Why do intricate features have a limited distance of merging?

Claim we have a function $\displaystyle f(z)=\sum_{n=0}^\infty a_n z^n$ with distance of merging $R>0$. Why is the distance of merging just $R$? Can we end that there must be a post, branch cut or stoppage for some $z_0$ with $|z_0|=R$? What does that mean for features like
$$f(z)=\begin{cases} 0 & \text{for $z=0$} \\\ e^{-\frac{1}{z^2}} & \text{for $z \neq 0$} \end{cases}$$
that have a distance of merging $0$?

2019-05-05 02:38:37
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Answers: 1

If the distance of merging is $R$, that suggests there is a single factor on the circle $|z| = R$. To put it simply, there is a factor $\xi$ on the circle of distance $R$ such that the function can not be expanded using "analytic extension" in an area of $\xi$. This is an uncomplicated application of density of the circle and also can be located in publications on complex analysis, as an example Rudin's.

Nonetheless, it does not suggest that there is a post, branch cut, or stoppage, though those would certainly create single values. Without a doubt, a "post" on the border would just make good sense if you can analytically proceed the power collection to some correct domain having the disk $D_R(0)$, and also this is usually difficult. As an example, the power collection $\sum z^{2^j}$ can not be proceeded at all outside the device disk, due to the fact that it is boundless along any kind of ray whose angle is a dyadic portion. The device circle is its all-natural border, though it does not make good sense to claim that the function has a branch factor or post there. (More usually, one can show that offered any kind of domain in the aircraft, there is a holomorphic function because domain which can not be expanded any kind of better, basically making use of variants on the very same motif.)

The function $\sum_j \frac{z^j}{j^2}$, by the way, is continual on the shut device disk, yet despite the fact that there is a single factor there. So connection might take place at single factors.

The last function you state does not have a power collection development in an area of absolutely no. Actually, it is not continual at absolutely no, due to the fact that it impacts up if you come close to absolutely no along the fictional axis.

2019-05-08 09:06:11