a
    RG5d                    @   s  d dl mZ d dlmZ d dlmZmZ d dlm	Z	 d dl
mZmZ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$m%Z%m&Z& d dl'm(Z(m)Z) d dl*m+Z,m-Z-m.Z. d dl/m0Z0m1Z1 d dl2m3Z3 d dl4m5Z5m6Z6m7Z7 d dl8m9Z9 d dl:m;Z; d dl<m=Z= d dl>m?Z? d dl@mAZA dd ZBG dd deZCdd ZDdAddZEG d d! d!eCZFG d"d# d#eCZGG d$d% d%eCZHG d&d' d'eCZIG d(d) d)eCZJG d*d+ d+eJZKG d,d- d-eJZLG d.d/ d/eZMG d0d1 d1eZNG d2d3 d3eNZOG d4d5 d5eNZPG d6d7 d7eNZQG d8d9 d9eNZRG d:d; d;eNZSG d<d= d=eNZTG d>d? d?eNZUd@S )B    )Tuple)Add)sympifycacheit)Expr)FunctionArgumentIndexError	PoleError
expand_mul)	fuzzy_notfuzzy_or	FuzzyBool	fuzzy_and)Mod)igcdexRationalpiIntegerFloat)NeEq)S)SymbolDummy)	factorialRisingFactorial)	bernoullieuler)argimre)logexp)floor)sqrtMinMax)	Piecewise)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   as_coefficientr   ImaginaryUnit)r    r/   d/var/www/html/django/DPS/env/lib/python3.9/site-packages/sympy/functions/elementary/trigonometric.py_imaginary_unit_as_coefficient   s    
r1   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sr/   r/   r0   _eval_is_rational1   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)r4   r5   r   r7   is_algebraic	_pi_coeffr6   )r9   r:   pi_coeffr/   r/   r0   _eval_is_algebraic9   s    z(TrigonometricFunction._eval_is_algebraicc                 K   s&   | j f d|i|\}}||tj  S )Ndeep)as_real_imagr   r.   )r9   rA   hintsre_partim_partr/   r/   r0   _eval_expand_complexD   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)r5   is_extended_realexpandr   ZerorB   )r9   rA   rC   r    r   r/   r/   r0   _as_real_imagH   s    "z#TrigonometricFunction._as_real_imagNc                 C   s   t | jd }|d u r$t|jd }||s4tjS ||kr@|S ||jv r|jrr||\}}||krr|t	| S |j
r||\}}|j|dd\}}||kr|t	| S tdd S )Nr   F)as_Addz%Use the periodicity function instead.)r
   r5   tuplefree_symbolshasr   rJ   is_Mulas_independentabsis_AddNotImplementedError)r9   Zgeneral_periodsymbolfghar/   r/   r0   _periodU   s$    

zTrigonometricFunction._period)T)T)N)__name__
__module____qualname____doc__
unbranchedr   ComplexInfinity_singularitiesr;   r@   rF   rK   rZ   r/   r/   r/   r0   r2   +   s   

r2   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   	make_argscoeffr   r6   appendHalf
is_integeris_even)r   r?   Z
rest_termsrY   Km1m2r/   r/   r0   _peeloff_pio   s    





rl      c                 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	|}	|	|k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

    rm   r   rb   N)r   r   OnerJ   rP   rd   as_coeff_Mulis_FloatrR   introundr!   evalfr   rg   rh   r   r7   )r   cyclescxcxrV   pmcmic2r/   r/   r0   r>      s@    %



r>   c                   @   s   e Zd ZdZd6ddZd7ddZedd	 Zee	d
d Z
d8d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d9d(d)Zd*d+ Zd:d,d-Zd.d/ Zd0d1 Zd2d3 Zd4d5 Zd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] http://dlmf.nist.gov/4.14
    .. [3] http://functions.wolfram.com/ElementaryFunctions/Sin
    .. [4] http://mathworld.wolfram.com/TrigonometryAngles.html

    Nc                 C   s   |  dt |S Nrb   rZ   r   r9   rU   r/   r/   r0   period  s    z
sin.periodrm   c                 C   s$   |dkrt | jd S t| |d S Nrm   r   )cosr5   r   r9   argindexr/   r/   r0   fdiff  s    z	sin.fdiffc                 C   s  ddl m} ddlm} |jrT|tju r.tjS |jr:tjS |tj	tj
fv rT|ddS |tju rdtjS t||rddlm} |j|j }}t|dt  }|tj
ur||d t  }|tj	ur||d t  }||||td ttdd tjur6||||ttd	d ttd
d tjur6|ddS ||||td ttdd tjurz|tt|t|dS ||||ttd	d ttdd tjur|dtt|t|S |tt|t|tt|t|S nt||r|| S | r| |  S t|}|d urDddlm}	 tj|	| S t|}
|
d urH|
j rdtjS d|
 j r|
j!du rtj"|
tj#  S |
j$s|
t }||kr| |S d S |
j$rH|
d }|dkr| |d t  S d| dkr| d| t S |
td	d d t }t%|}t|t%s*|S |
t |krD| |
t S d S |j&rt'|\}}|r|t }t|t%| t%|t|  S |jrtjS t|t(r|j)d S t|t*r|j)d }|t+d|d   S t|t,r|j)\}}|t+|d |d   S t|t-r,|j)d }t+d|d  S t|t.r^|j)d }dt+dd|d   |  S t|t/r||j)d }d| S t|t0r|j)d }t+dd|d   S d S )Nr   AccumBoundsSetExprrm   	FiniteSetrb               )sinhF)1!sympy.calculus.accumulationboundsr   sympy.sets.setexprr   	is_Numberr   NaNr7   rJ   InfinityNegativeInfinityr`   r,   sympy.sets.setsr   minmaxr#   r   intersectionr   EmptySetr%   r}   r&   
_eval_funccould_extract_minus_signr1   %sympy.functions.elementary.hyperbolicr   r.   r>   rg   rh   NegativeOnerf   is_Rationalr   rS   rl   asinr5   atanr$   atan2acosacotacscasec)clsr   r   r   r   r   r   di_coeffr   r?   nargrw   resultry   yr/   r/   r0   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   rb   rm   r   rJ   r   lenr   r   nrw   previous_termsrx   r/   r/   r0   taylor_term  s    zsin.taylor_termr   c                 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
r5   subsr!   rO   r   r   r`   r	   r   _eval_nseriesr9   rw   r   r   r   r   r/   r/   r0   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   HyperbolicFunctionrb   
r   r   r   r.   r,   r2   r4   r5   rewriter"   )r9   r   kwargsr   Ir/   r/   r0   _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   rb   r,   r!   r   r.   r5   r9   r   r   r   rw   r/   r/   r0   _eval_rewrite_as_Pow  s    

zsin._eval_rewrite_as_Powc                 K   s   t |td  ddS Nrb   Fevaluater   r   r9   r   r   r/   r/   r0   _eval_rewrite_as_cos  s    zsin._eval_rewrite_as_cosc                 K   s"   t tj| }d| d|d   S Nrb   rm   tanr   rf   r9   r   r   Ztan_halfr/   r/   r0   _eval_rewrite_as_tan  s    zsin._eval_rewrite_as_tanc                 K   s   t |t| t| S Nr}   r   r   r/   r/   r0   _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   rb   rm   T	cotr   rf   r'   r(   r   r   r   r   r9   r   r   Zcot_halfr/   r/   r0   _eval_rewrite_as_cot  s    $zsin._eval_rewrite_as_cotc                 K   s   |  t tS r   )r   r   powr   r/   r/   r0   _eval_rewrite_as_pow  s    zsin._eval_rewrite_as_powc                 K   s   |  t tS r   )r   r   r$   r   r/   r/   r0   _eval_rewrite_as_sqrt  s    zsin._eval_rewrite_as_sqrtc                 K   s   dt | S Nrm   cscr   r/   r/   r0   _eval_rewrite_as_csc  s    zsin._eval_rewrite_as_cscc                 K   s   dt |td  dd S )Nrm   rb   Fr   secr   r   r/   r/   r0   _eval_rewrite_as_sec  s    zsin._eval_rewrite_as_secc                 K   s   |t | S r   )sincr   r/   r/   r0   _eval_rewrite_as_sinc  s    zsin._eval_rewrite_as_sincc                 C   s   |  | jd  S Nr   r4   r5   	conjugater9   r/   r/   r0   _eval_conjugate  s    zsin._eval_conjugateTc                 K   sH   ddl m}m} | jf d|i|\}}t||| t||| fS Nr   coshr   rA   )r   r   r   rK   r}   r   r9   rA   rC   r   r   r    r   r/   r/   r0   rB     s    zsin.as_real_imagc                 K   s$  ddl m}m} | jd }d }|jr| \}}t|dd }t|dd }t|dd }	t|dd }
||
 ||	  S |j	r|j
dd\}}|jr|jrtj|d d  ||t| S ttj|d d  t| ||d t| dd	S t|}|d ur|jr| tS t|S )
Nr   )
chebyshevt
chebyshevuFr   Trationalrm   rb   )rA   )#sympy.functions.special.polynomialsr   r   r5   rS   as_two_termsr}   _eval_expand_trigr   rP   ro   
is_Integeris_oddr   r   r
   r>   r   r   r$   )r9   rC   r   r   r   rw   r   sxsyru   cyr   r?   r/   r/   r0   r     s2    
 

