Hello, On 6/26/23 11:52, Vladimir V. Kisil wrote:
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.
We are speaking about small x, not large. As when we say that e^x is 1+x+O(x^2).
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}).
It is intended to be meaningful in the context of dimensional regularization in practical QFT use, i.e. Laurent series expansions.
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)
Which makes sense if we interpret series() as "compute the terms of orders smaller than the x^N term, don't bother about higher orders, but tell so me if you think there are any", no?
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..
Well, Vitaly had me convinced that Order(x)^k -> Order(x^k) is useful and safe for integer k > 0. Is it? All my best, -richy. -- Richard B. Kreckel <https://in.terlu.de/~kreckel/>