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The logarithmic conjugation theorem and conformal equivalence between multiply connected domains

Löfström, Martin LU (2022) In Bachelor’s Theses in Mathematical Sciences MATK11 20221
Mathematics (Faculty of Engineering)
Mathematics (Faculty of Sciences)
Centre for Mathematical Sciences
Abstract
The existence of a primitive of analytic functions, of an analytic logarithm
of zero-free analytic functions, and of a harmonic conjugate to any harmonic
function are fundamental propositions of elementary complex analysis, holding on simply-connected domains. We find that with suitable corrections,
analogous statements can be made on multiply connected domains, foremost
of these is these being the logarithmic conjugation theorem. This latter theorem is used to prove that isolated singularities on Hardy spaces are removable,
as well as to prove the doubly connected mapping theorem. For more general
multiply connected domains, we will prove conformal equivalence with slit
domains, in order to lay the foundation to the multipy... (More)
The existence of a primitive of analytic functions, of an analytic logarithm
of zero-free analytic functions, and of a harmonic conjugate to any harmonic
function are fundamental propositions of elementary complex analysis, holding on simply-connected domains. We find that with suitable corrections,
analogous statements can be made on multiply connected domains, foremost
of these is these being the logarithmic conjugation theorem. This latter theorem is used to prove that isolated singularities on Hardy spaces are removable,
as well as to prove the doubly connected mapping theorem. For more general
multiply connected domains, we will prove conformal equivalence with slit
domains, in order to lay the foundation to the multipy connected mapping
theorem. (Less)
Popular Abstract (Swedish)
Vi behandlar i denna uppsats några generaliseringar av resultat i grundläggande komplex analys, såsom existens av logaritmiskt konjugat som generalisering av harmoniskt konjugat. Detta använder vi sedan bland annat för
att visa en sats om konforma avbildningar (d.v.s. avbildningar som bevarar
vinklar), som utgör en analogi med Riemanns avbildningssats för områden i
planet med ett hål. Detta resultat utgör sedan en bro till en ytterligare mer
generell sats, som behandlar sådana avbildningar mellan områden som har
n hål, och som kan visas med andra metoder.
Please use this url to cite or link to this publication:
author
Löfström, Martin LU
supervisor
organization
alternative title
Function theory in multiply connected domains
course
MATK11 20221
year
type
M2 - Bachelor Degree
subject
publication/series
Bachelor’s Theses in Mathematical Sciences
report number
LUNFMA-4136-2022
ISSN
1654-6229
other publication id
2022:K10
language
English
id
9169131
date added to LUP
2025-06-27 15:52:09
date last changed
2025-06-27 15:52:09
@misc{9169131,
  abstract     = {{The existence of a primitive of analytic functions, of an analytic logarithm
of zero-free analytic functions, and of a harmonic conjugate to any harmonic
function are fundamental propositions of elementary complex analysis, holding on simply-connected domains. We find that with suitable corrections,
analogous statements can be made on multiply connected domains, foremost
of these is these being the logarithmic conjugation theorem. This latter theorem is used to prove that isolated singularities on Hardy spaces are removable,
as well as to prove the doubly connected mapping theorem. For more general
multiply connected domains, we will prove conformal equivalence with slit
domains, in order to lay the foundation to the multipy connected mapping
theorem.}},
  author       = {{Löfström, Martin}},
  issn         = {{1654-6229}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Bachelor’s Theses in Mathematical Sciences}},
  title        = {{The logarithmic conjugation theorem and conformal equivalence between multiply connected domains}},
  year         = {{2022}},
}