non-sectoral real analytic functions

Hello all,

given a real analytic function f on the (real) open interval (0,a),
then f can be represented as power series in each x between 0 and a.
In each x the corresponding powerseries has a convergence radius r(x),
and we know that this power series then converges also in the complex
plane for |z-x|<r(x).
So we can extend f on (0,a) to some area containing (0,a) in the
complex plane.

But it seems that it is not always possible to extend f to an open
sector with tip at 0 containing (0,a).
Or merely define f on a sector with tip at 0 that intersects the real
line (and takes the values of the original f on that intersection).

So are there known counterexamples, i.e. functions f that are real
analytic on some interval (0,a),
but no restriction of f to a subinterval (0,b) can be extended to a
holomorphic function on an open sector with tip at 0 containing (0,b)?

.

Relevant Pages

  • Re: non-sectoral real analytic functions
    ... given a real analytic function f on the open interval, ... then f can be represented as power series in each x between 0 and a. ... But it seems that it is not always possible to extend f to an open ... Or merely define f on a sector with tip at 0 that intersects the real ...
    (sci.math.research)
  • Re: non-sectoral real analytic functions
    ... given a real analytic function f on the open interval, ... then f can be represented as power series in each x between 0 and a. ... But it seems that it is not always possible to extend f to an open ... Or merely define f on a sector with tip at 0 that intersects the real ...
    (sci.math.research)