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MathGroup Archive 2007

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Re: Integrate question

  • To: mathgroup at smc.vnet.net
  • Subject: [mg77656] Re: Integrate question
  • From: dimitris <dimmechan at yahoo.com>
  • Date: Thu, 14 Jun 2007 05:35:37 -0400 (EDT)
  • References: <f4omvd$7qh$1@smc.vnet.net>

Hi Andrzej and thanks for your message.
Let me state that I don't want to start a debate about Mathematica and
the other CAS.
It is obvious that they will have some important differences in their
fundamentals.
I just wanted to hear comments on a issue that impressed very much,
from much more experienced persons in mathematics and computer algebra
like you.

However, for reasons of completeness,

int(diff(y(x),x),x=a..b,continuous);

                             -y(a) + y(b)

Dimitris



Given a general functions f, without any special properties
Mathematica will assume that it is in fact an analytic function;
which is why you get:

  Series[f[x], {x, 0, 3}]
  SeriesData[x, 0, {f[0], Derivative[1][f][0], Derivative[2][f][0]/2,
      Derivative[3][f][0]/6}, 0, 4, 1]

etc. So in particular, an undefined function is always infinitely
differentiable everywhere and hence continuous.

Series has an option, which can be used to remove this assumption:

Series[f[x]*Sin[x], {x, 0, 3}, Analytic -> False]
f[x]*SeriesData[x, 0, {1, 0, -1/6}, 1, 4, 1]

In other situations, e.g with Integrate, or D, there is no analogous
option but then I can't see any situation in which it could be
useful
in the context of computer algebra. Can anyone?

Andrzej Kozlowski






although, to tell the truth, I can't think of any serious use for
this in the context of "computer algebra" (can anyone?)

Andrzej Kozlowski


 /  dimitris       :
> This question aose after replying in the forum for another CAS.
> In fact, it is not a question on "how I do something" but rather
> asking about some information about "why".
>
> In view of my reply I realized one fundamental difference
> between Integrate and the function Int from another CAS.
>
> int( diff(y(x),x)  , x=a..b);
>
>                               b
>                              /
>                             |   d
>                             |   -- y(x) dx
>                             |   dx
>                            /
>                              a
>
> In[11]:=
> Integrate[Derivative[1][f][x], {x, a, b}]
> Out[11]=
> -f[a]+f[b]
>
> I guess all we are familiar with the first fundamental theorem of
> calculus.
> It holds (http://mathworld.wolfram.com/
> FundamentalTheoremsofCalculus.html)
> if the function f[x] is continuous on the closed interval [a,b] (or at
> least piecewise
> continuous in Newton-Leibniz form with limits left and right of the
> finite discontinuities).
> I am with Mathematica default design. For example
>
> In[12]:=
> Integrate[Derivative[1][f][x]*Derivative[2][f][x], {x, a, b}]
>
> Out[12]=
> (1/2)*(-Derivative[1][f][a]^2 + Derivative[1][f][b]^2)
>
> In[13]:=
> Integrate[Derivative[1][f][x]*g[x] + f[x]*Derivative[1][g][x], {x, a,
> b}]
>
> Out[13]=
> (-f[a])*g[a] + f[b]*g[b]
>
> In[14]:=
> Integrate[Sin[f[x]]*Derivative[1][f][x], {x, a, b}]
>
> Out[14]=
> Cos[f[a]] - Cos[f[b]]
>
> I am neither software enginner, nor pure mathematician but this
> fundmental difference impressed me a lot! I am familiar
> with the issue of generic complex values in mathematica
> but here Mathematica "assumes" that typing Integrate[f'[x],{x,a,b}]
> f[x] is continuous in [a,b]?
>
> I will appreciate any comments on this issue.
>
> Greeting from the sunny Athens,
>
> Dimitris



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