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.
If we have a holomorphic function
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
This defines a new function: where is the unit circle.
Then it would be expected that the values of the extension of
onto the circle should be the limit of these functions, and so the question reduces to determining when
converges, and in what sense, as
, and how well defined is this limit.
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
(Note that the above theorem is just the special case of
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
must be contained in a wedge emanating from the limit point.
We summarize as follows: Definition.
be a continuous path such that
is the wedge inside the disk with angle
whose axis passes between
converges non-tangentially to
, or that it is a non-tangential limit, if there exists