B
    [                 @   s  d dl mZmZ d dlmZmZmZmZmZ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 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  d dl!m"Z"m#Z#m$Z$ d dl%m&Z&m'Z' d dl(m)Z) G dd deZ*G dd deZ+G dd deZ,G dd deZ-G dd deZ.G dd deZ/G dd deZ0G dd deZ1G dd deZ2G d d! d!eZ3d"d# Z4G d$d% d%eZ5d2d'd(Z6d3d*d+Z7d4d,d-Z8i d&fd.d/Z9d d0lm:Z; e-e;_<[;d1S )5    )print_functiondivision)SAddMulsympifySymbolDummyBasic)factor_terms)Function
DerivativeArgumentIndexErrorAppliedUndef)piIoo)sqrt)	Piecewise)Expr)Eq)	fuzzy_notfuzzy_or)exp	exp_polarlog)atanatan2)ceilingc               @   sV   e Zd ZdZdZdZedd ZdddZdd Z	d	d
 Z
dd Zdd Zdd ZdS )rea)  
    Returns real part of expression. This function performs only
    elementary analysis and so it will fail to decompose properly
    more complicated expressions. If completely simplified result
    is needed then use Basic.as_real_imag() or perform complex
    expansion on instance of this function.

    Examples
    ========

    >>> from sympy import re, im, I, E
    >>> from sympy.abc import x, y
    >>> re(2*E)
    2*E
    >>> re(2*I + 17)
    17
    >>> re(2*I)
    0
    >>> re(im(x) + x*I + 2)
    2

    See Also
    ========
    im
    Tc             C   sN  |t jkrt jS |t jkr t jS |jr*|S |js<t j| jrBt jS |jrT| d S |j	rrt
|trrt|jd S g g g   }}}t|}x||D ]t}|t j}|d k	r|js|| q|t js|jr|| q|j|d}| r||d  q|| qW t|t|krJdd |||gD \}	}
}| |	t|
 | S d S )Nr   )ignorec             s   s   | ]}t | V  qd S )N)r   ).0xs r#   Clib/python3.7/site-packages/sympy/functions/elementary/complexes.py	<genexpr>Y   s    zre.eval.<locals>.<genexpr>)r   NaNComplexInfinityis_realis_imaginaryImaginaryUnitZero	is_Matrixas_real_imagis_Function
isinstance	conjugater   argsr   	make_argsas_coefficientappendhaslenim)clsargincludedrevertedexcludedr1   termcoeff	real_imagabcr#   r#   r$   eval4   s8    



zre.evalc             K   s
   | t jfS )zE
        Returns the real number with a zero imaginary part.
        )r   r+   )selfdeephintsr#   r#   r$   r-   ]   s    zre.as_real_imagc             C   s`   |j s| jd j r*tt| jd |ddS |js<| jd jr\tj tt| jd |dd S d S )Nr   T)evaluate)r(   r1   r   r   r)   r   r*   r7   )rD   xr#   r#   r$   _eval_derivativec   s
    zre._eval_derivativec             C   s   | j d tjt| j d   S )Nr   )r1   r   r*   r7   )rD   r9   r#   r#   r$   _eval_rewrite_as_imj   s    zre._eval_rewrite_as_imc             C   s   | j d jS )Nr   )r1   is_algebraic)rD   r#   r#   r$   _eval_is_algebraicm   s    zre._eval_is_algebraicc             C   s   t | jd j| jd jgS )Nr   )r   r1   r)   is_zero)rD   r#   r#   r$   _eval_is_zerop   s    zre._eval_is_zeroc             C   s    dd l m} || jd  S )Nr   )sage.allallZ	real_partr1   _sage_)rD   sager#   r#   r$   rQ   t   s    z	re._sage_N)T)__name__
__module____qualname____doc__r(   
unbranchedclassmethodrC   r-   rI   rJ   rL   rN   rQ   r#   r#   r#   r$   r      s   )
r   c               @   sV   e Zd ZdZdZdZedd ZdddZdd Z	d	d
 Z
dd Zdd Zdd ZdS )r7   a/  
    Returns imaginary part of expression. This function performs only
    elementary analysis and so it will fail to decompose properly more
    complicated expressions. If completely simplified result is needed then
    use Basic.as_real_imag() or perform complex expansion on instance of
    this function.

