[GiNaC-devel] Power laws

[Vladimir V. Kisil]

	Dear Richard,

	Thank you for pointing out an issue with notmalisation of
  expressions. What about the attached _draft_ of the patch? It allows
  to reduce all suitable exponents with arguments different by a
  rational numeric factor to monomials of the same temporary
  variable. Thus it reduces

  (exp(2*x)-1)/(exp(x)-1)  to exp(x)+1

  as well as other more complicated cases like

  (exp(15*x)+exp(12*x)+2*exp(10*x)+2*exp(7*x))/(exp(5*x)+exp(2*x)) to
  exp(5*x)^2+2*exp(5*x)

  The patch modifies some signatures of functions, which however are not
  advertised as user interface. A footprint on the performance with
  expressions without exponents shall not be really noticeable.

  If the approach is suitable I can add a similar behaviour for powers,
  then the simplification

  (a^(2x)-1)/(a^x-1) to a^x+1

  will work as well.

  Some tests shall be added for the final version of the patch.

  Once this will be working, shall we add the automatic simplification
  (a^b)^c=a^(b*c) for suitable cases as well?

  Best wishes,
  Vladimir
-- 
Vladimir V. Kisil                 http://www.maths.leeds.ac.uk/~kisilv/
  Book:     Geometry of Mobius Transformations     http://goo.gl/EaG2Vu
  Software: Geometry of cycles          http://moebinv.sourceforge.net/
  Jupyter: https://github.com/vvkisil/MoebInv-notebooks
>>>>> On Thu, 9 Apr 2020 01:51:09 +0200, "Richard B. Kreckel" <kreckel at in.terlu.de> said:

    RK> Hi Vladimir!

    RK> On 06.04.20 14:34, Vladimir V. Kisil wrote:
    >> Coming back to our previous discussion (with a long history) on
    >> the power law (e^x)^a=e^(x*a). I am attaching a patch which does
    >> not break the automatic simplification exp(x)/exp(x)=1.

    RK> Your new patch is much better since it doesn't break any
    RK> existing test suite.

    RK> Playing around with it, it still seems to raise some fundamental
    RK> questions: What justifies treating exp(x)^a fundamentally
    RK> different than any other (b^x)^a with a (positive) base b? With
    RK> the patch, there seems to be this discrimination: exp(x)^5 is
    RK> rewritten to exp(5*x) but (b^x)^5 is _not_ rewritten to b^(5*x).

    RK> It's a nice pastime to fancy consequences of this. Let y=b^x,
    RK> then normal((y^2-1)/(y+1)) returns b^x-1. But if y=exp(x), the
    RK> patch prevents the normalization to exp(x)-1. Ugh.

    RK> Or, consider this gedankenexperiment: If we didn't have exp(x)
    RK> as a function but instead a symbol e, would it be justified to
    RK> have special re-writing rules for (e^x)^a but not for (b^x)^a?
    RK> I'm not sure...

    RK> Best wishes, -richy.  -- Richard B. Kreckel
    RK> <https://in.terlu.de/~kreckel/>

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