Thread: Analytic Function in a Region

Domain: open set in the complex plane. (interior of a closed curve)
Region: Domain plus its boundary. (interior of a closed curve plus its boundary (the curve))

There are references to functions analytic in a domain (no problem), and analytic in a region, which would require f'(z) be defined at a boundary, but I can't find any definition for f'(z) at a boundary. Does such a definition exist? It would be analagous to the definition of a one-sided derivative of a function of a real variable.

1) Not all open sets are interiors of closed curves (for example, a union of two disjoint open circles). What you mean is an open connected set.
2) I've never heard of that definition of a region, and wolfram cites both region and domain as the same object: Region -- from Wolfram MathWorld

The definition of region that I am familiar with is "a domain possibly together with some or all of its boundary points." I don't know if this definition is standard however. You can talk about limits as you approach a boundary point by restricting to paths that stay inside the region. In this way you can define f' on the boundary.

OK, closed region. Thanks for correction. I thought my definition and the subject made it clear: "(interior of a closed curve plus its boundary (the curve))."

The question arose wrt Cauchy's Integral theorem in the complex plane. Actually, the same question applies to Green's Theorem in the plane.

Kaplan (Advanced Calculus) and Ahlfors (Complex Analysis) get around the question by simply assuming the closed region is part of a larger domain or regon, so the question of partial derivatives and analyticity at a boundary does not come up.

Churchil (Complex Analysis) is ambiguous. He begins with the conditions for Green's theorem:

"..if two functions P(x,y) and Q(x,y), together with their first partial derivatives, are continuous throughout a closed region R consisting of the interior of the closed contour C together with the boundary C itself, then" -> Greens Theorem.

He then begins Cauchy's integral theorem with:
"If a function f is analytic at all points interior to and on a closed countor C, then" integral f(z)dz is zero around C. But then in the course of the proof he states: " Let f(z) be analytic at all points of a closed region R consiting of the interior of a closed contour C together with the points on C itself." comment: This is still ambiguous, becase the closed region could be part of a larger region in which f(z) is analytic, in which case he should really refer to a relativeley closed region.

It seems easy enough to define partial derivatives (and their continuity) at a boundary by taking only points in the neighborhood which are in the closed region, even if you have to include direction of approach to the limit. Has anybody done this carefully at a boundary?

My original post was misleading due to confusion on my part. Sorry. Hate to leave it at that, so:

An analytic function (complex variable) defined in a closed region will automatically be defined outside of that region and so the question of partial derivatives at the boundary is academic.
As for Greens theorem, a condition is that the function and its first partial derivatives be continuous in the region. This is misleading. It does not mean the partial derivatives are defined at a boundary outside of which the function is not defined, it means the function has to exist in a neighborhood of a boundary point, ie, the function is defined outside the boundary.

The same applies to partial differential equations in general. You are not looking for a function defined ONLY in a closed region with certain boundary conditions, you are looking for a generally defined function with specific boundary conditions evaluated at the boundary of the region, not the "boundary" of the function.

For example, potential equal constant is a solution of Laplace's equation everywhere. The potential evaluated at the inside of a conductor is constant and so the solution solves the boundary value problem and the potential is constant inside the hollow conductor.

With that, the academic question, more carefully posed, still remains:

If a function is defined in a closed region, and not outside of it, does a definition for partial derivative exist at the boundary?