    Examples
    ========

    >>> from sympy import re, im, E, I
    >>> from sympy.abc import x, y
    >>> im(2*E)
    0
    >>> re(2*I + 17)
    17
    >>> im(x*I)
    re(x)
    >>> im(re(x) + y)
    im(y)

    See Also
    ========

    re
    Tc             C   sX  |t jkrt jS |t jkr t jS |jr,t jS |js>t j| jrJt j | S |jr\| d S |j	r|t
|tr|t|jd  S g g g   }}}t|}x||D ]t}|t j}|d k	r|js|| n
|| q|t js|js|j|d}|r||d  q|| qW t|t|krTdd |||gD \}	}
}| |	t|
 | S d S )N   r   )r    c             s   s   | ]}t | V  qd S )N)r   )r!   r"   r#   r#   r$   r%      s    zim.eval.<locals>.<genexpr>)r   r&   r'   r(   r+   r)   r*   r,   r-   r.   r/   r0   r7   r1   r   r2   r3   r4   r5   r6   r   )r8   r9   r:   r;   r<   r1   r=   r>   r?   r@   rA   rB   r#   r#   r$   rC      s8    



zim.evalc             K   s
   | t jfS )z
        Return the imaginary part with a zero real part.

        Examples
        ========

        >>> from sympy.functions import im
        >>> from sympy import I
        >>> im(2 + 3*I).as_real_imag()
        (3, 0)
        )r   r+   )rD   rE   rF   r#   r#   r$   r-      s    zim.as_real_imagc             C   s`   |j s| jd j r*tt| jd |ddS |js<| jd jr\tj tt| jd |dd S d S )Nr   T)rG   )r(   r1   r7   r   r)   r   r*   r   )rD   rH   r#   r#   r$   rI      s
    zim._eval_derivativec             C   s    dd l m} || jd  S )Nr   )rO   rP   Z	imag_partr1   rQ   )rD   rR   r#   r#   r$   rQ      s    z	im._sage_c             C   s    t j | jd t| jd   S )Nr   )r   r*   r1   r   )rD   r9   r#   r#   r$   _eval_rewrite_as_re   s    zim._eval_rewrite_as_rec             C   s   | j d jS )Nr   )r1   rK   )rD   r#   r#   r$   rL      s    zim._eval_is_algebraicc             C   s   | j d jS )Nr   )r1   r(   )rD   r#   r#   r$   rN      s    zim._eval_is_zeroN)T)rS   rT   rU   rV   r(   rW   rX   rC   r-   rI   rQ   rZ   rL   rN   r#   r#   r#   r$   r7   y   s   (
r7   c               @   s   e Zd ZdZdZdZdd Ze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!S )"signaH  
    Returns the complex sign of an expression:

    If the expression is real the sign will be:

        * 1 if expression is positive
        * 0 if expression is equal to zero
        * -1 if expression is negative

    If the expression is imaginary the sign will be:

        * I if im(expression) is positive
        * -I if im(expression) is negative

    Otherwise an unevaluated expression will be returned. When evaluated, the
    result (in general) will be ``cos(arg(expr)) + I*sin(arg(expr))``.

    Examples
    ========

    >>> from sympy.functions import sign
    >>> from sympy.core.numbers import I

    >>> sign(-1)
    -1
    >>> sign(0)
    0
    >>> sign(-3*I)
    -I
    >>> sign(1 + I)
    sign(1 + I)
    >>> _.evalf()
    0.707106781186548 + 0.707106781186548*I

    See Also
    ========

    Abs, conjugate
    Tc             K   s,   | j d jdkr(| j d t| j d  S | S )Nr   F)r1   rM   Abs)rD   rF   r#   r#   r$   doit  s    z	sign.doitc       	      C   sB  |j r| \}}g }t|}xX|D ]P}|jr6| }q$|jr>q$t|}|jrj|jrj|tj	9 }|jrt| }q$|
| q$W |tjkrt|t|krd S || |j|  S |tjkrtjS |jrtjS |jrtjS |jrtjS |jrt|tr|S |jr>|jr|jtjkrtj	S tj	 | }|jr.tj	S |jr>tj	 S d S )N)is_Mulas_coeff_mulr[   is_negativeis_positiver7   r)   is_comparabler   r*   r4   Oner6   Z_new_rawargsr&   rM   r+   NegativeOner.   r/   is_Powr   ZHalf)	r8   r9   rB   r1   unksr@   Zaiarg2r#   r#   r$   rC     sJ    