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   rm   r   r   r5   r   cancelr   rg   as_leading_termr   r   r`   limitr    is_negativer   r   	is_finiter4   	r9   rw   r   r   r   r   x0r   ltr/   r/   r0   _eval_as_leading_term  s    


zsin._eval_as_leading_termc                 C   s   | j d jrdS d S Nr   Tr5   rH   r   r/   r/   r0   _eval_is_extended_real   s    zsin._eval_is_extended_realc                 C   s   | j d }|jrdS d S r  r  r9   r   r/   r/   r0   _eval_is_finite  s    
zsin._eval_is_finitec                 C   s"   t | jd \}}|jr|jS d S r   rl   r5   r7   rg   r9   restZpi_multr/   r/   r0   _eval_is_zero	  s    zsin._eval_is_zeroc                 C   s    | j d js| j d jrdS d S r  r5   rH   
is_complexr   r/   r/   r0   _eval_is_complex  s    
zsin._eval_is_complex)N)rm   )r   )T)Nr   )r[   r\   r]   r^   r   r   classmethodr   staticmethodr   r   r   r   r   r   r   r   r   r   r   r   r   r   r   rB   r   r  r  r  r  r  r/   r/   r/   r0   r}      s8   /


t


r}   c                   @   s   e Zd ZdZd4ddZd5ddZedd	 Zee	d
d Z
d6d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d7d&d'Zd(d) Zd8d*d+Zd,d- Zd.d/ Zd0d1 Zd2d3 ZdS )9r   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] http://dlmf.nist.gov/4.14
    .. [3] http://functions.wolfram.com/ElementaryFunctions/Cos

    Nc                 C   s   |  dt |S r~   r   r   r/   r/   r0   r   @  s    z
cos.periodrm   c                 C   s&   |dkrt | jd  S t| |d S r   )r}   r5   r   r   r/   r/   r0   r   C  s    z	cos.fdiffc              	   C   sn  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 tjtd
d d d}	|jr|j }
|j!d|
  }||
kr|d t }| | S d| |
krd| t }| | S ddddddddd}|
|v rd|t ||
 d  |t ||
 d   }}| || | }}d ||fv r<d S || | td | | td |   S |
dkrrd S |
|	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$|  }|td| d  S d S |j%rRt&|\}}|rR|t }t'|t'| t|t|  S |j	r`tj
S t|t(rv|j)d S t|t*r|j)d }dtd|d   S t|t+r|j)\}}|t|d |d   S t|t,r|j)d }td|d  S t|t-r"|j)d }dtdd|d    S t|t.rL|j)d }tdd|d   S t|t/rj|j)d }d| S d S )Nr   r   r   r   r   rm   rb   F)r   r      )r   r   r   r  r  r   r      r   
   r   r   r   r"        (   <      r&  r'        r)  r*  x   r,  )0r   r   r   r   r   r   r   r   r   r7   rn   r   r   r`   r,   r}   r   r   rH   r  r   r1   r   r   r>   rg   r   rh   rJ   r   rf   r$   qrx   rI   rq   rR   rS   rl   r   r   r5   r   r   r   r   r   r   )r   r   r   r   r   r   r   r?   r   cst_table_somer0  rx   table2rY   bnvalanvalbctsnvalrw   sign_cosry   r   r/   r/   r0   r   I  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   rb   rm   r   r   r   r/   r/   r0   r     s    zcos.taylor_termr   c                 C   sZ   | j d }|d ur"|t||}||dtjtjrFtd|  tj	| ||||dS r   r   r   r/   r/   r0   r     s    
zcos._eval_nseriesc                 K   sT   t j}ddlm} t|t|fr6||jd t	}t	|| t	| |  d S r   
r   r.   r   r   r,   r2   r4   r5   r   r"   )r9   r   r   r   r   r/   r/   r0   r     s
    zcos._eval_rewrite_as_expc                 K   s8   t |tr4tj}|jd }|| d ||  d  S d S r   r   r   r/   r/   r0   r     s    

zcos._eval_rewrite_as_Powc                 K   s   t |td  ddS r   )r}   r   r   r/   r/   r0   _eval_rewrite_as_sin   s    zcos._eval_rewrite_as_sinc                 K   s"   t tj| d }d| d|  S r   r   r   r/   r/   r0   r     s    zcos._eval_rewrite_as_tanc                 K   s   t |t| t | S r   r   r   r/   r/   r0   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 )Nrb   rm   r   Tr   r   r/   r/   r0   r   
  s    (zcos._eval_rewrite_as_cotc                 K   s
   |  |S r   )r   r   r/   r/   r0   r     s    zcos._eval_rewrite_as_powc                    s8  ddl m} fdddfdd	}t|}|d u r:d S |jrN| |t S |jsXd S dd }tjt	d	d
 d t	dt	d d t	dt	dt	d t	t	ddt	dt	d  d
t	d t	dt	d    dt	d  d   d  | d  fdd}|j
 v r8||j |j
 }|j
dk r4| }|S |j
d s|d }	t|	t t	}
|	d
 d }t|d r|dnd
}|t	d
|
 d  S ||j
}|r|||}dd t|tdD }ttdd |D  |}|t	S ||}dd t|tdD }ttdd |D  |}|S d S )Nr   r  c                    s   t | dkrd| d fS t | dkr6t| d | d S | dd  }t| d |d \} }t|g fdd|dd D  |g S )Nrm   r   rb   r   c                    s   g | ]} | qS r/   r/   .0r{   vr/   r0   
<listcomp>"      z>cos._eval_rewrite_as_sqrt.<locals>.migcdex.<locals>.<listcomp>)r   r   rM   )rw   rW   urX   migcdexr=  r0   rC    s    z*cos._eval_rewrite_as_sqrt.<locals>.migcdexc                    s   t trS t ts tdj djj kr<jS d |krb fddtj D }n fdd|D }t|dkrgS |}fddt|d d |D }t	|ksJ |S )	Nzr is not rationalrb   c                    s   g | ]\}} ||  qS r/   r/   )r<  rw   r   r   r/   r0   r?  .  r@  z@cos._eval_rewrite_as_sqrt.<locals>.ipartfrac.<locals>.<listcomp>c                    s   g | ]} | qS r/   r/   r<  rw   rD  r/   r0   r?  0  r@  rm   c                    s&   g | ]\}} j t||  j qS r/   )rx   r   r0  )r<  r{   j)rr/   r0   r?  4  r@  r   )
r,   rq   r   	TypeErrorr0  r)   itemsr   zipsum)rG  factorsrY   rX   ansrB  )r   rG  r0   	ipartfrac$  s     

 z,cos._eval_rewrite_as_sqrt.<locals>.ipartfracc            )      S   s  dd } dd }| dd\}}| |d\}}| |d\}}| |dd	| d
|   \}}	| |dd	| d
|   \}
}| |dd	| d
|   \}}| |dd	| d
|   \}}| |d|| | d
|
   \}}| |d|| | d
|   \}}| |d|| |	 d
|   \}}| |d|| |
 d
|   \}}| |	d||	 | d
|   \}}| |
d||
 | d
|   \}}| |d|| | d
|   \}}| |d|| | d
|	   \}}||d|| | |  } ||d|| | |  }!||d|| | |  }"||d|| | |  }#||d|| | |  }$||d|| | |  }%||  d|!|"   }&||# d|$|%   }'d||& d|'  }(t t d
t |(d  d tj S )a2   Express cos(pi/257) explicitly as a function of radicals
                Based upon the equations in
                http://math.stackexchange.com/questions/516142/how-does-cos2-pi-257-look-like-in-real-radicals
                See also http://www.susqu.edu/brakke/constructions/257-gon.m.txt
            c                 S   s0   | t | d |  d | t | d |  d fS r~   r$   rY   r3  r/   r/   r0   f1H  s    z8cos._eval_rewrite_as_sqrt.<locals>._cospi257.<locals>.f1c                 S   s   | t | d |  d S r~   rO  rP  r/   r/   r0   f2K  s    z8cos._eval_rewrite_as_sqrt.<locals>._cospi257.<locals>.f2r      @   r  r   rb   r   r   )r$   r   rf   ))rQ  rR  t1t2z1z3z2z4y1Zy5Zy6y2y3Zy7Zy8Zy4x1Zx9x2x10x3Zx11x4Zx12Zx5Zx13Zx6Zx14Zx15Zx7Zx8Zx16v1v2Zv3v4Zv5Zv6u1u2Zw1r/   r/   r0   	_cospi257B  s6    """"""""z,cos._eval_rewrite_as_sqrt.<locals>._cospi257r   rm   r  r-         rb   ir   "   )r   r   rj    c                    sJ   g } D ]<}t | |\}}|dkr|} || | dkrt|  S qdS )Nr   rm   F)divmodre   rM   )r   primesZp_iquotient	remainder)r1  r/   r0   _fermatCoordst  s    
z0cos._eval_rewrite_as_sqrt.<locals>._fermatCoordsrm  r   c                 S   s    g | ]}|d  |d t  fqS rm   r   r   rE  r/   r/   r0   r?    r@  z-cos._eval_rewrite_as_sqrt.<locals>.<listcomp>zc                 S   s   g | ]}|d  qS r   r/   rE  r/   r/   r0   r?    r@  c                 S   s    g | ]}|d  |d t  fqS rs  rt  rE  r/   r/   r0   r?    r@  c                 S   s   g | ]}|d  qS rv  r/   rE  r/   r/   r0   r?    r@  )N)r   r   r>   rg   r4   r   r   r   rf   r$   r0  rx   rI   r   r   rq   rJ  r+   rK  r   r   )r9   r   r   r   rN  r?   ri  rr  rvZpico2r7  rw   r8  ZFCdecompXZpclsr/   )r1  rC  r0   r     s`    '$$


