   r   r8   r/   r   r  r   r   )r   r   
acsc_tabler   r  r  r2   r2   r3   r     sN    



"



z	asec.evalrw   c                 C  sB   |dkr4d| j d d tdd| j d d     S t| |d S r  r8   r$   r   r   r2   r2   r3   r   4  s    ,z
asec.fdiffc                 C  s   t S rG  rA  r   r2   r2   r3   rI  :  s    zasec.inversec                 G  s   | dkrt jtd|  S | dk s.| d dkr4t jS t|}t|dkr| dkr|d }|| d | d   |d  d| d d   S | d }tt j||  }t||  d |  d }t j | | ||   d S d S Nr   rm   rw   r   r_   )	r   r0   r!   rJ   r   r   r   rp   r   r  r2   r2   r3   r   @  s    ,zasec.taylor_termc                 C  s  | j d }||d }|tju r4| ||S |dkrXtdt|tj | S |tj tj	fv r| 
tj|||dS |jrd|d  jr|||r|nd}t|jr|jr| | S n>t|jr|jrdt | | S n| 
tj|||d S | |S r  )r8   r   r  r   r   r7   r  r$   r{   rJ   r   r!   r  ru  r  r  r   r  r   rI   r  r2   r2   r3   r  R  s$    



zasec._eval_as_leading_termr   c                   s<  ddl m} | jd |d}|tju rtddd}ttj|d  t	
|dd| }tj| jd  }	|	|}
|	|
 |
 }ttj| j|||d}| t|
  }| ||  ||| | S |tju rtddd}ttj|d  t	
|dd| }tj| jd  }	|	|}
|	|
 |
 }ttj| j|||d}| t|
  }| ||  ||| | S t j|||d}|tju r|S |jr8d|d  jr8| jd ||r|nd}t|jr|jr8| S n:t|jr |jr8dt | S n| t	j||||d	S |S 
Nr   r  r  Tr  rm   rQ  rw   r  )r  r  r8   r   r   r{   r   r   r   r!   r  r   r  r$   r   r  rI   r  r   r\   ru  r  r  r   r  r   r  r   r2   r3   r   i  sB    
&
&&
&zasec._eval_nseriesc                 C  s2   | j d }|jdu rdS t|d j| d jfS r  )r8   rH   r   r  r  r2   r2   r3   r    s    

zasec._eval_is_extended_realc              	   K  s0   t d tjttj| tdd|d      S r   r  r   r2   r2   r3   r    s    zasec._eval_rewrite_as_logc                 K  s   t d td|  S r   r  r   r2   r2   r3   r    s    zasec._eval_rewrite_as_asinc                 K  s   t d| S r   )r   r   r2   r2   r3   r    s    zasec._eval_rewrite_as_acosc                 K  s8   t |d | }td d|  |tt |d d   S r   r$   r   r   r<   r   r   Zsx2xr2   r2   r3   r    s    zasec._eval_rewrite_as_atanc                 K  s<   t |d | }td d|  |tdt |d d    S r   r$   r   r   r  r2   r2   r3   r    s    zasec._eval_rewrite_as_acotc                 K  s   t d t| S r   r  r   r2   r2   r3   r    s    zasec._eval_rewrite_as_acsc)rw   )rw   )r   )rX   rY   rZ   r[   r%  r   r   rI  r&  r   r   r  r   r  r  r  r  r  r  r  r  r'  r2   r2   r   r3   r     s$   ;
0

(r   c                      s   e Zd ZdZedd ZdddZdddZee	d	d
 Z
dd Zd fdd	Zdd ZeZdd Zdd Zdd Zdd Zdd Z  ZS )r   aV  
    The inverse cosecant function.

    Returns the arc cosecant of x (measured in radians).

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

    ``acsc(x)`` will evaluate automatically in the cases
    $x \in \{\infty, -\infty, 0, 1, -1\}$` and for some instances when the
    result is a rational multiple of $\pi$ (see the ``eval`` class method).

    Examples
    ========

    >>> from sympy import acsc, oo
    >>> acsc(1)
    pi/2
    >>> acsc(-1)
    -pi/2
    >>> acsc(oo)
    0
    >>> acsc(-oo) == acsc(oo)
    True
    >>> acsc(0)
    zoo

    See Also
    ========

    sin, csc, cos, sec, tan, cot
    asin, acos, asec, atan, acot, atan2

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
    .. [2] https://dlmf.nist.gov/4.23
    .. [3] https://functions.wolfram.com/ElementaryFunctions/ArcCsc

    c                 C  s>  |j rtjS |jrH|tju r"tjS |tju r4td S |tju rHt d S |tjtj	tjfv rbtj
S | rv| |  S |jrtj
S |jr|  }||v r|| S t|tr|jd }|jr|dt ; }|tkrt| }|td krt| }|t d k rt | }|S t|tr:|jd }|jr:td t| S d S )Nrm   r   )r:   r   r\   r   r   r{   r   r   r   r   rJ   r   r  r  r  r/   r   r8   r  r   r   )r   r   r  r  r2   r2   r3   r     sD    






z	acsc.evalrw   c                 C  sB   |dkr4d| j d d tdd| j d d     S t| |d S r  r  r   r2   r2   r3   r     s    ,z
acsc.fdiffc                 C  s   t S rG  r   r   r2   r2   r3   rI    s    zacsc.inversec                 G  s   | dkr,t d tjtd  tjt|  S | dk s@| d dkrFtjS t|}t|dkr| dkr|d }|| d | d   |d  d| d d   S | d }ttj||  }t	||  d |  d }tj| | ||   d S d S r  )
r   r   r0   r!   rJ   r   r   r   rp   r   r  r2   r2   r3   r     s    $,zacsc.taylor_termc                 C  s  | j d }||d }|tju r4| ||S |tj tjtjfv rd| 	t
j|||d S |tju r|d| |S |jrd|d  jr|||r|nd}t|jr|jrt| | S n<t|jr|jrt | | S n| 	t
j|||d S | |S r  )r8   r   r  r   r   r7   r  r{   rJ   r   r!   r  rI   r\   ru  r  r  r   r  r   r  r2   r2   r3   r  $  s$    