z	sign.evalc             C   s   t | jd jrtjS d S )Nr   )r   r1   rM   r   rc   )rD   r#   r#   r$   	_eval_AbsF  s    zsign._eval_Absc             C   s   t t| jd S )Nr   )r[   r0   r1   )rD   r#   r#   r$   _eval_conjugateJ  s    zsign._eval_conjugatec             C   s   | j d jr>ddlm} dt| j d |dd || j d  S | j d jrddlm} dt| j d |dd |tj | j d   S d S )Nr   )
DiracDelta   T)rG   )r1   r(   'sympy.functions.special.delta_functionsrk   r   r)   r   r*   )rD   rH   rk   r#   r#   r$   rI   M  s    &zsign._eval_derivativec             C   s   | j d jrdS d S )Nr   T)r1   is_nonnegative)rD   r#   r#   r$   _eval_is_nonnegativeW  s    zsign._eval_is_nonnegativec             C   s   | j d jrdS d S )Nr   T)r1   is_nonpositive)rD   r#   r#   r$   _eval_is_nonpositive[  s    zsign._eval_is_nonpositivec             C   s   | j d jS )Nr   )r1   r)   )rD   r#   r#   r$   _eval_is_imaginary_  s    zsign._eval_is_imaginaryc             C   s   | j d jS )Nr   )r1   r(   )rD   r#   r#   r$   _eval_is_integerb  s    zsign._eval_is_integerc             C   s   | j d jS )Nr   )r1   rM   )rD   r#   r#   r$   rN   e  s    zsign._eval_is_zeroc             C   s&   t | jd jr"|jr"|jr"tjS d S )Nr   )r   r1   rM   
is_integeris_evenr   rc   )rD   otherr#   r#   r$   _eval_powerh  s    zsign._eval_powerc             C   s    dd l m} || jd  S )Nr   )rO   rP   Zsgnr1   rQ   )rD   rR   r#   r#   r$   rQ   p  s    zsign._sage_c             C   s&   |j r"td|dkfd|dk fdS d S )NrY   r   )r   T)r(   r   )rD   r9   r#   r#   r$   _eval_rewrite_as_Piecewiset  s    zsign._eval_rewrite_as_Piecewisec             C   s&   ddl m} |jr"||d d S d S )Nr   )	Heavisiderl   rY   )rm   rz   r(   )rD   r9   rz   r#   r#   r$   _eval_rewrite_as_Heavisidex  s    zsign._eval_rewrite_as_Heavisidec             C   s   |  | jd  S )Nr   )funcr1   Zfactor)rD   ZratioZmeasureZrationalZinverser#   r#   r$   _eval_simplify}  s    zsign._eval_simplifyN)rS   rT   rU   rV   	is_finiteZ
is_complexr]   rX   rC   ri   rj   rI   ro   rq   rr   rs   rN   rw   rQ   ry   r{   r}   r#   r#   r#   r$   r[      s$   '/
r[   c               @   s   e Zd ZdZdZdZdZd(ddZedd Z	d	d
 Z
dd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd  Zd!d" Zd#d$ Zd%d& Zd'S ))r\   a  
    Return the absolute value of the argument.

    This is an extension of the built-in function abs() to accept symbolic
    values.  If you pass a SymPy expression to the built-in abs(), it will
    pass it automatically to Abs().

    Examples
    ========

    >>> from sympy import Abs, Symbol, S
    >>> Abs(-1)
    1
    >>> x = Symbol('x', real=True)
    >>> Abs(-x)
    Abs(x)
    >>> Abs(x**2)
    x**2
    >>> abs(-x) # The Python built-in
    Abs(x)

    Note that the Python built-in will return either an Expr or int depending on
    the argument::

        >>> type(abs(-1))
        <... 'int'>
        >>> type(abs(S.NegativeOne))
        <class 'sympy.core.numbers.One'>

    Abs will always return a sympy object.

    See Also
    ========

    sign, conjugate
    TFrY   c             C   s$   |dkrt | jd S t| |dS )z
        Get the first derivative of the argument to Abs().

