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

**References**:**Re: Definite Integration in Mathematica***From:*"Michael Weyrauch" <michael.weyrauch@gmx.de>