 
 zcos._eval_rewrite_as_sqrtc                 K   s   dt | S r   r   r   r/   r/   r0   r     s    zcos._eval_rewrite_as_secc                 K   s   dt |t S r   )r   r   r   r   r/   r/   r0   r     s    zcos._eval_rewrite_as_cscc                 C   s   |  | jd  S r   r   r   r/   r/   r0   r     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   r/   r/   r0   rB     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|}|d ur|jr| tS t|S )Nr   r  Fr   Tr   )r   r   r5   rS   r   r}   r   r   rP   ro   r   r>   r   r   r$   )r9   rC   r   r   rw   r   r   r   ru   r   rd   termsr?   r/   r/   r0   r     s&    

zcos._eval_expand_trigc           	      C   s   ddl m} | jd }||d }|td  t }|jrd||t  td  |}tj	| | S |tj
u r|j|dt|jrdndd}|tjtjfv r|ddS |jr| |S | S )	Nr   r   rb   r   r   r  r   rm   r  r	  r/   r/   r0   r    s    


zcos._eval_as_leading_termc                 C   s   | j d jrdS d S r  r  r   r/   r/   r0   r    s    zcos._eval_is_extended_realc                 C   s   | j d }|jrdS d S r  r  r  r/   r/   r0   r    s    
zcos._eval_is_finitec                 C   s    | j d js| j d jrdS d S r  r  r   r/   r/   r0   r    s    
zcos._eval_is_complexc                 C   s6   t | jd \}}|r,t|tj j|jgS |jS d S r   )rl   r5   r   r   rf   rg   r7   r  r/   r/   r0   r    s    zcos._eval_is_zero)N)rm   )r   )T)Nr   )r[   r\   r]   r^   r   r   r  r   r  r   r   r   r   r   r:  r   r   r   r   r   r   r   r   rB   r   r  r  r  r  r  r/   r/   r/   r0   r     s:   +


 
 


r   c                   @   s   e Zd ZdZd8ddZd9ddZd:dd	Zed
d Ze	e
dd Zd;ddZdd Zdd Zd<ddZdd Zdd Zdd Zdd Zd d! Zd"d# Zd$d% Zd&d' Zd(d) Zd*d+ Zd=d,d-Zd.d/ Zd0d1 Zd2d3 Zd4d5 Zd6d7 Zd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] http://dlmf.nist.gov/4.14
    .. [3] http://functions.wolfram.com/ElementaryFunctions/Tan

    Nc                 C   s   |  t|S r   r   r   r/   r/   r0   r   	  s    z
tan.periodrm   c                 C   s$   |dkrt j| d  S t| |d S Nrm   rb   )r   rn   r   r   r/   r/   r0   r     s    z	tan.fdiffc                 C   s   t S z7
        Returns the inverse of this function.
        r   r   r/   r/   r0   inverse  s    ztan.inversec              	   C   s  ddl m} |jrL|tju r"tjS |jr.tjS |tjtjfv rL|tjtjS |tj	u r\tjS t
|| r|j|j }}t|t }|tjur||t  }|tjur||t  }ddlm} ||||td ttdd r|tjtjS |t|t|S | r| |  S t|}|d ur@ddlm} tj|| S t|d}	|	d ur|	jrbtjS |	js|	t }
|
|kr| |
S d S |	jr|	j}|	j| }tddtd d  tddtd  tddtd d  tddtd  d	}|d
v r0d| | }|dkr(d| }||  S || S |	jd s|	t d }
t|
t|
td   }}t
|tst
|ts|dkrtj	S d| ||  S ddddddddd}||v r| |t || d  | |t || d   }}d ||fv rd S || d||   S |	tj  d tj  t }
t|
t|
td   }}t
|tstt
|tst|dkrltj	S || S |
|kr| |
S |j!rt"|\}}|rt|t }|tj	u rt#| S t|S |jrtjS t
|t$r|j%d S t
|t&r|j%\}}|| S t
|t'r6|j%d }|td|d   S t
|t(r`|j%d }td|d  | S t
|t)r~|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   rb   r   )tanhrm   r   )rm   rb   r   r  r   r"  r"  r  r  r  r!  r#  r$  r%  r(  r+  ),r   r   r   r   r   r7   rJ   r   r   r`   r,   r   r   r#   r   r   r   r   r   r   r   r1   r   r  r.   r>   rg   r   r0  rx   r$   r   rf   rS   rl   r   r   r5   r   r   r   r   r   r   )r   r   r   r   r   r   r   r   r  r?   r   r0  rx   Ztable10r   cresultsresultr2  r4  r5  rw   ry   Ztanmr   r/   r/   r0   r     s    



$











2









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   rb   rm   )r   rJ   r   r   r   r   )r   rw   r   rY   r3  BFr/   r/   r0   r     s    ztan.taylor_termr   c                 C   sL   | j d |dd t }|r:|jr:| tj|||dS tj| |||dS )Nr   rb   r   r   )r5   r  r   r   r   r   r   r   r9   rw   r   r   r   r{   r/   r/   r0   r     s    
ztan._eval_nseriesc                 K   sF   t |trBtj}|jd }|||  ||   ||  ||   S d S r   r   r   r/   r/   r0   r     s    

ztan._eval_rewrite_as_Powc                 C   s   |  | jd  S r   r   r   r/   r/   r0   r     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 NrA   r   r   rb   	rK   r   r   r   r   r}   r4   r   rJ   r9   rA   rC   r    r   r   r   denomr/   r/   r0   rB     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 r/   nextr;  ZYgr/   r0   r?    r@  z)tan._eval_expand_trig.<locals>.<listcomp>rm   rb   r   r  Tr   dummyreal)r5   rS   r   r   r   re   r+   ranger*   r   listrJ  rP   ro   r   r   r.   r   rI   r   r    )r9   rC   r   rw   r   ZTXZtxr  rx   r{   rd   r{  r   ru  Pr/   r  r0   r     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   r9  )r9   r   r   r   r   neg_exppos_expr/   r/   r0   r     s    ztan._eval_rewrite_as_expc                 K   s   dt |d  t d|  S r~   r}   r9   rw   r   r/   r/   r0   r:    s    ztan._eval_rewrite_as_sinc                 K   s   t |td  ddt | S r   r   r  r/   r/   r0   r     s    ztan._eval_rewrite_as_cosc                 K   s   t |t| S r   r   r   r/   r/   r0   r     s    ztan._eval_rewrite_as_sincosc                 K   s   dt | S r   r   r   r/   r/   r0   r     s    ztan._eval_rewrite_as_cotc                 K   s$   t |t}t|t}|| S r   )r}   r   r   r   )r9   r   r   sin_in_sec_formcos_in_sec_formr/   r/   r0   r     s    ztan._eval_rewrite_as_secc                 K   s$   t |t}t|t}|| S r   )r}   r   r   r   )r9   r   r   sin_in_csc_formcos_in_csc_formr/   r/   r0   r     s    ztan._eval_rewrite_as_cscc                 K   s"   |  t t}|trd S |S r   r   r   r   rO   r9   r   r   r   r/   r/   r0   r     s    
ztan._eval_rewrite_as_powc                 K   s"   |  t t}|trd S |S r   r   r   r$   rO   r  r/   r/   r0   r     s    
ztan._eval_rewrite_as_sqrtc           
      C   s   ddl m} ddlm} | jd }||d }d| t }|jrl||t d  	|}	|j
rd|	S d|	 S |tju r|j|d||jrdndd}|tjtjfv r|tjtjS |jr| |S | S )	Nr   r   r    rb   r   r   r   r  r   r   $sympy.functions.elementary.complexesr    r5   r   r  r   rg   r  rh   r   r`   r  r  r   r   r  r4   
r9   rw   r   r   r   r    r   r
  r   r  r/   r/   r0   r    s    