        Examples
        ========

        >>> from sympy.abc import x
        >>> from sympy.functions import Abs
        >>> Abs(-x).fdiff()
        sign(x)
        rY   r   N)r[   r1   r   )rD   Zargindexr#   r#   r$   fdiff  s    z	Abs.fdiffc                s@  ddl m} ddlm} t dr6  }|d k	r6|S t tsPtdt	  | dd  j
rg }g }x< jD ]2}| |}t|| r||jd  qr|| qrW t| }|r| t| ddntj}|| S  tjkrtjS  tjkrtjS  jrֈ  \}	}
|	jr|
jr\|
jr" S |	tjkr4tjS t|	| rP|
tjkrP S t|	|
 S |	jrp|	t|
 S |	jr|	 t|
 ttj t|
  S d S |	 t!st"|	# \}}|t$|  }tt|
| S t trtt jd S t t%rd S  j&r<  tjtj'r<t(dd	  # D r<tjS  j)rJtj*S  jrV S  j+rd  S  j,rtj-   }|jr|S | . dd/t. /t. }|rt0 fd
d	|D rd S  kr<  kr< /t} 1dd |D }dd |j2D }|r,t0fdd	|D s<t3|  S d S )Nr   )signsimp)
expand_mulri   zBad argument type for Abs(): %sF)rG   c             s   s   | ]}|j V  qd S )N)Zis_infinite)r!   r@   r#   r#   r$   r%     s    zAbs.eval.<locals>.<genexpr>c             3   s   | ]}  |jd  V  qdS )r   N)r5   r1   )r!   i)r9   r#   r$   r%     s    c             S   s   i | ]}t d d|qS )T)real)r	   )r!   r   r#   r#   r$   
<dictcomp>  s    zAbs.eval.<locals>.<dictcomp>c             S   s   g | ]}|j d kr|qS )N)r(   )r!   r@   r#   r#   r$   
<listcomp>	  s    zAbs.eval.<locals>.<listcomp>c             3   s   | ]}  t|V  qd S )N)r5   r0   )r!   u)conjr#   r$   r%   
  s    )4Zsympy.simplify.simplifyr   sympy.core.functionr   hasattrri   r/   r   	TypeErrortyper^   r1   r4   r   r   rc   r&   r'   ZInfinityre   Zas_base_expr(   rt   ru   rd   r\   rn   r   r`   r   ZPir7   r5   r   r   r-   r   r   is_AddZNegativeInfinityanyrM   r+   rp   r)   r*   r0   atomsrP   Zxreplacefree_symbolsr   )r8   r9   r   r   objZknownrf   tZtnewbaseexponentr@   rA   zrh   Znew_conjr    Zabs_free_argr#   )r9   r   r$   rC     s    




"
zAbs.evalc             C   s   | j d jr| j d jS d S )Nr   )r1   r(   rt   )rD   r#   r#   r$   rs     s    zAbs._eval_is_integerc             C   s   t | jd jS )Nr   )r   _argsrM   )rD   r#   r#   r$   _eval_is_nonzero  s    zAbs._eval_is_nonzeroc             C   s   | j d jS )Nr   )r   rM   )rD   r#   r#   r$   rN     s    zAbs._eval_is_zeroc             C   s   | j }|d k	r| S d S )N)rM   )rD   Zis_zr#   r#   r$   _eval_is_positive  s    zAbs._eval_is_positivec             C   s   | j d jr| j d jS d S )Nr   )r1   r(   Zis_rational)rD   r#   r#   r$   _eval_is_rational  s    zAbs._eval_is_rationalc             C   s   | j d jr| j d jS d S )Nr   )r1   r(   ru   )rD   r#   r#   r$   _eval_is_even   s    zAbs._eval_is_evenc             C   s   | j d jr| j d jS d S )Nr   )r1   r(   Zis_odd)rD   r#   r#   r$   _eval_is_odd$  s    zAbs._eval_is_oddc             C   s   | j d jS )Nr   )r1   rK   )rD   r#   r#   r$   rL   (  s    zAbs._eval_is_algebraicc             C   sP   | j d jrL|jrL|jr&| j d | S |tjk	rL|jrL| j d |d  |  S d S )Nr   rY   )r1   r(   rt   ru   r   rd   Z
is_Integer)rD   r   r#   r#   r$   rw   +  s    zAbs._eval_powerc             C   sV   | j d |d }| j d j|||d}t|d}t||d|ft|| dfS )Nr   )nlogxT)r1   Zleadterm_eval_nseriesr   r   subsr[   )rD   rH   r   r   	directionrg   Zwhenr#   r#   r$   r   3  s    
zAbs._eval_nseriesc             C   s    dd l m} || jd  S )Nr   )rO   rP   Zabs_symbolicr1   rQ   )rD   rR   r#   r#   r$   rQ   <  s    z
Abs._sage_c             C   s   | j d js| j d jr>t| j d |ddtt| j d  S t| j d tt| j d |dd t| j d tt| j d |dd  t| j d  S )Nr   T)rG   )	r1   r(   r)   r   r[   r0   r   r7   r\   )rD   rH   r#   r#   r$   rI   @  s    "zAbs._eval_derivativec             C   s,   ddl m} |jr(|||||   S d S )Nr   )rz   )rm   rz   r(   )rD   r9   rz   r#   r#   r$   r{   H  s    zAbs._eval_rewrite_as_Heavisidec             C   s"   |j rt||dkf| dfS d S )Nr   T)r(   r   )rD   r9   r#   r#   r$   ry   O  s    zAbs._eval_rewrite_as_Piecewisec             C   s   |t | S )N)r[   )rD   r9   r#   r#   r$   _eval_rewrite_as_signS  s    zAbs._eval_rewrite_as_signN)rY   )rS   rT   rU   rV   r(   r`   rW   r   rX   rC   rs   r   rN   r   r   r   r   rL   rw   r   rQ   rI   r{   ry   r   r#   r#   r#   r$   r\     s*   $
Q	r\   c               @   s4   e Zd ZdZdZdZedd Zdd Zdd Z	d	S )
r9   a5  
    Returns the argument (in radians) of a complex number. For a positive
    number, the argument is always 0.

