Re: Equality of integrals

*To*: mathgroup at smc.vnet.net*Subject*: [mg18165] Re: [mg18145] Equality of integrals*From*: "Andrzej Kozlowski" <andrzej at tuins.ac.jp>*Date*: Sat, 19 Jun 1999 23:54:28 -0400*Sender*: owner-wri-mathgroup at wolfram.com

To the moderator: Dear Steve The reply I just posted to the message below contained a silly mistake. Could you please replace it by the one below? Thanks. _____________________________________________________________________ The integrals are equal in comlete generality (well, that is assuming that they are defined, of course. If they are not defined it is hard to talk of them being equal). The problem is that Mathematica does not see any transformations that it can apply to such general integrals. You have to help it yourself. The easiest way is to give the indefinite integral of f a name g. This g must exist if I1 and I2 are going to be defined at all! In[5]:= f=g'; and then In[4]:= I1-I2//Simplify Out[4]= 0 If you only want to prove that the integrals are equal there is also another way: In[3]:= D[I1-I2,q] Out[3]= 0 This shows that when considered as functions of q I1 and I2 differ by a constant. Now you only need to evaluate them at a q=0 (say) : In[4]:= (I1-I2)/.q->0 Out[4]= 0 -- Andrzej Kozlowski Toyama International University JAPAN http://sigma.tuins.ac.jp http://eri2.tuins.ac.jp ---------- >From: Bill Campbell <bill at metosrv2.umd.edu> To: mathgroup at smc.vnet.net >To: mathgroup at smc.vnet.net >Subject: [mg18165] [mg18145] Equality of integrals >Date: Fri, Jun 18, 1999, 1:51 PM > > I defined the following two integrals: > > I1=Integrate[f[x],{x,0,q}] > > and > > I2= 3*Integrate[f[3x],{x,0,q/3}] > > and asked if they were equal > > I1==I2//FullSimplify > > The result was returned unevaluated. Why doesn't Mathematica (3.0 for > Windows (for students) and 3.0 for DEC) know that the two are equal? Is > this result only true if q is real? If so, is there a way to let > Mathematica know q is real and thereby return True? If I issue the > command > > I1==I2/.f->Sin//FullSimplify > or > I1==I2/.f->ArcCot//FullSimplify > > or any of a variety of functions, Mathematica returns True. > > > -- > Bill Campbell > > > -- Andrzej Kozlowski Toyama International University JAPAN http://sigma.tuins.ac.jp http://eri2.tuins.ac.jp ---------- >From: Bill Campbell <bill at metosrv2.umd.edu> To: mathgroup at smc.vnet.net >To: mathgroup at smc.vnet.net >Subject: [mg18165] [mg18145] Equality of integrals >Date: Fri, Jun 18, 1999, 1:51 PM > > I defined the following two integrals: > > I1=Integrate[f[x],{x,0,q}] > > and > > I2= 3*Integrate[f[3x],{x,0,q/3}] > > and asked if they were equal > > I1==I2//FullSimplify > > The result was returned unevaluated. Why doesn't Mathematica (3.0 for > Windows (for students) and 3.0 for DEC) know that the two are equal? Is > this result only true if q is real? If so, is there a way to let > Mathematica know q is real and thereby return True? If I issue the > command > > I1==I2/.f->Sin//FullSimplify > or > I1==I2/.f->ArcCot//FullSimplify > > or any of a variety of functions, Mathematica returns True. > > > -- > Bill Campbell > > >