ztan._eval_as_leading_termc                 C   s   | j d jS r   r  r   r/   r/   r0   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<   r5   is_realr   r   rf   rg   r  r/   r/   r0   _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<   )r5   r  r   r   rf   rg   is_imaginaryr  r/   r/   r0   r  %  s
    
ztan._eval_is_finitec                 C   s"   t | jd \}}|jr|jS d S r   r  r  r/   r/   r0   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<   r  r  r/   r/   r0   r  3  s    
ztan._eval_is_complex)N)rm   )rm   )r   )T)Nr   ) r[   r\   r]   r^   r   r   r  r  r   r  r   r   r   r   r   rB   r   r   r:  r   r   r   r   r   r   r   r  r  r  r  r  r  r/   r/   r/   r0   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*d+Zd,d- Zd.d/ Zd0d1 Zd2d3 Zd4d5 Zd6d7 Zd8d9 Z dS )@r   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] http://dlmf.nist.gov/4.14
    .. [3] http://functions.wolfram.com/ElementaryFunctions/Cot

    Nc                 C   s   |  t|S r   r   r   r/   r/   r0   r   `  s    z
cot.periodrm   c                 C   s$   |dkrt j| d  S t| |d S r|  )r   r   r   r   r/   r/   r0   r   c  s    z	cot.fdiffc                 C   s   t S r}  r   r   r/   r/   r0   r  i  s    zcot.inversec              	   C   s  ddl m} |jrL|tju r"tjS |jr.tjS |tjtjfv rL|tjtjS |tju r\tjS t	||rxt
|td   S | r| |  S t|}|d urddlm} tj || S t|d}|d ur|jrtjS |js|t }||kr| |S d S |jr|jdv rt
td | S |jdkr|jd s|t d }t|t|td   }}t	|tst	|tsd| ||  S ddd	d
ddddd}	|j}
|j|
 }|
|	v r| |t |	|
 d  | |t |	|
 d   }}d ||fv rd S d||  ||  S |tj d tj t }t|t|td   }}t	|tsnt	|tsn|dkrftjS || S ||kr| |S |jrt|\}}|rt|t }|tju rt|S t
| S |jrtjS t	|tr|jd S t	|tr|jd }d| S t	|tr$|j\}}|| S t	|trN|jd }t d|d  | S t	|t!rx|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   rb   )cothr  rm   r  r  r  r!  r#  r$  r%  r(  r+  )$r   r   r   r   r   r7   r`   r   r   r,   r   r   r   r1   r   r  r.   r>   rg   r   r0  r   rx   rf   rS   rl   r   r   r5   r   r   r   r$   r   r   r   )r   r   r   r   r  r?   r   r  r  r2  r0  rx   r4  r5  rw   ry   Zcotmr   r/   r/   r0   r   o  s    







2









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   rm   rb   )r   r   rJ   r   r   r   )r   rw   r   r  r  r/   r/   r0   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   r  )r5   r  r   r   r   r   r   r   r  r/   r/   r0   r     s    
zcot._eval_nseriesc                 C   s   |  | jd  S r   r   r   r/   r/   r0   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 r  r  r  r/   r/   r0   rB     s    "zcot.as_real_imagc                 K   sf   ddl m} tj}t|t|fr6||jd t	}t	| | t	||  }}|||  ||  S r  r   )r9   r   r   r   r   r  r  r/   r/   r0   r     s    zcot._eval_rewrite_as_expc                 K   sH   t |trDtj}|jd }| ||  ||   ||  ||   S d S r   r   r   r/   r/   r0   r   	  s    

zcot._eval_rewrite_as_Powc                 K   s   t d| dt |d   S r~   r  r  r/   r/   r0   r:    s    zcot._eval_rewrite_as_sinc                 K   s   t |t |td  dd S r   r   r  r/   r/   r0   r     s    zcot._eval_rewrite_as_cosc                 K   s   t |t| S r   r   r}   r   r/   r/   r0   r     s    zcot._eval_rewrite_as_sincosc                 K   s   dt | S r   r   r   r/   r/   r0   r     s    zcot._eval_rewrite_as_tanc                 K   s$   t |t}t|t}|| S r   )r   r   r   r}   )r9   r   r   r  r  r/   r/   r0   r     s    zcot._eval_rewrite_as_secc                 K   s$   t |t}t|t}|| S r   )r   r   r   r}   )r9   r   r   r  r  r/   r/   r0   r      s    zcot._eval_rewrite_as_cscc                 K   s"   |  t t}|trd S |S r   r  r  r/   r/   r0   r   %  s    
zcot._eval_rewrite_as_powc                 K   s"   |  t t}|trd S |S r   r  r  r/   r/   r0   r   +  s    
zcot._eval_rewrite_as_sqrtc           
      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   r  rb   rm   r   r   r  r  r  r/   r/   r0   r  1  s    

zcot._eval_as_leading_termc                 C   s   | j d jS r   r  r   r/   r/   r0   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   r  c                    s   g | ]}t  qS r/   r  r;  r  r/   r0   r?  N  r@  z)cot._eval_expand_trig.<locals>.<listcomp>r   rb   r  rm   Tr   r  r  )r5   rS   r   r   r   re   r+   r  r*   r   r  rJ  rP   ro   r   r   r.   r   rI   r    r   )r9   rC   r   rw   r   ZCXru   r  rx   r{   rd   r{  r   ru  r  r/   r  r0   r   C  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<   )r5   r  r   rg   r  r  r/   r/   r0   r  ]  s
    
zcot._eval_is_finitec                 C   s&   | j d }|jr"|t jdu r"dS d S r<   r5   r  r   rg   r  r/   r/   r0   r  d  s    
zcot._eval_is_realc                 C   s&   | j d }|jr"|t jdu r"dS d S r<   r  r  r/   r/   r0   r  i  s    
zcot._eval_is_complexc                 C   s,   t | jd \}}|r(|jr(|tj jS d S r   )rl   r5   r7   r   rf   rg   )r9   r  Zpimultr/   r/   r0   r  n  s    
zcot._eval_is_zeroc                 C   s6   | j d }|||}||kr.|t jr.tjS t|S r   )r5   r   r   rg   r   r`   r   )r9   oldnewr   Zargnewr/   r/   r0   
_eval_subss  s
    
zcot._eval_subs)N)rm   )rm   )r   )T)Nr   )!r[   r\   r]   r^   r   r   r  r  r   r  r   r   r   r   rB   r   r   r:  r   r   r   r   r   r   r   r  r  r   r  r  r  r  r  r/   r/   r/   r0   r   :  s<   %



p

	
r   c                   @   s   e Zd ZdZdZejfZdZdZ	e
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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d0d(d)Zd*d+ Zd1d,d-ZdS )2ReciprocalTrigonometricFunctionz@Base class for reciprocal functions of trigonometric functions. Nc                 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 )Nrb   rm   r  r   c                 s   s   | ]}t |tV  qd S r   )r,   r   r;  r/   r/   r0   	<genexpr>  r@  z7ReciprocalTrigonometricFunction.eval.<locals>.<genexpr>c                 s   s   | ]}t |tV  qd S r   )r,   r}   r;  r/   r/   r0   r    r@  )r   _is_even_is_oddr>   rg   r   r0  rx   r   hasattrr  r5   _reciprocal_ofr   anyr   r   r   )r   r   r?   r0  rx   r   tr/   r/   r0   r     s@    