    Examples
    ========

    >>> from sympy.functions import arg
    >>> from sympy import I, sqrt
    >>> arg(2.0)
    0
    >>> arg(I)
    pi/2
    >>> arg(sqrt(2) + I*sqrt(2))
    pi/4

    Tc             C   s   t |trt|tS |jsRt| \}}|jrDtdd |j	D  }t
|| }n|}|trdd S | \}}t||}|jr|S ||kr| |ddS d S )Nc             S   s$   g | ]}t |d kr|nt |qS ))rx   rY   )r[   )r!   r@   r#   r#   r$   r   t  s   zarg.eval.<locals>.<listcomp>F)rG   )r/   r   periodic_argumentr   is_Atomr   Zas_coeff_Mulr^   r   r1   r[   r   r   r-   r   	is_number)r8   r9   rB   Zarg_rH   yrvr#   r#   r$   rC   m  s"    



zarg.evalc             C   sF   | j d  \}}|t||dd |t||dd  |d |d   S )Nr   T)rG   rl   )r1   r-   r   )rD   r   rH   r   r#   r#   r$   rI     s    zarg._eval_derivativec             C   s   | j d  \}}t||S )Nr   )r1   r-   r   )rD   r9   rH   r   r#   r#   r$   _eval_rewrite_as_atan2  s    zarg._eval_rewrite_as_atan2N)
rS   rT   rU   rV   r(   r~   rX   rC   rI   r   r#   r#   r#   r$   r9   W  s   r9   c               @   sL   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dS )r0   a,  
    Returns the `complex conjugate` Ref[1] of an argument.
    In mathematics, the complex conjugate of a complex number
    is given by changing the sign of the imaginary part.

    Thus, the conjugate of the complex number
    :math:`a + ib` (where a and b are real numbers) is :math:`a - ib`

    Examples
    ========

    >>> from sympy import conjugate, I
    >>> conjugate(2)
    2
    >>> conjugate(I)
    -I

    See Also
    ========

    sign, Abs

    References
    ==========

    .. [1] http://en.wikipedia.org/wiki/Complex_conjugation
    c             C   s   |  }|d k	r|S d S )N)rj   )r8   r9   r   r#   r#   r$   rC     s    zconjugate.evalc             C   s   t | jd ddS )Nr   T)rG   )r\   r1   )rD   r#   r#   r$   ri     s    zconjugate._eval_Absc             C   s   t | jd S )Nr   )	transposer1   )rD   r#   r#   r$   _eval_adjoint  s    zconjugate._eval_adjointc             C   s
   | j d S )Nr   )r1   )rD   r#   r#   r$   rj     s    zconjugate._eval_conjugatec             C   sB   |j rtt| jd |ddS |jr>tt| jd |dd S d S )Nr   T)rG   )r(   r0   r   r1   r)   )rD   rH   r#   r#   r$   rI     s    zconjugate._eval_derivativec             C   s   t | jd S )Nr   )adjointr1   )rD   r#   r#   r$   _eval_transpose  s    zconjugate._eval_transposec             C   s   | j d jS )Nr   )r1   rK   )rD   r#   r#   r$   rL     s    zconjugate._eval_is_algebraicN)rS   rT   rU   rV   rX   rC   ri   r   rj   rI   r   rL   r#   r#   r#   r$   r0     s   r0   c               @   s4   e Zd ZdZedd Zdd Zdd Zdd	 Zd
S )r   z#
    Linear map transposition.
    c             C   s   |  }|d k	r|S d S )N)r   )r8   r9   r   r#   r#   r$   rC     s    ztranspose.evalc             C   s   t | jd S )Nr   )r0   r1   )rD   r#   r#   r$   r     s    ztranspose._eval_adjointc             C   s   t | jd S )Nr   )r   r1   )rD   r#   r#   r$   rj     s    ztranspose._eval_conjugatec             C   s
   | j d S )Nr   )r1   )rD   r#   r#   r$   r     s    ztranspose._eval_transposeN)	rS   rT   rU   rV   rX   rC   r   rj   r   r#   r#   r#   r$   r     s
   r   c               @   sF   e Zd ZdZedd Zdd Zdd Zdd	 ZdddZ	dd Z
d
S )r   z5
    Conjugate transpose or Hermite conjugation.
    c             C   s0   |  }|d k	r|S | }|d k	r,t|S d S )N)r   r   r0   )r8   r9   r   r#   r#   r$   rC     s    zadjoint.evalc             C   s
   | j d S )Nr   )r1   )rD   r#   r#   r$   r     s    zadjoint._eval_adjointc             C   s   t | jd S )Nr   )r   r1   )rD   r#   r#   r$   rj     s    zadjoint._eval_conjugatec             C   s   t | jd S )Nr   )r0   r1   )rD   r#   r#   r$   r     s    zadjoint._eval_transposeNc             G   s2   | | jd }d| }|r.d|| |f }|S )Nr   z%s^{\dagger}z\left(%s\right)^{%s})_printr1   )rD   printerr   r1   r9   Ztexr#   r#   r$   _latex  s
    zadjoint._latexc             G   sF   ddl m} |j| jd f| }|jr6||d }n||d }|S )Nr   )
prettyFormu   †+)Z sympy.printing.pretty.stringpictr   r   r1   Z_use_unicode)rD   r   r1   r   Zpformr#   r#   r$   _pretty  s    zadjoint._pretty)N)rS   rT   rU   rV   rX   rC   r   rj   r   r   r   r#   r#   r#   r$   r     s   	
r   c               @   s4   e Zd ZdZdZdZedd Zdd Zdd	 Z	d
S )
polar_lifta4  
    Lift argument to the Riemann surface of the logarithm, using the
    standard branch.

