a
    ޶Df2                     @   s   d dl 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 d dlmZ d dlmZmZmZmZ d dlmZmZ d dlmZ d d	lmZmZ d d
lmZ ddl m Z  dddZ!dd Z"G dd deZ#dS )    )AccumBounds)SSymbolAddsympifyExpr	PoleErrorMul)factor_terms)Float_illegal)	factorial)Abssignargre)explog)gamma)PolynomialErrorfactor)Order   )gruntz+c                 C   s   t | |||jddS )aQ  Computes the limit of ``e(z)`` at the point ``z0``.

    Parameters
    ==========

    e : expression, the limit of which is to be taken

    z : symbol representing the variable in the limit.
        Other symbols are treated as constants. Multivariate limits
        are not supported.

    z0 : the value toward which ``z`` tends. Can be any expression,
        including ``oo`` and ``-oo``.

    dir : string, optional (default: "+")
        The limit is bi-directional if ``dir="+-"``, from the right
        (z->z0+) if ``dir="+"``, and from the left (z->z0-) if
        ``dir="-"``. For infinite ``z0`` (``oo`` or ``-oo``), the ``dir``
        argument is determined from the direction of the infinity
        (i.e., ``dir="-"`` for ``oo``).

    Examples
    ========

    >>> from sympy import limit, sin, oo
    >>> from sympy.abc import x
    >>> limit(sin(x)/x, x, 0)
    1
    >>> limit(1/x, x, 0) # default dir='+'
    oo
    >>> limit(1/x, x, 0, dir="-")
    -oo
    >>> limit(1/x, x, 0, dir='+-')
    zoo
    >>> limit(1/x, x, oo)
    0

    Notes
    =====

    First we try some heuristics for easy and frequent cases like "x", "1/x",
    "x**2" and similar, so that it's fast. For all other cases, we use the
    Gruntz algorithm (see the gruntz() function).

    See Also
    ========

     limit_seq : returns the limit of a sequence.
    F)deep)Limitdoit)ezz0dir r"   `/nfs/NAS7/SABIOD/METHODE/ermites/ermites_venv/lib/python3.9/site-packages/sympy/series/limits.pylimit   s    3r$   c                 C   s:  d}|t ju r<t| |d| |t jd}t|tr8dS n| jsV| jsV| j	sV| j
r6g }ddlm} | jD ]}t||||}|t jr|jdu rt| trt| }	t|	ts||	}	t|	tst| }	t|	trt|	|||  S  dS  dS t|tr dS |t ju r dS || ql|r6| j| }|t ju r| jrtdd |D rg }
g }t|D ]2\}}t|tr|
| n|| j|  qjt|dkrt|  }t||||}|t|
  }|t ju r6zddlm} || }W n ty   Y dS 0 |t ju s$|| kr(dS t||||S |S )	a+  Computes the limit of an expression term-wise.
    Parameters are the same as for the ``limit`` function.
    Works with the arguments of expression ``e`` one by one, computing
    the limit of each and then combining the results. This approach
    works only for simple limits, but it is fast.
    Nr   r   r   )togetherc                 s   s   | ]}t |tV  qd S N)
isinstancer   ).0rrr"   r"   r#   	<genexpr>h       zheuristics.<locals>.<genexpr>)ratsimp) r   Infinityr$   subsZeror'   r   is_MulZis_Addis_PowZis_Functionsympy.simplify.simplifyr%   argshas	is_finiter   r
   r	   r   
heuristicsNaNappendfuncany	enumerater   lenZsimplifyZsympy.simplify.ratsimpr,   r   )r   r   r    r!   rvrr%   almZr2e2iirvalZe3r,   Zrat_er"   r"   r#   r6   C   sb    







(r6   c                   @   s6   e Zd ZdZdddZedd Zdd Zd	d
 ZdS )r   a  Represents an unevaluated limit.

    Examples
    ========

    >>> from sympy import Limit, sin
    >>> from sympy.abc import x
    >>> Limit(sin(x)/x, x, 0)
    Limit(sin(x)/x, x, 0, dir='+')
    >>> Limit(1/x, x, 0, dir="-")
    Limit(1/x, x, 0, dir='-')

    r   c                 C   s   t |}t |}t |}|tjtjtj fv r4d}n|tjtjtj fv rNd}||rhtd||f t|tr|t	|}nt|t	st
dt| t|dvrtd| t| }||||f|_|S )N-r   z@Limits approaching a variable point are not supported (%s -> %s)z6direction must be of type basestring or Symbol, not %s)r   rE   +-z1direction must be one of '+', '-' or '+-', not %s)r   r   r-   ZImaginaryUnitNegativeInfinityr4   NotImplementedErrorr'   strr   	TypeErrortype
ValueErrorr   __new___args)clsr   r   r    r!   objr"   r"   r#   rM      s0    