z$ReciprocalTrigonometricFunction.evalc                 O   s$   |  | jd }t|||i |S r   )r  r5   getattr)r9   method_namer5   r   or/   r/   r0   _call_reciprocal  s    z0ReciprocalTrigonometricFunction._call_reciprocalc                 O   s,   | j |g|R i |}|d ur(d| S |S r   )r  )r9   r  r5   r   r  r/   r/   r0   _calculate_reciprocal  s    z5ReciprocalTrigonometricFunction._calculate_reciprocalc                 C   s.   |  ||}|d ur*|| |kr*d| S d S r   )r  r  )r9   r  r   r  r/   r/   r0   _rewrite_reciprocal  s    z3ReciprocalTrigonometricFunction._rewrite_reciprocalc                 C   s   t | jd }| ||S r   )r
   r5   r  r   )r9   rU   rV   r/   r/   r0   rZ     s    z'ReciprocalTrigonometricFunction._periodrm   c                 C   s   |  d| | d  S )Nr   rb   r  r   r/   r/   r0   r     s    z%ReciprocalTrigonometricFunction.fdiffc                 K   s   |  d|S )Nr   r  r   r/   r/   r0   r     s    z4ReciprocalTrigonometricFunction._eval_rewrite_as_expc                 K   s   |  d|S )Nr   r  r   r/   r/   r0   r     s    z4ReciprocalTrigonometricFunction._eval_rewrite_as_Powc                 K   s   |  d|S )Nr:  r  r   r/   r/   r0   r:    s    z4ReciprocalTrigonometricFunction._eval_rewrite_as_sinc                 K   s   |  d|S )Nr   r  r   r/   r/   r0   r     s    z4ReciprocalTrigonometricFunction._eval_rewrite_as_cosc                 K   s   |  d|S )Nr   r  r   r/   r/   r0   r     s    z4ReciprocalTrigonometricFunction._eval_rewrite_as_tanc                 K   s   |  d|S )Nr   r  r   r/   r/   r0   r     s    z4ReciprocalTrigonometricFunction._eval_rewrite_as_powc                 K   s   |  d|S )Nr   r  r   r/   r/   r0   r     s    z5ReciprocalTrigonometricFunction._eval_rewrite_as_sqrtc                 C   s   |  | jd  S r   r   r   r/   r/   r0   r     s    z/ReciprocalTrigonometricFunction._eval_conjugateTc                 K   s"   d|  | jd  j|fi |S r   )r  r5   rB   )r9   rA   rC   r/   r/   r0   rB     s    z,ReciprocalTrigonometricFunction.as_real_imagc                 K   s   | j di |S )Nr   )r   r  )r9   rC   r/   r/   r0   r     s    z1ReciprocalTrigonometricFunction._eval_expand_trigc                 C   s   |  | jd  S r   )r  r5   r  r   r/   r/   r0   r    s    z6ReciprocalTrigonometricFunction._eval_is_extended_realr   c                 C   s   d|  | jd  |S r   )r  r5   r  )r9   rw   r   r   r/   r/   r0   r    s    z5ReciprocalTrigonometricFunction._eval_as_leading_termc                 C   s   d|  | jd  jS r   )r  r5   r  r   r/   r/   r0   r    s    z/ReciprocalTrigonometricFunction._eval_is_finitec                 C   s   d|  | jd  |||S r   )r  r5   r   r9   rw   r   r   r   r/   r/   r0   r     s    z-ReciprocalTrigonometricFunction._eval_nseries)rm   )T)Nr   )r   )r[   r\   r]   r^   r  r   r`   ra   r  r  r  r   r  r  r  rZ   r   r   r   r:  r   r   r   r   r   rB   r   r  r  r  r   r/   r/   r/   r0   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eedd Zd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] http://dlmf.nist.gov/4.14
    .. [3] http://functions.wolfram.com/ElementaryFunctions/Sec

    TNc                 C   s
   |  |S r   rZ   r   r/   r/   r0   r     s    z
sec.periodc                 K   s    t |d d }|d |d  S r   r  )r9   r   r   Zcot_half_sqr/   r/   r0   r     s    zsec._eval_rewrite_as_cotc                 K   s   dt | S r   r   r   r/   r/   r0   r   !  s    zsec._eval_rewrite_as_cosc                 K   s   t |t|t |  S r   r   r   r/   r/   r0   r   $  s    zsec._eval_rewrite_as_sincosc                 K   s   dt |t S r   )r   r   r}   r   r/   r/   r0   r:  '  s    zsec._eval_rewrite_as_sinc                 K   s   dt |t S r   )r   r   r   r   r/   r/   r0   r   *  s    zsec._eval_rewrite_as_tanc                 K   s   t td | ddS r   )r   r   r   r/   r/   r0   r   -  s    zsec._eval_rewrite_as_cscrm   c                 C   s2   |dkr$t | jd t| jd  S t| |d S r   )r   r5   r   r   r   r/   r/   r0   r   0  s    z	sec.fdiffc                 C   s,   | j d }|jr(|t tj jdu r(dS d S r<   )r5   r  r   r   rf   rg   r  r/   r/   r0   r  6  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 r  )r   rJ   r   r   r   r   r   rw   r   kr/   r/   r0   r   <  s
    zsec.taylor_termr   c           
      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   r  rb   r   r   r  r   r   r  r    r5   r   r  r   rg   r  r   r   r`   r  r  r   r   r  r4   r  r/   r/   r0   r  H  s    

zsec._eval_as_leading_term)N)rm   )Nr   )r[   r\   r]   r^   r   r  r  r   r   r   r   r:  r   r   r   r  r  r   r   r  r/   r/   r/   r0   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dZdd Zeedd Zd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] http://dlmf.nist.gov/4.14
    .. [3] http://functions.wolfram.com/ElementaryFunctions/Csc

    TNc                 C   s
   |  |S r   r  r   r/   r/   r0   r     s    z
csc.periodc                 K   s   dt | S r   r  r   r/   r/   r0   r:    s    zcsc._eval_rewrite_as_sinc                 K   s   t |t|t |  S r   r  r   r/   r/   r0   r     s    zcsc._eval_rewrite_as_sincosc                 K   s    t |d }d|d  d|  S r   r  r   r/   r/   r0   r     s    zcsc._eval_rewrite_as_cotc                 K   s   dt |t S r   )r}   r   r   r   r/   r/   r0   r     s    zcsc._eval_rewrite_as_cosc                 K   s   t td | ddS r   r   r   r/   r/   r0   r     s    zcsc._eval_rewrite_as_secc                 K   s   dt |t S r   )r}   r   r   r   r/   r/   r0   r     s    zcsc._eval_rewrite_as_tanrm   c                 C   s4   |dkr&t | jd  t| jd  S t| |d S r   )r   r5   r   r   r   r/   r/   r0   r     s    z	csc.fdiffc                 C   s&   | j d }|jr"|t jdu r"dS d S r<   r  r  r/   r/   r0   r    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 r  )r   r   rJ   r   r   r   r  r/   r/   r0   r     s    $

zcsc.taylor_termr   c           
      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   r  r   r   r  r  r  r/   r/   r0   r    s    

zcsc._eval_as_leading_term)N)rm   )Nr   )r[   r\   r]   r^   r}   r  r  r   r:  r   r   r   r   r   r   r  r  r   r   r  r/   r/   r/   r0   r   X  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

    rm   c                 C   s<   | j d }|dkr.t|| t||d   S t| |d S r  )r5   r   r}   r   )r9   r   rw   r/   r/   r0   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~   )r7   r   rn   r   r   r   rJ   r   r`   r   r>   rg   r   r   rf   )r   r   r?   r/   r/   r0   r     s$    




z	sinc.evalr   c                 C   s    | j d }t|| |||S r   )r5   r}   r   r  r/   r/   r0   r     s    
zsinc._eval_nseriesc                 K   s   ddl m} |d|S )Nr   )jn)sympy.functions.special.besselr  )r9   r   r   r  r/   r/   r0   _eval_rewrite_as_jn  s    zsinc._eval_rewrite_as_jnc                 K   s&   t t|| t|tjftjtjfS r   )r'   r}   r   r   rJ   rn   truer   r/   r/   r0   r:  !  s    zsinc._eval_rewrite_as_sinc                 C   sL   | j d jrdS t| j d \}}|jr8t|j|jgS |jrH|jrHdS d S )Nr   TF)r5   is_infiniterl   r7   r   rg   
is_nonzeror   r  r/   r/   r0   r  $  s    zsinc._eval_is_zeroc                 C   s    | j d js| j d jrdS d S r  )r5   rH   r  r   r/   r/   r0   r  -  s    zsinc._eval_is_realN)rm   )r   )r[   r\   r]   r^   r   r`   ra   r   r  r   r   r  r:  r  r  r  r/   r/   r/   r0   r     s   4


	r   c                   @   sT   e Zd ZdZejejejejfZ	e
edd Ze
edd Ze
edd ZdS )	InverseTrigonometricFunctionz/Base class for inverse trigonometric functions.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   rb   r  rm   r   r   r   r"  r,  )r$   r   r   r   rf   r/   r/   r/   r0   _asin_table=  s,     &
  "z(InverseTrigonometricFunction._asin_tablec                   C   s  t dd td dt d td t dtd t dd td dt d t d dt d ttdd t ddt d  td t ddt d  ttdd t ddt d d  td t ddt d d  ttdd dt d td d	t d t d dt d ttdd iS )
Nr   r   rm   rb   r   r   r"  r,  r   r$   r   r   r/   r/   r/   r0   _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 )Nrb   r   r  r   rm   r   r   r"  r,  r  r/   r/   r/   r0   _acsc_tablep  s$    ""(z(InverseTrigonometricFunction._acsc_tableN)r[   r\   r]   r^   r   rn   r   rJ   r`   ra   r  r   r  r  r  r/   r/   r/   r0   r  9  s   r  c                   @   s   e Zd ZdZd%ddZdd Zdd Zd	d
 Zedd Z	e
edd Zd&ddZd'ddZdd Zdd Zdd ZeZdd Zdd Zdd  Zd!d" Zd(d#d$ZdS ))r   ab  
    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] http://dlmf.nist.gov/4.23
    .. [3] http://functions.wolfram.com/ElementaryFunctions/ArcSin