    >>> from sympy import Symbol, polar_lift, I
    >>> p = Symbol('p', polar=True)
    >>> x = Symbol('x')
    >>> polar_lift(4)
    4*exp_polar(0)
    >>> polar_lift(-4)
    4*exp_polar(I*pi)
    >>> polar_lift(-I)
    exp_polar(-I*pi/2)
    >>> polar_lift(I + 2)
    polar_lift(2 + I)

    >>> polar_lift(4*x)
    4*polar_lift(x)
    >>> polar_lift(4*p)
    4*p

    See Also
    ========

    sympy.functions.elementary.exponential.exp_polar
    periodic_argument
    TFc             C   s   ddl m} |jrH||}|dtd t d tfkrHtt| t| S |jrV|j}n|g}g }g }g }x:|D ]2}|j	r||g7 }qn|j
r||g7 }qn||g7 }qnW t|t|k r|rt||  tt|  S |rt||  S t| td S d S )Nr   )r9   rl   )Z$sympy.functions.elementary.complexesr9   r   r   r   r   absr^   r1   is_polarra   r6   r   r   )r8   r9   Zargumentarr1   r:   r<   Zpositiver#   r#   r$   rC   &  s.    
zpolar_lift.evalc             C   s   | j d |S )z. Careful! any evalf of polar numbers is flaky r   )r1   _eval_evalf)rD   precr#   r#   r$   r   H  s    zpolar_lift._eval_evalfc             C   s   t | jd ddS )Nr   T)rG   )r\   r1   )rD   r#   r#   r$   ri   L  s    zpolar_lift._eval_AbsN)
rS   rT   rU   rV   r   rb   rX   rC   r   ri   r#   r#   r#   r$   r     s   "r   c               @   s0   e Zd ZdZedd Zedd Zdd ZdS )	r   a  
    Represent the argument on a quotient of the Riemann surface of the
    logarithm. That is, given a period P, always return a value in
    (-P/2, P/2], by using exp(P*I) == 1.

    >>> from sympy import exp, exp_polar, periodic_argument, unbranched_argument
    >>> from sympy import I, pi
    >>> unbranched_argument(exp(5*I*pi))
    pi
    >>> unbranched_argument(exp_polar(5*I*pi))
    5*pi
    >>> periodic_argument(exp_polar(5*I*pi), 2*pi)
    pi
    >>> periodic_argument(exp_polar(5*I*pi), 3*pi)
    -pi
    >>> periodic_argument(exp_polar(5*I*pi), pi)
    0

