Re: Re: More Integrate woes
- To: mathgroup at smc.vnet.net
- Subject: [mg9706] Re: [mg9650] Re: More Integrate woes
- From: David Withoff <withoff>
- Date: Fri, 21 Nov 1997 01:31:54 -0500
- Sender: owner-wri-mathgroup at wolfram.com
> > In any statistical setting, calculation of the Gaussian distribution
> > function is extremely important. Under Mathematica v3, this is
> > something of a disaster.
> >
> >
> > To summarise the problem:
> >
> > Under v3:
> > ________
> >
> > Expressions such as:
> >
> > aa = Integrate[Exp[-x^2],{x,-Infinity,y},
> > GenerateConditions->False]
> >
> > return output *of form*:
> >
> > 1 - Erf[Sqrt[y^2]]
> >
> > Our user now seeks numerical output. S/he enters:
> >
> > (aa /. y -> 3) == (aa/. y -> -3)
> > True
> >
> > This is clearly FALSE.
>
>
> Very strange!,
> if you omit the "GenerateConditions->False" from the command, the output
> from Mathematica 3 reads:
>
> If[y < 0, -(1/2)*Sqrt[Pi]*(Erf[Sqrt[y^2]] - 1),
> Integrate[Exp[-x^2], x, -Infinity, y]]
>
> which implies that the Erf solution is only valid for negative y, and
> now
>
> (aa /. y -> 3) == (aa/. y -> -3)
>
> returns False.
>
> However, for positive y Mathematica 3 does not seem to be able to
> perform the integration!
>
> This may well be classified as a bug in this version of Mathematica.
>
> Cheers,
>
> Bill Bertram
> ANSTO
If you tell Mathematica that y is positive, then the integral works
fine:
In[5]:= Integrate[Exp[-x^2],{x,-Infinity,y}, Assumptions -> {y > 0}]
Sqrt[Pi] (1 + Erf[y])
Out[5]= ---------------------
2
If you don't tell Mathematica that y is positive, then, as your examples
show, Integrate returns the result in a form which is only valid when y
is negative, so the condition y < 0 is necessary.
The results are all mathematically correct, but since it is possible to
return a simple result which is valid for all y, it would obviously be
desirable to do so. This will probably change for some future version
of Integrate.
Results with unnecessary conditions arise in a fairly understandable
way. If you do a calculation by hand, for example, and make an
assumption about a parameter, it sometimes turns out that the opposite
assumption would have led to an equivalent result, but the only way to
find that out is to do the calculation a second time, or to do the
calculation in a different way. The Integrate function only does the
calculation once. It doesn't go back and do the calculation a second
time to see if the assumptions that it made are necessary.
Dave Withoff
Wolfram Research