On Mon, 26 Jun 2023 10:26:55 +0200, "Richard B. Kreckel" <kreckel@in.terlu.de> said: RK> Now, I may be missing something: Isn't RK> Order(x^k*sin(1/x))==Order(x^k) at x==0? And isn't RK> Order(x^-1)==Order(x)^-1 not finite either but still fine in a RK> Laurent series?
I think a lot of confusion is coming because our function Order is not clearly defined. First of all, if we speak on Order(x^4) do we mean the asymptotic behaviour for x→0 or x→∞? For simple cases like series(sin(x),x==0,4) = x-1/6*x^3+Order(x^4) both orders are the same, but this is not true in general. Ginsh answer series(sin(x)+x^10,x==0,4) = 1*x+(-1/6)*x^3+Order(x^4) suggests that we are speaking for x→0 only. Next, either GiNaC::Order() is only meaningful in the context of series expansions of analytic functions or it is a sort of big-O concept? For the latter take f(x) = x^k * ( sin(1/x) +1) +x^m with k > m. Then for x→0 we have f(x) = O(x^k) but 1/f(x) = O(x^{-m}). Finally, if we only consider power expansion of analytic functions then having a zero of an integer order n for f(x) at some point implies that 1/f(x) has a pole of the same order n there. But I am not sure that it will be safe to translate this into some properties of GiNaC::Order(). Currently we have:
series(x^(100),x==0,4); Order(x^4) series(x^(-100),x==0,4); 1*x^(-100)
The root of the issue is that things like Order() are not about identities, they are merely about inequalities—which are not implemented very much in GiNaC presently.. -- Vladimir V. Kisil http://www1.maths.leeds.ac.uk/~kisilv/ Book: Geometry of Mobius Maps https://doi.org/10.1142/p835 Soft: Geometry of cycles http://moebinv.sourceforge.net/ Jupyter notebooks: https://github.com/vvkisil?tab=repositories