cauchy integral formula for asymptotic differentiability
Henryk Trappmann
a curve g
has usually the precondition that f must be holomorphic on the closure
of D (i.e. holomorphic on some open set containing the closure of D).
However for me it seems that if there are some points on the curve
where f is not analytic but only asymptotically analytic from inside D
the Cauchy integral formula would still hold.
Does someone know more about this topic?
David C. Ullrich
I'm not sure exactly what "asymptotically analytic from inside D"
means, but there are certainly many results like this.
For the disk it's trivial that continuous on the closure and
holomorphic in the interior is enough, and in fact the weaker
condition "f in H^1" suffices - you could see various texts,
for example Rudin "Real and Complex Analysis".
In fact if the boundary of D is a rectifiable closed curve
then continuity on the closure and holomorphy in the
interior is enough, and I believe that assuming f in H^1
is also enough, where that's defined by saying that |f|
has a harmonic majorant. I'm not certain of a reference
for these facts - one way to prove them is by the Riemann
mapping theorem:
Say U is the unit disk and phi : U -> D is a conformal
mapping, where the boundary of D is a rectifiable simple
closed curve. Caratheodory says that phi extends to
a homeomoprhism of the cloures. Now the rectifiability
of the boundary of D says that the restriction of phi to
the boundary has bounded variation. So it has a measure
for a derivative; F&M Riesz then says that the derivative
of phi is actually an H^1 function in the disk, and that
should suffice to transfer the Cauchy formula from the
disk to D.
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
Henryk Trappmann
> I'm not sure exactly what "asymptotically analytic from inside D"
> means, but there are certainly many results like this.
I mean that f has an asymptotic powerseries, i.e. that (assume 0 being
the point on the boundary) there is a formal powerseries at 0 with
coefficients c_n such that
(f(z) - sum(k=0..n, c_k * z^k ))/z^n -> 0 for every n for z -> 0 for z
inside D.
> For the disk it's trivial that continuous on the closure and
> holomorphic in the interior is enough, and in fact the weaker
> condition "f in H^1" suffices - you could see various texts,
> for example Rudin "Real and Complex Analysis".
Thank you for the reference. Indeed I was unaware of such theorems.