Re: Re: Trouble with Integrate
- To: mathgroup at smc.vnet.net
- Subject: [mg39316] Re: [mg39268] Re: [mg39264] Trouble with Integrate
- From: "Marko Vojinovic" <vojinovi at panet.co.yu>
- Date: Tue, 11 Feb 2003 04:41:28 -0500 (EST)
- References: <200302070807.DAA26439@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Thanks to DrBob, Andrzej, David and Daniel for their responces.
Andrzej has found one workaround for the problem. Just to note that there is
also another way:
Let
g = Sqrt[1+x^(2a)]-x^a
where if we put the parameter a to be 2 we recover the original function, f.
Now,
Integrate[g,{x,0,Infinity}] /. a->2
gives the correct answer, in terms of Gamma. Also, we could choose not to
set a->2, but just to use the assumption Re[a]>0, and then
Integrate[g,{x,0,Infinity}, Assumptions->Re[a]>0]
gives a (nice) answer for general a (also in terms of Gamma).
Another (rather minor and unrelated) question arises: one can write down the
result in many various forms using the identity
Gamma(1+x) = x * Gamma(x)
(I am not using Mathematica notation here), and FullSimplify knows about
this identity, but applies it only "from right to left", i.e. 1/2 Gamma(1/2)
reduces to Gamma(3/2), but never the other way around, which is sometimes
wanted (in my case, particularly). Since I am not very familiar with
FullSimplify, is there any way to explain to FullSimplify that I want only
Gamma[1/4] to appear in the result?
Best regards and thanks again,
Marko
----- Original Message -----
From: "Andrzej Kozlowski" <akoz at mimuw.edu.pl>
To: mathgroup at smc.vnet.net
Subject: [mg39316] [mg39268] Re: [mg39264] Trouble with Integrate
>
> On Thursday, February 6, 2003, at 05:08 PM, Marko Vojinovic wrote:
>
> > Consider the function:
> >
> > f = Sqrt[1+x^4] -x^2
> >
> > Upon asking to
> >
> > Integrate[f,{x,0,Infinity}]
> >
> > Mathematica 4.0 answers:
> >
> > -Infinity
> >
> > which is not correct. However,
> >
> > NIntegrate[f,{x,0,Infinity}]
> >
> > gives the correct (numerical) answer:
> > 1.23605
> >
> > The correct (analytical, i.e.. exact) answer to the integral is:
> >
> > Gamma[1/4] Gamma[1/4] / 6 Sqrt[Pi]
> >
> > which can be obtained after some paperwork. However, if I ask
> >
> > Integrate[1/(Sqrt[1+x^4] + x^2),{x,0,Infinity}]
> >
> > (this integrand is equivalent to f) one gets a complicated answer in
> > terms
> > of EllipticF. Meanwhile, when I ask Mathematica 3.0 the same set of
> > questions, I get correct answers, and analytical integration gives
> > answer in
> > terms of Gamma. Two questions:
> >
> > 1) Why does version 4.0 give so fairly incorrect result "-Infinity"
> > for the
> > first integral?
> > 2) How can I 'switch off' the use of elliptic functions and/or 'force'
> > Mathematica to use Gamma?
> >
> > Thanks,
> > Marko
> >
>
> 1). Well, it's a bug. The problem seems to be that attempts to fix bugs
> and improve the capabilities of Integrate in each new version of
> Mathematica tend to result in previously "good" integrals getting
> broken. In this case Mathematica's use of elliptic functions seems to
> be the culprit.
> 2). Unfortunately there is no "official" way to turn off the use of
> elliptic functions or anything else in Integrate. I have always
> considered this to be a fundamental error in design: it seems to me
> that Integrate should have been designed in such a way that you could
> turn off and on the use of certain methods, which expand the number of
> integrals Mathematica can manage but at the cost of increasing the risk
> of getting incorrect answers.
>
> Having said that, there is a way that sometimes works, and which
> luckily includes your case. Here is how. First we use a limit for
> Integrate and force Integrate to generate conditions. Since I do not
> wish to see the condition generated (it is simply b>0) I use Simplify
> with the appropriate assumption:
>
> In[1]:=
> Simplify[Integrate[Sqrt[1 + x^4] - x^2, {x, 0, b},
> GenerateConditions -> True], b > 0]
>
> Out[1]=
> -(b^3/3) + b*Hypergeometric2F1[-(1/2), 1/4, 5/4, -b^4]
>
> Note that we got an answer without elliptic functions (which you would
> get if you did not set GenerateConditions to True). So now we try to
> use Limit:
>
> In[2]:=
> Limit[%, b -> Infinity]
>
> Out[2]=
> -((Gamma[-(3/4)]*Gamma[5/4])/(2*Sqrt[Pi]))
>
> Numerically this seems to agree with your answer (given by Mathematica
> 3.0) although Mathematica 4.2 does not seem to be able to prove that
> the two answers are equivalent.
>
>
>
> Andrzej Kozlowski
> Yokohama, Japan
> http://www.mimuw.edu.pl/~akoz/
> http://platon.c.u-tokyo.ac.jp/andrzej/
>
>
>
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