Asymptotic powerseries development

From Hyperoperations Wiki
(Redirected from Asymptotic development)
Jump to: navigation, search

Let $f$ be holomorphic on $D$ and let $a$ be a point on the boundary $\partial D$. We say that $f$ has the asymptotic powerseries (development) $\sum_{n=0}^\infty c_n (z-a)^n$ at $a$, iff $$\lim_{z\to a} \frac{1}{(z-a)^N}\left(f(z)-\sum_{n=0}^N c_n (z-a)^n\right) = 0$$ for all $N\in\N_0$.

Every function can have at most one asymptotic powerseries at $a$ because the coefficients $c_n$ are determined by: $$c_0=\lim_{z\to a} f(z),\quad c_{N+1}=\lim_{z\to a} \frac{1}{(z-a)^{N+1}}\left(f(z)-\sum_{n=0}^{N} c_n (z-a)^n\right)$$

However there may be different functions with the same asymptotic powerseries: for example $e^{-1/x}$ and $f(x)=0$ both have the asymptotic powerseries $c_n=0$ at 0.

The development (the coefficients) may depend on the region $D$.

Theorem. Let $D$ be of the following simplicity: for each $z\in D$ there is a sequence $a_n\to a$, such that $[a_n,a]\subset D$. If all the limits $c_n=\lim_{z\to a} \frac{f^{(n)}(z)}{n!}$ exist then $f$ has the asymptotic development with coefficients $c_n$ in $a$.

Theorem of Ritt. For any proper sector $S$ at $a$ and for every given sequence of $c_n$ there exists a holomorphic function $f$ on $S$ having the asymptotic powerseries $\sum_{n=0}^\infty c_n (z-a)^n$ at $a$.