Otj VddlmZddlmZddlmZddlmZmZddl m Z m Z dZ dZ d S) )Mul)S)default_sort_key) DiracDelta Heaviside)Integral integratec g}d}|\}}t|t}|||D]}|jrWt |jtr=|| |j|j dz |j}|-t |tr| |r|}|||sg}|D]}t |tr+|| d|B|jrct |jtrI|| |j d||j ||||krt| }nd}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)keyrT diracdeltawrt)args_cncsortedrextendis_Pow isinstancebaserappendfuncexp is_simpleexpandr) nodexnew_argsdiraccnc sorted_argsargnnodes \/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/sympy/integrals/deltafunctions.py change_mulr%sNH E MMOOEAr 0111Kr!! : *SXz::  OOCHHSXsw{;; < < <(C =jj99=cmmA>N>N=EE OOC   % %C#z** % d B BCCCC % 38Z @ @ %Da)P)PRURY Z Z[[[[$$$$ { " "N))++EEEe} 3> ""cR|tsdS|jtkr|d|}||kr||rt |jdks|jddkrt|jdSt|jd|jddz |jd z Snt||}|S|j s|j r|}||kr+t||}|t|ts|SnBt||\}}|s|rt||}|Snddlm}|d|}|j rt||\}}||z}||jd|d} t |jdkrdn |jd} d} | dkrt$j| z||| || z} | jr | dz} | dz } n1| dkr| t|| z zS| t|| dz zS| dkt$jSdS)a deltaintegrate(f, x) Explanation =========== The idea for integration is the following: - If we are dealing with a DiracDelta expression, i.e. DiracDelta(g(x)), we try to simplify it. If we could simplify it, then we integrate the resulting expression. We already know we can integrate a simplified expression, because only simple DiracDelta expressions are involved. If we couldn't simplify it, there are two cases: 1) The expression is a simple expression: we return the integral, taking care if we are dealing with a Derivative or with a proper DiracDelta. 2) The expression is not simple (i.e. DiracDelta(cos(x))): we can do nothing at all. - If the node is a multiplication node having a DiracDelta term: First we expand it. If the expansion did work, then we try to integrate the expansion. If not, we try to extract a simple DiracDelta term, then we have two cases: 1) We have a simple DiracDelta term, so we return the integral. 2) We didn't have a simple term, but we do have an expression with simplified DiracDelta terms, so we integrate this expression. Examples ======== >>> from sympy.abc import x, y, z >>> from sympy.integrals.deltafunctions import deltaintegrate >>> from sympy import sin, cos, DiracDelta >>> deltaintegrate(x*sin(x)*cos(x)*DiracDelta(x - 1), x) sin(1)*cos(1)*Heaviside(x - 1) >>> deltaintegrate(y**2*DiracDelta(x - z)*DiracDelta(y - z), y) z**2*DiracDelta(x - z)*Heaviside(y - z) See Also ======== sympy.functions.special.delta_functions.DiracDelta sympy.integrals.integrals.Integral NTr rr)solve)hasrrrrlenargsras_polyLCr is_Mulrrr r% sympy.solversr(r NegativeOnediffsubsis_zeroZero) frhfhg deltaterm rest_multr( rest_mult_2pointnmrs r$deltaintegrater@Qsp 55  t v HH!H , , 66{{1~~ 2KK1$$q Q$QVAY///&qvay!&)a-@@q ))++..0012  21aBI -QX- HHJJ 661aB~jX&>&>~ $.a#3#3 Iy# "9a00BI0/////%,,!,DD #6-7 1-E-E*I{ )+ 5IinQ/33A6in--q00QQinQ6G1ff q(1)=)=)B)B1e)L)LLAy7QQ66#$Yq5y%9%9#99#$Z!A#%6%6#661ffv 4r&N)sympy.core.mulrsympy.core.singletonrsympy.core.sortingrsympy.functionsrr integralsr r r%r@r&r$rGs""""""//////11111111********F#F#F#Rxxxxxr&