[GiNaC-list] series of x^n around x=0

[Feng Feng]

Of course, the issue here is mathematics, not GiNaC. I totally agree with that. :)


And thanks a lot for pointing out the method to find the leading power.

Since there is also log(x)-terms in my expression, and leading power seems not be able to capture these log terms.


Currently, I am trying to walk around by finding the all pattern: x^n, since n is rational number,

and multiply n by lcm of all those denominators, and transform all rational powers to integer powers …


BTW, when the input contains x^n (n as an integer) and log(x), it seems that series works fine around x=0.


Thanks again!


Best regards!

Feng

 原始邮件
发件人: Vladimir V. Kisil<kisilv at maths.leeds.ac.uk>
收件人: Feng Feng<F.Feng at outlook.com>
抄送: GiNaC discussion list<ginac-list at ginac.de>; Vladimir V. Kisil<V.Kisil at leeds.ac.uk>
发送时间: 2019年12月22日(周日) 06:17
主题: Re: [GiNaC-list] series of x^n around x=0


>>>>> On Sat, 21 Dec 2019 11:14:58 +0000, Feng Feng <F.Feng at outlook.com<mailto:F.Feng at outlook.com>> said:

    FF> Thanks very much for the reply.!  And yes, it is not
    FF> well-defined in mathematical sense to taylor expand x^(3/2)
    FF> around x=0.  What I exactly want is to get the asyptotic
    FF> behaviour of a function f(x) around x=0.  For example, f(x) =
    FF> x^(3/2) (1 + x + x^2 + …), the asyptotic behaviour at leading
    FF> order (LO) and next-to-leading order (NLO) are x^(3/2) and
    FF> x^(5/3), respectively.

    FF> So I wonder there is a way to get the asyptotic behaviour of a
    FF> function f(x), for example, Input: x^(3/2) * (1+x+x^2+x^3), and
    FF> request the terms at LO and NLO Output: x^(3/2) + x^(5/3)

    I am not sure that is NLO, but the leading term can be obtained as
  follows:

ex e=pow(x, numeric(3,2))*(1+x+x^2+x^3);

cout << (log(e).diff(x)*x).expand().subs(x==0);
// -> 3/2

  So now e can be multiplied by x^(2/3) to remove the singularity.
  See for explanations:

  https://en.wikipedia.org/wiki/Logarithmic_derivative

  So, the issue here is mathematics, not GiNaC. Once you know the right
  method, it shall be easier to implement in the code.
--
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 (Colab):         https://github.com/vvkisil/MoebInv-notebooks
  Jupyter (CodeOcean):       https://codeocean.com/capsule/7952650/tree

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