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.
Partial Derivatives at a 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?