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.

Motivation and statement of theorem

If we have a holomorphic function f defined on the open unit disk, 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 r. This defines a new function: where is the unit circle. Then it would be expected that the values of the extension of f onto the circle should be the limit of these functions, and so the question reduces to determining when f_r converges, and in what sense, as r\to 1, and how well defined is this limit. In particular, if the L^p norms of these f_r are well behaved, we have an answer: Now, notice that this pointwise limit is a radial limit. That is, the limit being taken is along a straight line from the center of the disk to the boundary of the circle, and the statement above hence says that The natural question is, with this boundary function defined, will we converge pointwise to this function by taking a limit in any other way? That is, suppose instead of following a straight line to the boundary, we follow an arbitrary curve converging to some point e^{i\theta} on the boundary. Will f converge to ? (Note that the above theorem is just the special case of ). It turns out that the curve \gamma needs to be non-tangential, meaning that the curve does not approach its target on the boundary in a way that makes it tangent to the boundary of the circle. In other words, the range of \gamma must be contained in a wedge emanating from the limit point. We summarize as follows: Definition. Let be a continuous path such that. Define That is, is the wedge inside the disk with angle 2\alpha whose axis passes between e^{i\theta} and zero. We say that \gamma converges non-tangentially to e^{i\theta}, or that it is a non-tangential limit, if there exists such that \gamma is contained in and.

Discussion