o 8Va@sHddlmZddlmZmZddlmZddlmZddZ ddZ d S) )Mul) DiracDelta Heaviside)default_sort_key)Sc Cs6g}d}|\}}t|td}|||D]1}|jr2t|jtr2|| |j|j d|j}|durCt|trC| |rC|}q||q|sg}|D]1}t|tra||j d|dqO|jr{t|jtr{|| |jj d|d|j qO||qO||krt | }d|fSd}d|fS|t |fS)achange_mul(node, x) Rearranges the operands of a product, bringing to front any simple DiracDelta expression. Explanation =========== If no simple DiracDelta expression was found, then all the DiracDelta expressions are simplified (using DiracDelta.expand(diracdelta=True, wrt=x)). Return: (dirac, new node) Where: o dirac is either a simple DiracDelta expression or None (if no simple expression was found); o new node is either a simplified DiracDelta expressions or None (if it could not be simplified). Examples ======== >>> from sympy import DiracDelta, cos >>> from sympy.integrals.deltafunctions import change_mul >>> from sympy.abc import x, y >>> change_mul(x*y*DiracDelta(x)*cos(x), x) (DiracDelta(x), x*y*cos(x)) >>> change_mul(x*y*DiracDelta(x**2 - 1)*cos(x), x) (None, x*y*cos(x)*DiracDelta(x - 1)/2 + x*y*cos(x)*DiracDelta(x + 1)/2) >>> change_mul(x*y*DiracDelta(cos(x))*cos(x), x) (None, None) See Also ======== sympy.functions.special.delta_functions.DiracDelta deltaintegrate N)keyTZ diracdeltaZwrt)Zargs_cncsortedrextendis_Pow isinstancebaserappendfuncZexp is_simpleexpandr) Znodexnew_argsZdiraccZncZ sorted_argsargZnnoder@/usr/lib/python3/dist-packages/sympy/integrals/deltafunctions.py change_muls4'     "   rcCs|tsdSddlm}m}ddlm}|jtkr^|jd|d}||krW| |rUt |j dks8|j ddkr?t |j dSt|j d|j dd|j d SdS|||}|S|jsd|jr|}||kr~|||}|dur|t||s||SdSt||\}} |s| r|| |}|SdS|jd|d}|jrt||\}} | | } ||j d|d} t |j dkrdn|j d} d} | dkrd| | || || }|jr| d8} | d7} n| dkr|t || S|t|| dS| dkstjSdS) a deltaintegrate(f, x) Explanation =========== The idea for integration is the following: - If we are dealing with a DiracDelta expression, i.e. DiracDelta(g(x)), we try to simplify it. If we could simplify it, then we integrate the resulting expression. We already know we can integrate a simplified expression, because only simple DiracDelta expressions are involved. If we couldn't simplify it, there are two cases: 1) The expression is a simple expression: we return the integral, taking care if we are dealing with a Derivative or with a proper DiracDelta. 2) The expression is not simple (i.e. DiracDelta(cos(x))): we can do nothing at all. - If the node is a multiplication node having a DiracDelta term: First we expand it. If the expansion did work, then we try to integrate the expansion. If not, we try to extract a simple DiracDelta term, then we have two cases: 1) We have a simple DiracDelta term, so we return the integral. 2) We didn't have a simple term, but we do have an expression with simplified DiracDelta terms, so we integrate this expression. Examples ======== >>> from sympy.abc import x, y, z >>> from sympy.integrals.deltafunctions import deltaintegrate >>> from sympy import sin, cos, DiracDelta >>> deltaintegrate(x*sin(x)*cos(x)*DiracDelta(x - 1), x) sin(1)*cos(1)*Heaviside(x - 1) >>> deltaintegrate(y**2*DiracDelta(x - z)*DiracDelta(y - z), y) z**2*DiracDelta(x - z)*Heaviside(y - z) See Also ======== sympy.functions.special.delta_functions.DiracDelta sympy.integrals.integrals.Integral Nr)Integral integrate)solveTr r)ZhasrZsympy.integralsrrZ sympy.solversrrrrlenargsrZas_polyZLCZis_Mulr r rZdiffZsubsZis_zerorZZero)frrrrhZfhgZ deltatermZ rest_multZ rest_mult_2ZpointnmrrrrdeltaintegratePs^ 8   2   (     r&N) Z sympy.corerZsympy.functionsrrZsympy.core.compatibilityrZsympy.core.singletonrrr&rrrrs    I