+ i V^RIHt^RIHt^RIHt^RIHtHt^RI H t H t Rt Rt R#) )Mul)S)default_sort_key) DiracDelta Heaviside)Integral integratec .pRpVP4wrE\V\R7pVPV4VFpVP'di\ VP \4'dIVPVPVP VP^, 44VP pVf2\ V\4'dVPV4'dTpKVPV4K V'g.pVFp\ V\4'd%VPVPRVR74K=VP'di\ VP \4'dIVPVPVP PRVR7VP44KVPV4K W&8wd\V!P4pRV3#RpRV3#V\V!3#)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)keyT diracdeltawrt)args_cncsortedrextendis_Pow isinstancebaserappendfuncexp is_simpleexpandr) nodexnew_argsdiraccnc sorted_argsargnnodes && ~/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/integrals/deltafunctions.py change_mulr$sNH E MMOEA 01Kr :::*SXXz:: OOCHHSXXsww{; <((C =jj99cmmA>N>NE OOC  C#z** d BC 388Z @ @Da)PRURYRY Z[$   "N))+Ee}Ee} 3> ""cVP\4'gR#VP\8XdVPRVR7pW 8XdVP V4'd\ VP 4^8:gVP ^,^8Xd\VP ^,4#\VP ^,VP ^,^, 4VP ^,P4P4, #R#\W!4pV#VP'gVP'EdVP4pW8wd*\WA4pVe\V\4'gV#R#\W4wrVV'gV'd\Wa4pV#R#^RIHpVPRVR7pVP'd\WQ4wrXWh,pV!VP ^,V4^,p \ VP 4^8Xd^MVP ^,p ^p V ^8d\$P&V ,VP)W4P+W4,p V P,'dV ^,p V ^, p KiV ^8XdV \W, 4,#V \W^, 4,#\$P.#R#)aK 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 NTr )solve)hasrrrrlenargsras_polyLCr is_Mulrrrr$ sympy.solversr'r NegativeOnediffsubsis_zeroZero) frhfhg deltaterm rest_multr' rest_mult_2pointnmrs && r#deltaintegrater?QsSp 55   vv HH!H , 6{{1~~K1$q Q$QVVAY//&qvvay!&&)a-@q ))+..012 n a1BI QXXX HHJ 61B~jX&>&> R M$.a#3 I"90BIF ?0%,,!,D ###-7 -E*I ) 5IinnQ/3A6inn-q0QinnQ6G1f q()=)B)B1)LLAyyyQQ6#$Yqy%9#99#$ZA#%6#66vv r%N)sympy.core.mulrsympy.core.singletonrsympy.core.sortingrsympy.functionsrr integralsrr r$r?r%r#rFs!"/1*F#Rxr%