\h TSSKJr SSKJr SSKJr SSKJrJr SSK J r J r Sr Sr g ) )Mul)S)default_sort_key) DiracDelta Heaviside)Integral integratec /nSnUR5upE[U[S9nURU5 UHnUR(ac[ UR [5(aDURURUR URS- 55 UR nUc/[ U[5(aURU5(aUnMURU5 M U(d/nUHn[ U[5(a!URURSUS95 M9UR(ad[ UR [5(aEURURUR RSUS9UR55 MURU5 M X&:wa[U6R5nSU4$SnSU4$U[U64$)achange_mul(node, x) Rearranges the operands of a product, bringing to front any simple DiracDelta expression. Explanation =========== If no simple DiracDelta expression was found, then all the DiracDelta expressions are simplified (using DiracDelta.expand(diracdelta=True, wrt=x)). Return: (dirac, new node) Where: o dirac is either a simple DiracDelta expression or None (if no simple expression was found); o new node is either a simplified DiracDelta expressions or None (if it could not be simplified). Examples ======== >>> from sympy import DiracDelta, cos >>> from sympy.integrals.deltafunctions import change_mul >>> from sympy.abc import x, y >>> change_mul(x*y*DiracDelta(x)*cos(x), x) (DiracDelta(x), x*y*cos(x)) >>> change_mul(x*y*DiracDelta(x**2 - 1)*cos(x), x) (None, x*y*cos(x)*DiracDelta(x - 1)/2 + x*y*cos(x)*DiracDelta(x + 1)/2) >>> change_mul(x*y*DiracDelta(cos(x))*cos(x), x) (None, None) See Also ======== sympy.functions.special.delta_functions.DiracDelta deltaintegrate N)keyrT diracdeltawrt)args_cncsortedrextendis_Pow isinstancebaserappendfuncexp is_simpleexpandr) nodexnew_argsdiraccnc sorted_argsargnnodes V/var/www/auris/envauris/lib/python3.13/site-packages/sympy/integrals/deltafunctions.py change_mulr%sNH E MMOEA 01Kr ::*SXXz:: OOCHHSXXsww{; <((C =jj99cmmA>N>NE OOC  C#z** d BC 388Z @ @Da)PRURYRY Z[$   "N))+Ee}Ee} 3> ""c<UR[5(dgUR[:XaURSUS9nX :XaUR U5(a[ UR 5S::dUR SS:Xa[UR S5$[UR SUR SS- 5UR SR5R5- $g[X!5nU$UR(dUR(GawUR5nX:wa&[XA5nUb[U[5(dU$g[X5upVU(dU(a [Xa5nU$gSSKJn URSUS9nUR(a[XQ5upXXh-nU"UR SU5Sn [ UR 5S:XaSOUR Sn Sn U S:a[$R&U -UR)X5R+X5-n U R,(a U S-n U S- n O'U S:XaU [X- 5-$U [XS- 5-$U S:aM[$R.$g)aK deltaintegrate(f, x) Explanation =========== The idea for integration is the following: - If we are dealing with a DiracDelta expression, i.e. DiracDelta(g(x)), we try to simplify it. If we could simplify it, then we integrate the resulting expression. We already know we can integrate a simplified expression, because only simple DiracDelta expressions are involved. If we couldn't simplify it, there are two cases: 1) The expression is a simple expression: we return the integral, taking care if we are dealing with a Derivative or with a proper DiracDelta. 2) The expression is not simple (i.e. DiracDelta(cos(x))): we can do nothing at all. - If the node is a multiplication node having a DiracDelta term: First we expand it. If the expansion did work, then we try to integrate the expansion. If not, we try to extract a simple DiracDelta term, then we have two cases: 1) We have a simple DiracDelta term, so we return the integral. 2) We didn't have a simple term, but we do have an expression with simplified DiracDelta terms, so we integrate this expression. Examples ======== >>> from sympy.abc import x, y, z >>> from sympy.integrals.deltafunctions import deltaintegrate >>> from sympy import sin, cos, DiracDelta >>> deltaintegrate(x*sin(x)*cos(x)*DiracDelta(x - 1), x) sin(1)*cos(1)*Heaviside(x - 1) >>> deltaintegrate(y**2*DiracDelta(x - z)*DiracDelta(y - z), y) z**2*DiracDelta(x - z)*Heaviside(y - z) See Also ======== sympy.functions.special.delta_functions.DiracDelta sympy.integrals.integrals.Integral NTr rr)solve)hasrrrrlenargsras_polyLCr is_Mulrrr r% sympy.solversr(r NegativeOnediffsubsis_zeroZero) frhfhg deltaterm rest_multr( rest_mult_2pointnmrs r$deltaintegrater@QsUp 55   vv HH!H , 6{{1~~K1$q Q$QVVAY//&qvvay!&&)a-@q ))+..012 n a1BI QXXX HHJ 61B~jX&>&> R M$.a#3 I"90BIF ?0%,,!,D ##-7 -E*I ) 5IinnQ/3A6inn-q0QinnQ6G1f q()=)B)B1)LLAyyQQ6#$Yqy%9#99#$ZA#%6#661fvv r&N)sympy.core.mulrsympy.core.singletonrsympy.core.sortingrsympy.functionsrr integralsr r r%r@r&r$rGs!"/1*F#Rxr&