Re: Re: piecewise integration

*To*: mathgroup at smc.vnet.net*Subject*: [mg67055] Re: [mg66999] Re: piecewise integration*From*: "Chris Chiasson" <chris at chiasson.name>*Date*: Thu, 8 Jun 2006 04:53:36 -0400 (EDT)*References*: <20060605102611.774$dR_-_@newsreader.com> <200606061028.GAA20748@smc.vnet.net> <acbec1a40606071502r7aea9e4ahcea554c39976f739@mail.gmail.com> <B4EA9689-6BEF-4F01-AD21-FC9C6EAFC212@mimuw.edu.pl> <acbec1a40606072149na61bec9ofb16ce601628afbb@mail.gmail.com>*Sender*: owner-wri-mathgroup at wolfram.com

And on a related note, does anyone know why Mathematica handles DiracDelta'[x] in this way: In[1]:= D[UnitStep[x],{x,2}] Integrate[%,{x,-1,1}] Out[1]= Derivative[1][DiracDelta][x] Out[2]= 0 On 6/7/06, Chris Chiasson <chris at chiasson.name> wrote: > Andrzej, > > I had no idea it could be so convoluted! > > Just, let's say, for kicks... what if the treatment of Integrate was > to always exclude the DiracDelta from the integral if it occurs on an > endpoint? Do you think that would be as "consistent" as Maxim's > approach? > > On 6/7/06, Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote: > > *This message was transferred with a trial version of CommuniGate(tm) Pro* > > > > On 8 Jun 2006, at 07:02, Chris Chiasson wrote: > > > > >> I confess that I never noticed that Integrate and DiracDelta in > > >> Mathematica behaved like this at end points. It seems to me that the > > >> Piecewise approach, which assumes that boundary points are treated > > >> the same as interior points, is the more natural. But Chris obviously > > >> was not interested in the answer to this particular problem but in > > >> more general matters. It is trivial to modify the behaviour of the > > >> package in this respect (by adding ones own rules for handling > > >> DiracDelta) to make it conform with what Mathematica does, if one > > >> really wanted to. But my main point was that the package is > > >> interesting in its own right and it seems to me that anyone seriously > > >> interested in this topic would have already taken a look at it. Why, > > >> even people not seriously interested in it, like myself, have done so > > >> and found interesting and instructive things in it. > > >> > > >> Andrzej > > > > > > I think Mathematica's Integrate does this to preserve the identity: > > > Integrate[f[x],{x,a,c}]==Integrate[f[x],{x,a,b}]+Integrate[f[x], > > > {x,b,c}] > > > > > > There is a danger of violating this when using PiecewiseIntegrate: > > > > > > In[1]:= > > > <<PiecewiseIntegrate.m > > > load[x_]=-9*10^3*DiracDelta[x]- > > > Piecewise[{{x*10*(10^3/3),0\[LessEqual]x\[LessEqual]3}}]-6*10^3* > > > DiracDelta[x-5] > > > Integrate[load[x],{x,-10,10}] > > > PiecewiseIntegrate[load[x],{x,-10,10}] > > > {PiecewiseIntegrate[load[x],{x,-10,0}],PiecewiseIntegrate[load[x], > > > {x,0,5}], > > > PiecewiseIntegrate[load[x],{x,5,10}]} > > > Plus@@% > > > Out[2]= > > > -6000*DiracDelta[-5 + x] - 9000*DiracDelta[x] - > > > Piecewise[{{(10000*x)/3, 0 <= x <= 3}}] > > > Out[3]= > > > -30000 > > > Out[4]= > > > -30000 > > > Out[5]= > > > {-9000,-30000,-6000} > > > Out[6]= > > > -45000 > > > -- > > > http://chris.chiasson.name/ > > > > Chris > > > > I realised this. However, I do not consider this identity correct in > > this context. Mathematica often treats generalised functions as if > > they were ordinary functions and this often lead to contradictions, > > or at least unpleasant behaviour. Even if we for the time being > > ignore the mathematical meaning of distributions you can get > > inconsistent results such as this: > > > > > > {Integrate[DiracDelta[x], {x, -1, 0}], > > Integrate[DiracDelta[x], {x, -1, t}] /. t -> 0} > > > > > > {1/2, 1} > > > > Compare this with: > > > > <<Piecewise` > > > > > > {PiecewiseIntegrate[DiracDelta[x], {x, -1, 0}], > > PiecewiseIntegrate[DiracDelta[x], {x, -1, t}] /. t -> 0} > > > > > > {1,1} > > > > Or looking at it form another angle: > > > > > > Limit[PiecewiseIntegrate[DiracDelta[x], {x, -1, t}], > > t -> 0, Direction -> 1] > > > > 0 > > > > > > Limit[PiecewiseIntegrate[DiracDelta[x], {x, -1, t}], > > t -> 0, Direction -> -1] > > > > 1 > > > > So the Integral returned by Mathematica is not continuous either form > > the left or from the right. The integral returned by > > Piecewiseintegrate is continuous from the right, which is what is the > > usual assumption in mathematical texts. But there is something even > > worse about he Mathematica implementation. > > > > In fact, if you write out the formula for the sum of integrals using > > limits you will see that it holds for PiecwiseIntegrate too. > > In fact, generalised functions are not "functions" (they have no > > values at points) but are usually defined as functionals on a certain > > space of functions on real line (there is also another approach in > > terms of non-standard analysis, but that is not implemented in > > Mathematica). These functionals are defined in terms of integrals > > over the entire real line so formally you cannot integrate the > > DiracDelta over a finite interval; you can only integrate it from - > > Infinity to +Infinity, because that is how the functional is defined. > > Integrals over finite intervals can then be defined as integrals over > > the whole real line of products of DiracDelta and the characteristic > > functions of these intervals. But look what happens when you use > > Mathematica: > > > > > > {Integrate[DiracDelta[x]*Boole[0 <= x <= 2], > > {x, -Infinity, Infinity}], Integrate[DiracDelta[x], > > {x, 0, 2}]} > > > > > > {1, 1/2} > > > > > > But these ought to be the same by definition! Again compare this with > > PiecewiseIntegrate > > > > > > {PiecewiseIntegrate[DiracDelta[x]*Boole[0 <= x <= 2], > > {x, -Infinity, Infinity}], PiecewiseIntegrate[DiracDelta[x], > > {x, 0, 2}]} > > > > > > {1,1} > > > > > > From the mathematical point of view I have no doubt here: Maxim got > > it right and Mathematica got it wrong. Of course one can always say > > that these are only different conventions and as is usual with > > various conventions, once you understand them and get used to them > > you usually find that you can use them equally successfully. Still, > > there are just too many departures form standard mathematics for me > > to be comfortable with the way Mathematica behaves in the above > > examples. > > > > Andrzej Kozlowski > > > > > -- > http://chris.chiasson.name/ > -- http://chris.chiasson.name/

**References**:**Re: piecewise integration***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>

**Re: Re: piecewise integration**

**Re: Two questions (1) Sollve and (2) Precision**

**Re: Re: piecewise integration**

**Re: piecewise integration**