    See Also
    ========

    sympy.functions.elementary.exponential.exp_polar
    polar_lift : Lift argument to the Riemann surface of the logarithm
    principal_branch
    c             C   s   |j r|j}n|g}d}x|D ]}|js6|t|7 }qt|trT||j d 7 }q|jr|j \}}||t	|j
 |tt|j
  7 }qt|tr|t|jd 7 }qd S qW |S )Nr   rY   )r^   r1   r   r9   r/   r   r   r-   re   unbranched_argumentr   r   r   r   )r8   r   r1   rW   r@   r   r7   r#   r#   r$   _getunbranchedk  s"    

 
z periodic_argument._getunbranchedc             C   s   |j s
d S |tkr&t|tr&t|j S t|trL|dt krLt|jd |S |jrdd |jD }t	|t	|jkrtt
| |S | |}|d krd S |tttrd S |tkr|S |tkrt|| tdd  | }|ts|| S d S )Nrl   r   c             S   s   g | ]}|j s|qS r#   )ra   )r!   rH   r#   r#   r$   r     s    z*periodic_argument.eval.<locals>.<listcomp>rY   )ra   r   r/   principal_branchr   r1   r   r   r^   r6   r   r   r5   r   r   r   r   )r8   r   periodZnewargsrW   r   r#   r#   r$   rC     s*    


zperiodic_argument.evalc             C   sh   | j \}}|tkr2t|}|d kr(| S ||S t|t|}|t|| tdd  |  |S )NrY   rl   )r1   r   r   r   r   r   r   )rD   r   r   r   rW   ubr#   r#   r$   r     s    


zperiodic_argument._eval_evalfN)rS   rT   rU   rV   rX   r   rC   r   r#   r#   r#   r$   r   P  s   r   c             C   s
   t | tS )N)r   r   )r9   r#   r#   r$   r     s    r   c               @   s,   e Zd ZdZdZdZedd Zdd ZdS )	r   a  
    Represent a polar number reduced to its principal branch on a quotient
    of the Riemann surface of the logarithm.

    This is a function of two arguments. The first argument is a polar
    number `z`, and the second one a positive real number of infinity, `p`.
    The result is "z mod exp_polar(I*p)".

    >>> from sympy import exp_polar, principal_branch, oo, I, pi
    >>> from sympy.abc import z
    >>> principal_branch(z, oo)
    z
    >>> principal_branch(exp_polar(2*pi*I)*3, 2*pi)
    3*exp_polar(0)
    >>> principal_branch(exp_polar(2*pi*I)*3*z, 2*pi)
    3*principal_branch(z, 2*pi)

    See Also
    ========

    sympy.functions.elementary.exponential.exp_polar
    polar_lift : Lift argument to the Riemann surface of the logarithm
    periodic_argument
    TFc                s(  ddl m}m}m}m}mm  t|r:t|j	d |S ||krF|S t
||}t
||}||kr|t
s|t
s|}	 fdd}
|	|
}	t
|	|}|	s||kr||||  |	 }n|	}|js||s||d9 }|S |js|d }}n|j|j \}}g }x*|D ]"}|jr2||9 }n
||g7 }qW t|}t
||}|t
rdd S |jrt||ks|dkr|dkr|dkr|dkrt|t|| | S t||| ||  |t| S |jr$t||d k dks||d kr$|dkr$||| t| S d S )	Nr   )r   r   r   r   r   r   c                s   t |  s| S | S )N)r/   )expr)r   r   r#   r$   mr  s    
z!principal_branch.eval.<locals>.mrr#   rY   rl   T)sympyr   r   r   r   r   r   r/   r   r1   r   r5   replacer   r   r_   ra   tupler   r   r   )rD   rH   r   r   r   r   r   r   ZbargZplr   resrB   mZothersr   r9   r#   )r   r   r$   rC     sP     









",
zprincipal_branch.evalc             C   sb   ddl m}m}m} | j\}}t|||}t||ksD|| krH| S t||||  |S )Nr   )r   r   r   )r   r   r   r   r1   r   r   r   )rD   r   r   r   r   r   r   pr#   r#   r$   r     s    
zprincipal_branch._eval_evalfN)	rS   rT   rU   rV   r   rb   rX   rC   r   r#   r#   r#   r$   r     s
   4r   Fc       
         s>  ddl m} | jr| S | jr(s(t| S t| trBsB rBt| S | jrL| S | jr|| j	 fdd| j
D  } rxt|S |S | jr| j	 fdd| j
D  S t| |rt| j d}g }xN| j
dd  D ]<}t|d dd	}t|dd   d	}	||f|	  qW ||ft|  S | j	 fd
d| j
D  S d S )Nr   )Integralc                s   g | ]}t | d dqS )T)pause)	_polarify)r!   r9   )liftr#   r$   r     s    z_polarify.<locals>.<listcomp>c                s   g | ]}t | d dqS )F)r   )r   )r!   r9   )r   r#   r$   r     s    )r   rY   F)r   r   c                s(   g | ] }t |tr t| d n|qS ))r   )r/   r   r   )r!   r9   )r   r   r#   r$   r   !  s   )r   r   r   r   r   r/   r   r   r   r|   r1   r.   r   Zfunctionr4   r   )
eqr   r   r   rr|   Zlimitslimitvarrestr#   )r   r   r$   r     s4    
r   Tc             C   sN   |rd}t t| |} |s| S dd | jD }| |} | dd | D fS )a  
    Turn all numbers in eq into their polar equivalents (under the standard
    choice of argument).

