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- W2952303194 abstract "Let $f:mathbb{D}tomathbb{C}$ be a bounded analytic function. A set $Ksubsetmathbb{D}$ which contains the point $1$ in its boundary is called a convergence set for $f$ at $1$ if $f(z)$ converges to some value $zeta$ as $zto1$ with $zin K$. $K$ is called a path divergence set for $f$ at $1$ if $f$ diverges along every path $gamma$ which lies in $K$ and approaches $1$. In this article, we show that for a path $gamma$ through the unit disk from $-1$ to $1$, if $f$ fails to converge along $gamma$, then either the region above $gamma$ or the region below $gamma$ is a path divergence set for $f$. On the other hand, if $gamma_1$ and $gamma_2$ are two such paths, and $f$ converges along both $gamma_1$ and $gamma_2$, then the region between $gamma_1$ and $gamma_2$ is a convergence set for $f$. This latter fact is immediate when $gamma_1$ and $gamma_2$ do not intersect except at their end-points, but becomes non-trivial when $gamma_1$ and $gamma_2$ are highly intersecting. We conclude the paper with an examination of the convergence sets for the function $e^{frac{z+1}{z-1}}$ at $1$." @default.
- W2952303194 created "2019-06-27" @default.
- W2952303194 creator A5056181415 @default.
- W2952303194 date "2016-09-20" @default.
- W2952303194 modified "2023-09-27" @default.
- W2952303194 title "Boundary convergence and path divergence sets for bounded analytic functions in the disk" @default.
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