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Z
 ddlmZmZmZ ddlmZ ddlmZmZ dd	lmZmZ dd
lmZmZmZ ddlmZ ddlmZ dd„ Zdd„ Z dd„ Z!dd„ Z"dd„ Z#dS )aO  
This module contains the implementation of the 2nd_hypergeometric hint for
dsolve. This is an incomplete implementation of the algorithm described in [1].
The algorithm solves 2nd order linear ODEs of the form

.. math:: y'' + A(x) y' + B(x) y = 0\text{,}

where `A` and `B` are rational functions. The algorithm should find any
solution of the form

.. math:: y = P(x) _pF_q(..; ..;\frac{\alpha x^k + \beta}{\gamma x^k + \delta})\text{,}

where pFq is any of 2F1, 1F1 or 0F1 and `P` is an "arbitrary function".
Currently only the 2F1 case is implemented in SymPy but the other cases are
described in the paper and could be implemented in future (contributions
welcome!).

References
==========

.. [1] L. Chan, E.S. Cheb-Terrab, Non-Liouvillian solutions for second order
       linear ODEs, (2004).
       https://arxiv.org/abs/math-ph/0402063
é    )ÚSÚPow)Úexpand)ÚEq)ÚSymbolÚWild)ÚexpÚsqrtÚhyper)ÚIntegral)ÚrootsÚgcd)ÚcancelÚfactor)ÚcollectÚsimplifyÚ
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
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|d	d
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€z3equivalence_hypergeometric.<locals>._power_countingT©ÚforceÚ2F1)ÚI0ÚkÚ
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& 

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4(
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h" "@(.rx   c              	   C   s”   | dkr|ddgg d¢fv rdS d S | dkr*|g d¢g d¢ddgddgfv r(dS d S | dkrH|g d¢ddgg d¢ddgdgddgddgfv rHdS d S )Nr   )r   r   r   r>   r2   )r2   r   r   r   )r2   r2   r   r   )rP   rR   r   r   r   rJ   Ò   s   ù ü0rJ   c                 C   s  |j d }ddlm} ddlm} t| dd\}}|d }|d }	|d }
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is_integerr
   r   r   r   r!   rC   r   r   Úis_zeror   )r&   r'   Úmatch_objectr(   ry   r   ÚC0ÚC1rU   rV   rW   r.   ÚsolÚy2rC   ÚdtdxÚ_BÚ_AÚeÚe1r   r   r   Úget_sol_2F1_hypergeometricç   sF   

N^ h * "&
"
rˆ   N)$Ú__doc__Ú
sympy.corer   r   Úsympy.core.functionr   Úsympy.core.relationalr   Úsympy.core.symbolr   r   Úsympy.functionsr   r	   r
   Úsympy.integralsr   Úsympy.polysr   r   r{   r   r   Úsympy.simplifyr   r   r   Úsympy.simplify.powsimpr   Úsympy.solvers.ode.oder   r0   rT   rx   rJ   rˆ   r   r   r   r   Ú<module>   s"    IK