Abstract
Based on the axiom of choice we revisit a method to prove in simply connected domains the existence of a holomorphic function with prescribed zeros.
Acknowledgments
We thank Peter Pflug for reference [Citation6]. We learned from Robert B. Burckel that this approach (=meromorphic to holomorphic) of the Weierstrass theorem for simply connected domains is due to Mittag-Leffler and is mentioned as an exercise in his book [Citation3, p. 391], the hints given there being more involved, though, and that the general version is in [Citation2, p. 248]. We thank him for this information and for having caught some typos. Finally, we thank the referee(s) for their valuable two-page long report.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Notes
1 This step needs the axiom of choice: on the set of all polygonial arcs in which have
as the initial point consider the equivalence relation
if and only if both
have a common endpoint. Then from each equivalence class one chooses one member.
2 This is the only place where we need the assumption on the simply connectedness.
3 If we replace in (Equation1(1)
(1) ) the closed path γ by the closed path
, where
is another path connecting
with z, then we actually see that the function
is independent of the path chosen that connects
with z.
4 We think that this is also a magical behaviour since for any function m holomorphic in a domain Ω excepted at some isolated singularities , the function
has an essential singularity at
. As explained above, this ‘magical behaviour’ comes from the fact that G may not even by measurable.