    rm   c                 C   s0   |dkr"dt d| jd d   S t| |d S Nrm   r   rb   r$   r5   r   r   r/   r/   r0   r     s    z
asin.fdiffc                 C   s4   | j | j }|j | j kr*|jd jr0dS n|jS d S r3   r4   r5   r6   r8   r/   r/   r0   r;     s
    zasin._eval_is_rationalc                 C   s   |   o| jd jS r   )r  r5   is_positiver   r/   r/   r0   _eval_is_positive  s    zasin._eval_is_positivec                 C   s   |   o| jd jS r   )r  r5   r  r   r/   r/   r0   _eval_is_negative  s    zasin._eval_is_negativec                 C   s  |j rt|tju rtjS |tju r,tjtj S |tju rBtjtj S |jrNtjS |tju r`t	d S |tj
u rtt	 d S |tju rtjS | r| |  S |jr|  }||v r|| S t|}|d urddlm} tj|| S |jrtjS t|tr\|jd }|jr\|dt	 ; }|t	kr(t	| }|t	d kr>t	| }|t	 d k rXt	 | }|S t|tr|jd }|jrt	d t| S d S )Nrb   r   )asinh)r   r   r   r   r   r.   r7   rJ   rn   r   r   r`   r   	is_numberr  r1   r   r  r,   r}   r5   is_comparabler   r   )r   r   
asin_tabler   r  angr/   r/   r0   r     sT    










z	asin.evalc                 G   s   | dk s| d dkrt jS t|}t|dkrb| dkrb|d }|| d d  | | d   |d  S | d d }tt j|}t|}|| ||   |  S d S r   )r   rJ   r   r   r   rf   r   r   rw   r   rx   r  Rr  r/   r/   r0   r    	  s    $zasin.taylor_termNr   c                 C   s   | j d }||d }|jr*||S |tj tjtjfv rZ| t	j
|||d S |dkrn|||}t|dk r|jr|tjk rt | | S t|dkr|jr|tjkrt| | S | |S Nr   r   r   )r5   r   r  r7   r  r   rn   r`   r   r!   r  rI   r  r   r  r   r   r4   r9   rw   r   r   r   r
  r/   r/   r0   r  	  s    

zasin._eval_as_leading_termc                 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kr2| jd ||}t|dk r^|jr^|tjk r^t | S t|dkr|jr|tjkrt| S |S 	Nr   Or  Tpositiverb   rm   r  )sympy.series.orderr  r5   r   r   rn   r   r   r   r!   nseriesr  is_meromorphicr   r$   r   removeOrI   powsimpr   r   r`   r  r   r  r9   rw   r   r   r   r  arg0r  serarg1rV   rW   res1resr/   r/   r0   r    	  sD    &
$&&
(&
"
"zasin._eval_nseriesc                 K   s   t d t| S r~   r   r   r  r/   r/   r0   _eval_rewrite_as_acosF	  s    zasin._eval_rewrite_as_acosc                 K   s    dt |dtd|d     S r   )r   r$   r  r/   r/   r0   _eval_rewrite_as_atanI	  s    zasin._eval_rewrite_as_atanc                 K   s&   t j tt j| td|d    S r|  r   r.   r!   r$   r  r/   r/   r0   _eval_rewrite_as_logL	  s    zasin._eval_rewrite_as_logc                 K   s    dt dtd|d   |  S r   )r   r$   r   r/   r/   r0   _eval_rewrite_as_acotP	  s    zasin._eval_rewrite_as_acotc                 K   s   t d td|  S r   r   r   r   r/   r/   r0   _eval_rewrite_as_asecS	  s    zasin._eval_rewrite_as_asecc                 K   s   t d| S r   )r   r   r/   r/   r0   _eval_rewrite_as_acscV	  s    zasin._eval_rewrite_as_acscc                 C   s   | j d }|jodt| jS Nr   rm   r5   rH   rR   is_nonnegativer9   rw   r/   r/   r0   r  Y	  s    
zasin._eval_is_extended_realc                 C   s   t S r}  r  r   r/   r/   r0   r  ]	  s    zasin.inverse)rm   )Nr   )r   )rm   )r[   r\   r]   r^   r   r;   r  r  r  r   r  r   r   r  r   r  r  r  _eval_rewrite_as_tractabler  r   r  r  r  r/   r/   r/   r0   r     s*   *

6

&r   c                   @   s   e Zd ZdZd%ddZdd Zedd Zee	d	d
 Z
d&ddZdd Zdd Zd'ddZdd ZeZdd Zdd Zd(ddZdd Zdd  Zd!d" Zd#d$ ZdS ))r   a  
    The inverse cosine function.

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

    Examples
    ========

    ``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] http://dlmf.nist.gov/4.23
    .. [3] http://functions.wolfram.com/ElementaryFunctions/ArcCos

    rm   c                 C   s0   |dkr"dt d| jd d   S t| |d S Nrm   r   r   rb   r  r   r/   r/   r0   r   	  s    z
acos.fdiffc                 C   s4   | j | j }|j | j kr*|jd jr0dS n|jS d S r3   r  r8   r/   r/   r0   r;   	  s
    zacos._eval_is_rationalc                 C   sV  |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 t|tr$|jd }|jr$|dt ; }|tkr dt | }|S t|trR|jd }|jrRtd t| S d S Nrb   r   )r   r   r   r   r.   r   r7   r   rn   rJ   r   r`   r  r  r1   r   r,   r   r5   r  r}   )r   r   r  r   r  r/   r/   r0   r   	  sF    









z	acos.evalc                 G   s   | dkrt d S | dk s$| d dkr*tjS t|}t|dkrr| dkrr|d }|| d d  | | d   |d  S | d d }ttj|}t|}| | ||   |  S d S r   )r   r   rJ   r   r   r   rf   r   r  r/   r/   r0   r   	  s    $zacos.taylor_termNr   c                 C   s   | j d }||d }|dkr>tdttj| | S |tj tjfv rf| t	j
|||dS |dkrz|||}t|dk r|jr|tjk rdt | | S t|dkr|jr|tjkr| | S | |S Nr   rm   rb   r  )r5   r   r  r$   r   rn   r  r`   r   r!   r  r  r   r  r   r   r4   r  r/   r/   r0   r  	  s    
zacos._eval_as_leading_termc                 C   s   | j d }|jodt| jS r  r  r  r/   r/   r0   r  	  s    
zacos._eval_is_extended_realc                 C   s   |   S r   )r  r   r/   r/   r0   _eval_is_nonnegative	  s    zacos._eval_is_nonnegativec                 C   s|  ddl m} | jd |d}|tju rtddd}ttj|d  t	
|dd| }tj| jd  }	|	|}
|	|
 |
 }||ds|dkr|dS |t|S ttj| j|||d}| t|
  }| ||  ||| | S |tju rtddd}ttj|d  t	
|dd| }tj| jd  }	|	|}
|	|
 |
 }||ds|dkr|dS t|t| S ttj| j|||d}| t|
  }| ||  ||| | S tj| |||d}|tju r|S |dkr"| jd ||}t|dk rP|jrP|tjk rPdt | S t|dkrx|jrx|tjkrx| S |S r  )r  r  r5   r   r   rn   r   r   r   r!   r  r  r  r$   r   r  rI   r  r   r   r   r`   r  r   r  r  r/   r/   r0   r   	  sD    
&
&&
"&
""zacos._eval_nseriesc                 K   s,   t d tjttj| td|d     S r   r   r   r.   r!   r$   r  r/   r/   r0   r  
  s    
zacos._eval_rewrite_as_logc                 K   s   t d t| S r~   r   r   r  r/   r/   r0   _eval_rewrite_as_asin
  s    zacos._eval_rewrite_as_asinc                 K   s8   t td|d  | td d|td|d      S r|  )r   r$   r   r  r/   r/   r0   r  !
  s    zacos._eval_rewrite_as_atanc                 C   s   t S r}  r  r   r/   r/   r0   r  $
  s    zacos.inversec                 K   s(   t d dtdtd|d   |   S r   )r   r   r$   r   r/   r/   r0   r  *
  s    zacos._eval_rewrite_as_acotc                 K   s   t d| S r   )r   r   r/   r/   r0   r   -
  s    zacos._eval_rewrite_as_asecc                 K   s   t d td|  S r   r   r   r   r/   r/   r0   r  0
  s    zacos._eval_rewrite_as_acscc                 C   sN   | j d }| | j d  }|jdu r,|S |jrJ|d jrJ|d jrJ|S d S Nr   Frm   )r5   r4   r   rH   r  is_nonpositive)r9   ru  rG  r/   r/   r0   r   3
  s    

zacos._eval_conjugate)rm   )Nr   )r   )rm   )r[   r\   r]   r^   r   r;   r  r   r  r   r   r  r  r
  r   r  r  r  r  r  r  r   r  r   r/   r/   r/   r0   r   d	  s*   +