    Note that no attempt is made to guess a formal convention of adding
    polar numbers, expressions like 1 + x will generally not be altered.

    Note also that this function does not promote exp(x) to exp_polar(x).

    If ``subs`` is True, all symbols which are not already polar will be
    substituted for polar dummies; in this case the function behaves much
    like posify.

    If ``lift`` is True, both addition statements and non-polar symbols are
    changed to their polar_lift()ed versions.
    Note that lift=True implies subs=False.

    >>> from sympy import polarify, sin, I
    >>> from sympy.abc import x, y
    >>> expr = (-x)**y
    >>> expr.expand()
    (-x)**y
    >>> polarify(expr)
    ((_x*exp_polar(I*pi))**_y, {_x: x, _y: y})
    >>> polarify(expr)[0].expand()
    _x**_y*exp_polar(_y*I*pi)
    >>> polarify(x, lift=True)
    polar_lift(x)
    >>> polarify(x*(1+y), lift=True)
    polar_lift(x)*polar_lift(y + 1)

    Adds are treated carefully:

    >>> polarify(1 + sin((1 + I)*x))
    (sin(_x*polar_lift(1 + I)) + 1, {_x: x})
    Fc             S   s   i | ]}t |jd d|qS )T)Zpolar)r	   name)r!   rg   r#   r#   r$   r   O  s    zpolarify.<locals>.<dictcomp>c             S   s   i | ]\}}||qS r#   r#   )r!   rg   r   r#   r#   r$   r   Q  s    )r   r   r   r   items)r   r   r   Zrepsr#   r#   r$   polarify%  s    %
r   c                sF  t | tr| jr| S |st | tr2tt| j S t | tr^| jd dt kr^t| jd  S | j	s| j
s| js| jr| jdkrd| jks| jdkr| j fdd| jD  S t | trt| jd  S | jrt| j }t| j |jo|  }|| S | jr,t| jddr,| j fd	d| jD  S | j fd
d| jD  S )NrY   rl   r   )z==z!=c                s   g | ]}t | qS r#   )_unpolarify)r!   rH   )exponents_onlyr#   r$   r   c  s    z_unpolarify.<locals>.<listcomp>rW   Fc                s   g | ]}t |  qS r#   )r   )r!   rH   )r   r#   r$   r   n  s   c                s   g | ]}t | d qS )T)r   )r!   rH   )r   r#   r$   r   q  s    )r/   r
   r   r   r   r   r   r1   r   r   r^   Z
is_BooleanZis_RelationalZrel_opr|   r   re   r   rt   r.   getattr)r   r   r   Zexpor   r#   )r   r$   r   T  s.    


r   c             C   s   t | tr| S t| } |i kr,t| |S d}d}|r<d}x6|rrd}t| ||}|| krbd}|} t |tr>|S q>W |tddtddiS )a  
    If p denotes the projection from the Riemann surface of the logarithm to
    the complex line, return a simplified version eq' of `eq` such that
    p(eq') == p(eq).
    Also apply the substitution subs in the end. (This is a convenience, since
    ``unpolarify``, in a certain sense, undoes polarify.)

    >>> from sympy import unpolarify, polar_lift, sin, I
    >>> unpolarify(polar_lift(I + 2))
    2 + I
    >>> unpolarify(sin(polar_lift(I + 7)))
    sin(7 + I)
    TFr   rY   )r/   boolr   
unpolarifyr   r   r   r   )r   r   r   Zchangedr   r   r#   r#   r$   r   t  s$    

r   )basicN)F)TF)F)=Z
__future__r   r   Z
sympy.corer   r   r   r   r   r	   r
   Zsympy.core.exprtoolsr   r   r   r   r   r   Zsympy.core.numbersr   r   r   Z(sympy.functions.elementary.miscellaneousr   Z$sympy.functions.elementary.piecewiser   Zsympy.core.exprr   Zsympy.core.relationalr   Zsympy.core.logicr   r   Z&sympy.functions.elementary.exponentialr   r   r   Z(sympy.functions.elementary.trigonometricr   r   Z#sympy.functions.elementary.integersr   r   r7   r[   r\   r9   r0   r   r   r   r   r   r   r   r   r   r   r   _Zabs_r#   r#   r#   r$   <module>   sB   $cm  W59,JXZ

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