zacsc._eval_as_leading_termr   c                   s<  ddl m} | jd |d}|tju rtddd}ttj|d  t	
|dd| }tj| jd  }	|	|}
|	|
 |
 }ttj| j|||d}| t|
  }| ||  ||| | S |tju rtddd}ttj|d  t	
|dd| }tj| jd  }	|	|}
|	|
 |
 }ttj| j|||d}| t|
  }| ||  ||| | S t j|||d}|tju r|S |jr8d|d  jr8| jd ||r|nd}t|jr |jr8t| S n8t|jr |jr8t | S n| t	j||||d	S |S r  )r  r  r8   r   r   r{   r   r   r   r!   r  r   r  r$   r   r  rI   r  r   r\   ru  r  r  r   r  r   r  r   r2   r3   r   ;  sB    
&
&&
&
zacsc._eval_nseriesc                 K  s*   t j tt j| tdd|d     S rF  r  r   r2   r2   r3   r  c  s    zacsc._eval_rewrite_as_logc                 K  s   t d| S r   )r   r   r2   r2   r3   r  h  s    zacsc._eval_rewrite_as_asinc                 K  s   t d td|  S r   r  r   r2   r2   r3   r  k  s    zacsc._eval_rewrite_as_acosc                 K  s,   t |d | td tt |d d   S r   r  re  r2   r2   r3   r  n  s    zacsc._eval_rewrite_as_atanc                 K  s0   t |d | td tdt |d d    S r   r  r   r2   r2   r3   r  q  s    zacsc._eval_rewrite_as_acotc                 K  s   t d t| S r   r  r   r2   r2   r3   r  t  s    zacsc._eval_rewrite_as_asec)rw   )rw   )r   )rX   rY   rZ   r[   r%  r   r   rI  r&  r   r   r  r   r  r  r  r  r  r  r  r'  r2   r2   r   r3   r     s"   *
,

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    The function ``atan2(y, x)`` computes `\operatorname{atan}(y/x)` taking
    two arguments `y` and `x`.  Signs of both `y` and `x` are considered to
    determine the appropriate quadrant of `\operatorname{atan}(y/x)`.
    The range is `(-\pi, \pi]`. The complete definition reads as follows:

    .. math::

        \operatorname{atan2}(y, x) =
        \begin{cases}
          \arctan\left(\frac y x\right) & \qquad x > 0 \\
          \arctan\left(\frac y x\right) + \pi& \qquad y \ge 0, x < 0 \\
          \arctan\left(\frac y x\right) - \pi& \qquad y < 0, x < 0 \\
          +\frac{\pi}{2} & \qquad y > 0, x = 0 \\
          -\frac{\pi}{2} & \qquad y < 0, x = 0 \\
          \text{undefined} & \qquad y = 0, x = 0
        \end{cases}

    Attention: Note the role reversal of both arguments. The `y`-coordinate
    is the first argument and the `x`-coordinate the second.

    If either `x` or `y` is complex:

    .. math::

        \operatorname{atan2}(y, x) =
            -i\log\left(\frac{x + iy}{\sqrt{x^2 + y^2}}\right)

    Examples
    ========

    Going counter-clock wise around the origin we find the
    following angles:

    >>> from sympy import atan2
    >>> atan2(0, 1)
    0
    >>> atan2(1, 1)
    pi/4
    >>> atan2(1, 0)
    pi/2
    >>> atan2(1, -1)
    3*pi/4
    >>> atan2(0, -1)
    pi
    >>> atan2(-1, -1)
    -3*pi/4
    >>> atan2(-1, 0)
    -pi/2
    >>> atan2(-1, 1)
    -pi/4

    which are all correct. Compare this to the results of the ordinary
    `\operatorname{atan}` function for the point `(x, y) = (-1, 1)`

    >>> from sympy import atan, S
    >>> atan(S(1)/-1)
    -pi/4
    >>> atan2(1, -1)
    3*pi/4

    where only the `\operatorname{atan2}` function returns what we expect.
    We can differentiate the function with respect to both arguments:

    >>> from sympy import diff
    >>> from sympy.abc import x, y
    >>> diff(atan2(y, x), x)
    -y/(x**2 + y**2)

    >>> diff(atan2(y, x), y)
    x/(x**2 + y**2)

    We can express the `\operatorname{atan2}` function in terms of
    complex logarithms:

    >>> from sympy import log
    >>> atan2(y, x).rewrite(log)
    -I*log((x + I*y)/sqrt(x**2 + y**2))

    and in terms of `\operatorname(atan)`:

    >>> from sympy import atan
    >>> atan2(y, x).rewrite(atan)
    Piecewise((2*atan(y/(x + sqrt(x**2 + y**2))), Ne(y, 0)), (pi, re(x) < 0), (0, Ne(x, 0)), (nan, True))

    but note that this form is undefined on the negative real axis.

    See Also
    ========

    sin, csc, cos, sec, tan, cot
    asin, acsc, acos, asec, atan, acot

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
    .. [2] https://en.wikipedia.org/wiki/Atan2
    .. [3] https://functions.wolfram.com/ElementaryFunctions/ArcTan2

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