zLimit.__new__c                 C   s8   | j d }|j}|| j d j || j d j |S )Nr   r      )r3   free_symbolsdifference_updateupdate)selfr   Zisymsr"   r"   r#   rR      s
    
zLimit.free_symbolsc           
      C   s   | j \}}}}|j|j }}||sBt|t| ||}t|S t|||}t|||}	|	tju r|tjtj	fv rt||d  ||}t|S |	tj	u r|tju rtj
S d S )Nr   )r3   baser   r4   r$   r   r   Oner-   rG   ComplexInfinity)
rU   r   _r   r    b1e1resZex_limZbase_limr"   r"   r#   pow_heuristics   s    

zLimit.pow_heuristicsc                    s  | j \} t dkrt|dd}t|dd}t|trft|trf|j d |j d krf| S ||krr|S |jr|jrtjS td||f tju rt	djrt
}|t| }|| }d tj|dd	r|jf i |}jf i |jf i ||kr&S |s6|S tju rHtjS |jt rX| S |jrtt|jg|j d
d R  S d}t dkrd
}nt dkrd} fdd|trddlm} ||}|}|rtju r|d
 }| }n| }z|j|d\}}	W n tyT   Y nd0 |	dkrftjS |	dkrt|S |d
kst|	d
@ stjt
| S |dkrtjt
| S tjS tju r|jrt|}|d
 }| }n| }z|j|d\}}	W n tt	t fy   ddl!m"}
 |
|}|j#r`| $|}|dur`| Y S zL|j%|d}||kr|t&rt'|dt(|j)rdndW  Y S W n tt	t fy   Y n0 Y n0 t|t*r|	tjkr|S |tjtjtjtjr| S |s|	j+r"tjS |	dkr0|S |	j)r|d
krPtjt
| S |dkrxtjt
| tj,tj-|	   S tjS nt	d|	 j.r|/t0t1}d}z0t'| }|tju s|tju rt  W nD t tfy   |dur t2| }|du r|  Y S Y n0 |S )aP  Evaluates the limit.

        Parameters
        ==========

        deep : bool, optional (default: True)
            Invoke the ``doit`` method of the expressions involved before
            taking the limit.

        hints : optional keyword arguments
            To be passed to ``doit`` methods; only used if deep is True.
        rF   r   )r!   rE   r   zMThe limit does not exist since left hand limit = %s and right hand limit = %sz.Limits at complex infinity are not implementedr   Tr   Nc                    s   | j s
| S tfdd| j D }|| j kr6| j| } t| t}t| t}t| t}|s`|s`|rt| j d  }|jrtd| j d   }|j	r|dk dkr|r| j d  S |rt
jS t
jS |dkdkr|r| j d S |rt
jS t
jS | S )Nc                 3   s   | ]} |V  qd S r&   r"   )r(   r   )	set_signsr"   r#   r*     r+   z0Limit.doit.<locals>.set_signs.<locals>.<genexpr>r   r   T)r3   tupler9   r'   r   r   r   r$   is_zeroZis_extended_realr   NegativeOnePirW   r/   )exprZnewargsZabs_flagZarg_flagZ	sign_flagsigr!   r_   r   r    r"   r#   r_   
  s,    




zLimit.doit.<locals>.set_signs)	nsimplify)cdir)powsimpzNot sure of sign of %s)3r3   rI   r$   r'   r   is_infiniter   rX   rL   rH   r   absr.   r-   getr   r4   r7   r   Zis_Orderr   rd   r   r2   rg   Zis_meromorphicZleadtermr/   intrG   r0   r
   r   Zsympy.simplify.powsimpri   r1   r]   Zas_leading_termr   r   r   Zis_negativer   Zis_positiverb   rW   Zis_extended_positiveZrewriter   r   r6   )rU   hintsr   r>   r@   rh   rg   ZneweZcoeffexri   r"   rf   r#   r      s    

$




(





z
Limit.doitN)r   )	__name__
__module____qualname____doc__rM   propertyrR   r]   r   r"   r"   r"   r#   r      s   

r   N)r   )$Z!sympy.calculus.accumulationboundsr   Z
sympy.corer   r   r   r   r   r   r	   Zsympy.core.exprtoolsr
   Zsympy.core.numbersr   r   Z(sympy.functions.combinatorial.factorialsr   Z$sympy.functions.elementary.complexesr   r   r   r   Z&sympy.functions.elementary.exponentialr   r   Z'sympy.functions.special.gamma_functionsr   Zsympy.polysr   r   Zsympy.series.orderr   r   r$   r6   r   r"   r"   r"   r#   <module>   s   $
6?