+

&
r   c                       s   e Zd ZU dZee ed< ejej fZ	d*ddZ
dd Zdd	 Zd
d Zdd Zdd Zedd Zeedd Zd+ddZd,ddZdd ZeZ fddZd-ddZd d! Zd"d# Zd$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] http://dlmf.nist.gov/4.23
    .. [3] http://functions.wolfram.com/ElementaryFunctions/ArcTan

    r5   rm   c                 C   s,   |dkrdd| j d d   S t| |d S r  r5   r   r   r/   r/   r0   r   g
  s    z
atan.fdiffc                 C   s4   | j | j }|j | j kr*|jd jr0dS n|jS d S r3   r  r8   r/   r/   r0   r;   m
  s
    zatan._eval_is_rationalc                 C   s   | j d jS r   )r5   is_extended_positiver   r/   r/   r0   r  u
  s    zatan._eval_is_positivec                 C   s   | j d jS r   )r5   is_extended_nonnegativer   r/   r/   r0   r
  x
  s    zatan._eval_is_nonnegativec                 C   s   | j d jS r   )r5   r7   r   r/   r/   r0   r  {
  s    zatan._eval_is_zeroc                 C   s   | j d jS r   r  r   r/   r/   r0   r  ~
  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 )Nrb   r  r   r   )atanh)r   r   r   r   r   r   r7   rJ   rn   r   r`   r   r   r   r  r  r1   r   r  r.   r,   r   r5   r  r   r   )r   r   r   
atan_tabler   r  r  r/   r/   r0   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 r  )r   rJ   r   r   r   rw   r   r/   r/   r0   r   
  s    zatan.taylor_termNr   c                 C   s   | j d }||d }|jr*||S |tj tjtjfv rZ| t	j
|||d S |dkrn|||}t|dk rt|jrt|tjkr| |t S t|dkrt|jrt|tjk r| |t S | |S r  )r5   r   r  r7   r  r   r.   r`   r   r!   r  rI   r  r    r   rn   r4   r   r   r  r/   r/   r0   r  
  s    

$$zatan._eval_as_leading_termc                 C   s   | j d |d}tj| |||d}|dkr>| j d ||}|tju r`t|dkr\|t S |S t|dk rt|j	rt
|tjkr|t S t|dkrt|j	rt
|tjk r|t S |S r  )r5   r   r   r   r  r   r`   r    r   r7   r   rn   r   r9   rw   r   r   r   r  r  r/   r/   r0   r   
  s    
$$zatan._eval_nseriesc                 K   s2   t jd tt jt j|  tt jt j|    S r~   )r   r.   r!   rn   r  r/   r/   r0   r  
  s    zatan._eval_rewrite_as_logc                    s|   |d t ju r2td td| jd   |||S |d t ju rft d td| jd   |||S t ||||S d S r  )	r   r   r   r   r5   r   r   super_eval_aseriesr9   r   args0rw   r   	__class__r/   r0   r  
  s
    $&zatan._eval_aseriesc                 C   s   t S r}  r  r   r/   r/   r0   r  
  s    zatan.inversec                 K   s0   t |d | td tdt d|d     S r   r$   r   r   r   r/   r/   r0   r  
  s    zatan._eval_rewrite_as_asinc                 K   s(   t |d | tdt d|d    S r   r$   r   r   r/   r/   r0   r  
  s    zatan._eval_rewrite_as_acosc                 K   s   t d| S r   r  r   r/   r/   r0   r  
  s    zatan._eval_rewrite_as_acotc                 K   s$   t |d | tt d|d   S r   r$   r   r   r/   r/   r0   r   
  s    zatan._eval_rewrite_as_asecc                 K   s,   t |d | td tt d|d    S r   r$   r   r   r   r/   r/   r0   r  
  s    zatan._eval_rewrite_as_acsc)rm   )Nr   )r   )rm   )!r[   r\   r]   r^   tTupler   __annotations__r   r.   ra   r   r;   r  r
  r  r  r  r   r  r   r   r  r   r  r  r  r  r  r  r  r   r  __classcell__r/   r/   r  r0   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dZd)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] http://dlmf.nist.gov/4.23
    .. [2] http://functions.wolfram.com/ElementaryFunctions/ArcCot

    rm   c                 C   s,   |dkrdd| j d d   S t| |d S r  r  r   r/   r/   r0   r   .  s    z
acot.fdiffc                 C   s4   | j | j }|j | j kr*|jd jr0dS n|jS d S r3   r  r8   r/   r/   r0   r;   4  s
    zacot._eval_is_rationalc                 C   s   | j d jS r   )r5   r  r   r/   r/   r0   r  <  s    zacot._eval_is_positivec                 C   s   | j d jS r   )r5   r  r   r/   r/   r0   r  ?  s    zacot._eval_is_negativec                 C   s   | j d jS r   r  r   r/   r/   r0   r  B  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 )Nrb   r  r   )acoth)r   r   r   r   rJ   r   r7   r   rn   r   r`   r   r  r  r1   r   r%  r.   rf   r,   r   r5   r  r   r   )r   r   r  r  r   r%  r/   r/   r0   r   E  sX    










z	acot.evalc                 G   sT   | dkrt d S | dk s$| d dkr*tjS t|}tj| d d  ||   |  S d S r  )r   r   rJ   r   r   r  r/   r/   r0   r   {  s    zacot.taylor_termNr   c                 C   s   | j d }||d }|tju r2d| |S |dkrF|||}|tj tjtjfv rv| 	t
j|||d S t|dkrt|jrt|tjkrt|tjk r| |t S t|dk rt|jrt|tjk rt|tjkr| |t S | |S )Nr   rm   r  )r5   r   r  r   r`   r  r  r.   rJ   r   r!   r  rI   r    r7   r   rn   r4   r   r   r  r/   r/   r0   r    s    

22zacot._eval_as_leading_termc                 C   s   | j d |d}tj| |||d}|tju r2|S |dkrL| j d ||}|jrjt|dk rf|t	 S |S t|dkrt|jrt
|tjkrt
|tjk r|t	 S t|dk rt|jrt
|tjk rt
|tjkr|t	 S |S r  )r5   r   r   r   r   r`   r  r7   r    r   r   rJ   rn   r   r  r/   r/   r0   r     s    
22zacot._eval_nseriesc                    s   |d t ju r2td td| jd   |||S |d t ju rjttdd td| jd   |||S tt	| 
||||S d S )Nr   rb   rm   r   )r   r   r   r   r5   r   r   r   r  r   r  r  r  r/   r0   r    s
    $*zacot._eval_aseriesc                 K   s.   t jd tdt j|  tdt j|    S r   )r   r.   r!   r  r/   r/   r0   r    s    zacot._eval_rewrite_as_logc                 C   s   t S r}  r  r   r/   r/   r0   r    s    zacot.inversec                 K   s@   |t d|d   td tt |d  t |d  d    S r|  r  r   r/   r/   r0   r    s    *zacot._eval_rewrite_as_asinc                 K   s8   |t d|d   tt |d  t |d  d   S r|  r  r   r/   r/   r0   r    s    zacot._eval_rewrite_as_acosc                 K   s   t d| S r   r~  r   r/   r/   r0   r    s    zacot._eval_rewrite_as_atanc                 K   s0   |t d|d   tt d|d  |d   S r|  r   r   r/   r/   r0   r     s    zacot._eval_rewrite_as_asecc                 K   s8   |t d|d   td tt d|d  |d    S r|  r!  r   r/   r/   r0   r    s    zacot._eval_rewrite_as_acsc)rm   )Nr   )r   )rm   )r[   r\   r]   r^   r   r.   ra   r   r;   r  r  r  r  r   r  r   r   r  r   r  r  r  r  r  r  r  r   r  r$  r/   r/   r  r0   r     s.   *

