Fatou's theorem

Fatou's theorem Not to be confused with Fatou's lemma.

In mathematics, specifically in complex analysis, Fatou's theorem, named after Pierre Fatou, is a statement concerning holomorphic functions on the unit disk and their pointwise extension to the boundary of the disk.

Contents 1 Motivation and statement of theorem 2 Discussion 3 See also 4 References Motivation and statement of theorem If we have a holomorphic function {displaystyle f} defined on the open unit disk {displaystyle mathbb {D} ={z:|z|<1}} , it is reasonable to ask under what conditions we can extend this function to the boundary of the unit disk. To do this, we can look at what the function looks like on each circle inside the disk centered at 0, each with some radius {displaystyle r} . This defines a new function: {displaystyle {begin{cases}f_{r}:S^{1}to mathbb {C} \f_{r}(e^{itheta })=f(re^{itheta })end{cases}}} where {displaystyle S^{1}:={e^{itheta }:theta in [0,2pi ]}={zin mathbb {C} :|z|=1},} is the unit circle. Then it would be expected that the values of the extension of {displaystyle f} onto the circle should be the limit of these functions, and so the question reduces to determining when {displaystyle f_{r}} converges, and in what sense, as {displaystyle rto 1} , and how well defined is this limit. In particular, if the {displaystyle L^{p}} norms of these {displaystyle f_{r}} are well behaved, we have an answer: Theorem. Let {displaystyle f:mathbb {D} to mathbb {C} } be a holomorphic function such that {displaystyle sup _{0

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