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MathGroup Archive 2007

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Re: Re: Definite Integration in Mathematica

  • To: mathgroup at smc.vnet.net
  • Subject: [mg74585] Re: [mg74564] Re: Definite Integration in Mathematica
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Tue, 27 Mar 2007 04:01:59 -0500 (EST)
  • References: <etqo3f$10i$1@smc.vnet.net> <200703260708.CAA11567@smc.vnet.net>

On 26 Mar 2007, at 09:08, Michael Weyrauch wrote:

> Hello,
>
>    thanks, Dimitris, for bringing up this issue. (By the way, as I  
> just realized, Michael Trott
> in his book on symbolics has quite some interesting things to say  
> related to this issue.)
>
> My remark in this thread clearly was written in the spirit of  
> "functions of one real
> variable". As I see my statements are clearly incorrect in the  
> sense of  "functions of a complex
> variable".
>
> However, in many practical applications people want to have an  
> integral in the
> former sense, and it is very disconcerting to fall down a  
> discontinuity step along
> an integral of a continous integrand on the real line due to a  
> singularity of the
> integrand somewhere in the complex plane.  Therefore, I guess, the  
> designers of "the other
> CAS", which Dimitris uses for comparison, obviously opted to return  
> an integral, which is continuous along
> the real axis (if such an integral is available in the set of  
> possible solutions).
>
> This is not just a matter of aestetics or simplicity as Dimitris  
> remark in this thread may suggest
>
>> Do you prefer the extend antiderivative over the compact one obtained
>> directly
>> by Mathematica only because is real in the real axis?
>> As regards myself, no!
>
> In applications we have to use that solution which makes practical  
> sense, and in many
> applications it's the integral which is continouus (and  
> differentiable) for a continouus integrand
> on the real axis. Of course, it's my task as a physicist or  
> engineer to see if a mathematical
> solution serves my purposes or not, however, in such rather  
> intriguing cases, it would be
> desirable that  the designers of Mathematica would help me by  
> providing an Option
> to Integrate, in which I could ask for an integral which is  
> continuous on the real axis if such an integral
> exists for a particular integrand.
> (As I understand the remark of Daniel Lichtblau in some future  
> version of Mathematica
> they may provide such an Option).
>
> Thanks again
>
> Michael Weyrauch
>


I am not convinced that there is a genuine need for this. This may be  
due my lack of knowledge of applied mathematics but I cannot see any  
advantage in having a complicated (non-analytic)  anti-derivative,  
even a continuous one, on the real axis over simply doing this:


g = First[g /. NDSolve[{Derivative[1][g][x] == (4 + 2*x + x^2)/(17 +  
2*x - 7*x^2 + x^4), g[0] == 0}, g,
       {x, 0, 4}]];

You get a smooth function with which you can perform any computations  
you like with excellent accuracy, as you can see from:


Plot[{Derivative[1][g][x], (4 + 2*x + x^2)/(17 + 2*x - 7*x^2 + x^4)},  
{x, 0, 4},
   PlotStyle -> {Green, Red}]

What is there that is useful in applications and that you can do with  
a symbolic anti-derivative but you can't with this interpolating  
function?
(The situation is quite different in the complex case, where having a  
function that is actually complex analytic even in some part of  the  
complex plane can be a huge advantage  over just having an  
interpolating function).

Andrzej Kozlowski


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