5	


r   c                   @   s   e Zd ZdZedd ZdddZdddZdddZd 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d	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] http://dlmf.nist.gov/4.23
    .. [3] http://functions.wolfram.com/ElementaryFunctions/ArcSec
    .. [4] http://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jd S t|tr|jd }|jr|dt ; }|tkrdt | }|S t|tr|jd }|jrtd t| S d S r  )r7   r   r`   r   r   rn   rJ   r   r   r   r   r  r  r  Pir,   r   r5   r  r   r   r   r   Z
acsc_tabler  r/   r/   r0   r     s<    







z	asec.evalrm   c                 C   sB   |dkr4d| j d d tdd| j d d     S t| |d S r  r5   r$   r   r   r/   r/   r0   r   -  s    ,z
asec.fdiffc                 C   s   t S r}  rz  r   r/   r/   r0   r  3  s    zasec.inverseNr   c                 C   s   | j d }||d }|dkr>tdt|tj | S |tj tjfv rf| t	j
|||dS |dkrz|||}t|dk r|jr|tjkr|tjk r| | S t|dkr|jr|tjk r|tjkrdt | | S | |S r	  )r5   r   r  r$   r   rn   r  rJ   r   r!   r  r  r   r  r4   r   r   r  r/   r/   r0   r  9  s    
&&zasec._eval_as_leading_termc                 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 |dkr| jd ||}t|dk r|jr|tjkr|tjk r| S t|dkr8|jr8|tjk r8|tjkr8dt | S |S Nr   r  r  Tr  rb   r  )r  r  r5   r   r   rn   r   r   r   r!   r  r   r  r$   r   r  rI   r  r   r`   r  r   r  rJ   r   r  r/   r/   r0   r   I  s<    
&
&&
&
..zasec._eval_nseriesc                 C   s2   | j d }|jdu rdS t|d j| d jfS r  )r5   rH   r   r  r  r/   r/   r0   r  k  s    

zasec._eval_is_extended_realc              	   K   s0   t d tjttj| tdd|d      S r   r  r   r/   r/   r0   r  q  s    zasec._eval_rewrite_as_logc                 K   s   t d td|  S r   r  r   r/   r/   r0   r  u  s    zasec._eval_rewrite_as_asinc                 K   s   t d| S r   )r   r   r/   r/   r0   r  x  s    zasec._eval_rewrite_as_acosc                 K   s8   t |d | }td d|  |tt |d d   S r   r$   r   r   r9   rw   r   Zsx2xr/   r/   r0   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+  r/   r/   r0   r    s    zasec._eval_rewrite_as_acotc                 K   s   t d t| S r~   r  r   r/   r/   r0   r    s    zasec._eval_rewrite_as_acsc)rm   )rm   )Nr   )r   )r[   r\   r]   r^   r  r   r   r  r  r   r  r  r  r  r  r  r  r  r/   r/   r/   r0   r     s   ;
%



"r   c                   @   sx   e Zd ZdZedd ZdddZdddZdddZdddZ	dd Z
e
Zdd Zdd Zdd Zdd Zdd Zd	S )r   aT  
    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] http://dlmf.nist.gov/4.23
    .. [3] http://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 r  )r7   r   r`   r   r   rn   r   r   r   r   rJ   r   r  r  r  r,   r   r5   r  r   r   r'  r/   r/   r0   r     sD    






z	acsc.evalrm   c                 C   sB   |dkr4d| j d d tdd| j d d     S t| |d S r  r(  r   r/   r/   r0   r     s    ,z
acsc.fdiffc                 C   s   t S r}  r   r   r/   r/   r0   r    s    zacsc.inverseNr   c                 C   s   | j d }||d }|tj tjtjfv rJ| tj|||d	 S |tj
u rbd| |S |dkrv|||}t|dk r|jr|tjkr|tjk rt| | S t|dkr|jr|tjk r|tjkrt | | S | |S )Nr   r  rm   )r5   r   r  r   rn   rJ   r   r!   r  rI   r`   r  r  r   r  r   r4   r   r  r/   r/   r0   r    s    

&&zacsc._eval_as_leading_termc                 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 |dkr| jd ||}t|dk r |jr |tjkr |tjk r t| S t|dkr8|jr8|tjk r8|tjkr8t | S |S r)  )r  r  r5   r   r   rn   r   r   r   r!   r  r   r  r$   r   r  rI   r  r   r`   r  r   r  rJ   r   r  r/   r/   r0   r     s<    
&
&&
&
..
zacsc._eval_nseriesc                 K   s*   t j tt j| tdd|d     S r|  r  r   r/   r/   r0   r    s    zacsc._eval_rewrite_as_logc                 K   s   t d| S r   )r   r   r/   r/   r0   r  !  s    zacsc._eval_rewrite_as_asinc                 K   s   t d td|  S r   r  r   r/   r/   r0   r  $  s    zacsc._eval_rewrite_as_acosc                 K   s,   t |d | td tt |d d   S r   r*  r  r/   r/   r0   r  '  s    zacsc._eval_rewrite_as_atanc                 K   s0   t |d | td tdt |d d    S r   r,  r   r/   r/   r0   r  *  s    zacsc._eval_rewrite_as_acotc                 K   s   t d t| S r~   r  r   r/   r/   r0   r   -  s    zacsc._eval_rewrite_as_asec)rm   )rm   )Nr   )r   )r[   r\   r]   r^   r  r   r   r  r  r   r  r  r  r  r  r  r   r/   r/   r/   r0   r     s   *
,



"r   c                       s\   e Zd ZdZedd Zdd Zdd Zdd	 Zd
d Z	dd Z
dd Z fddZ  ZS )r   a
  
    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 reurns 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] http://functions.wolfram.com/ElementaryFunctions/ArcTan2

    c                 C   s  ddl m} |tju r8|jr tS dt |t| t S |tju rHtjS |j	rp|j	rp|j
rp|j
rpt|}t|}|jr|jr|jrt|| S |jr|jrt|| t S |jrt|| t S n0|jr|jrtd S |jrt d S |jrtjS |jrD|jrttj||  S |j
rDttt|dk fdt|dftjdfS |j
r|j
rtj t|tj|  t|d |d    S d S )Nr   )	Heavisiderb   T)'sympy.functions.special.delta_functionsr-  r   r   r7   r   r    r   rJ   r  r  r   rH   r  r   r  r  r   is_extended_nonzerorn   r'   r   r.   r!   r$   )r   r   rw   r-  r/   r/   r0   r     sJ    


 z
atan2.evalc                 K   s.   t j t|t j|  t|d |d    S r~   r  r9   r   rw   r   r/   r/   r0   r    s    zatan2._eval_rewrite_as_logc              	   K   sT   t dt||t|d |d     t|dftt|dk fdt|dftjdfS )Nrb   r   T)r'   r   r$   r   r   r    r   r   r0  r/   r/   r0   r    s
    .zatan2._eval_rewrite_as_atanc                 K   sj   |j r|j rt||tj  S |tj|  }|d |d  }t|t| tjtt|tt|   S r~   )rH   arg_fr   r.   r$   r!   rR   )r9   r   rw   r   r   r   r/   r/   r0   _eval_rewrite_as_arg  s
    zatan2._eval_rewrite_as_argc                 C   s   | j d jo| j d jS r  r  r   r/   r/   r0   r    s    zatan2._eval_is_extended_realc                 C   s    |  | jd  | jd  S r  r   r   r/   r/   r0   r     s    zatan2._eval_conjugatec                 C   sR   | j \}}|dkr&||d |d   S |dkrD| |d |d   S t| |d S r|  r  )r9   r   r   rw   r/   r/   r0   r     s    
zatan2.fdiffc                    s&   | j \}}|jr"|jr"t |S d S r   )r5   rH   r  _eval_evalf)r9   precr   rw   r  r/   r0   r3    s    
zatan2._eval_evalf)r[   r\   r]   r^   r  r   r  r  r2  r  r   r   r3  r$  r/   r/   r  r0   r   1  s   f
'r   N)rm   )Vtypingr   r"  Zsympy.core.addr   sympy.core.basicr   r   sympy.core.exprr   sympy.core.functionr   r   r	   r
   sympy.core.logicr   r   r   r   Zsympy.core.modr   sympy.core.numbersr   r   r   r   r   sympy.core.relationalr   r   sympy.core.singletonr   sympy.core.symbolr   r   (sympy.functions.combinatorial.factorialsr   r   %sympy.functions.combinatorial.numbersr   r   r  r   r1  r   r    &sympy.functions.elementary.exponentialr!   r"   #sympy.functions.elementary.integersr#   (sympy.functions.elementary.miscellaneousr$   r%   r&   $sympy.functions.elementary.piecewiser'   sympy.logic.boolalgr(   sympy.ntheoryr)   sympy.polys.specialpolysr*   sympy.utilities.iterablesr+   r1   r2   rl   r>   r}   r   r   r   r  r   r   r   r  r   r   r   r   r   r   r   r/   r/   r/   r0   <module>   sp   D%
J  8   R  Y  Cxee|Q [ Y F K = +