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Normal Families and Picard's Great Theorem

Jahic, Ena LU (2022) In Bachelor's Theses in Mathematical Sciences MATK11 20221
Mathematics (Faculty of Engineering)
Mathematics (Faculty of Sciences)
Abstract
We establish the theory of normal families of meromorphic functions taking values in the extended complex plane, which can be regarded as a metric space equipped with the spherical metric. Using the notion of spherical derivatives, we state and prove Marty's theorem on normal families of meromorphic functions. As a result, we deduce the classical Montel's theorem on relatively compact families of analytic functions. With this knowledge, we obtain the fundamental normality test - Montel's three value theorem and prove it using the lemma of Zalcman. Applying these results, the proof of the celebrated Picard's great theorem easily follows.
Popular Abstract
At the heart of the complex function theory lies the notion of a normal family of meromorphic functions. The subject has permeated through Picard's theorems, the Riemann mapping theorem, and many modern results such as the Bloch principle.
This thesis deals with the concept of convergence of sequences of meromorphic functions taking values in the extended complex plane. A family of meromorphic functions is said to be normal if each sequence in the family converges uniformly in the chordal metric on each compact subset of the domain.
This work culminates in proving different normality criteria for families of meromorphic functions, such as Marty's theorem (a criterion that uses the notion of spherical derivatives); Montel's three-value... (More)
At the heart of the complex function theory lies the notion of a normal family of meromorphic functions. The subject has permeated through Picard's theorems, the Riemann mapping theorem, and many modern results such as the Bloch principle.
This thesis deals with the concept of convergence of sequences of meromorphic functions taking values in the extended complex plane. A family of meromorphic functions is said to be normal if each sequence in the family converges uniformly in the chordal metric on each compact subset of the domain.
This work culminates in proving different normality criteria for families of meromorphic functions, such as Marty's theorem (a criterion that uses the notion of spherical derivatives); Montel's three-value theorem (fundamental normality test); and the great Picard's theorem. (Less)
Please use this url to cite or link to this publication:
author
Jahic, Ena LU
supervisor
organization
course
MATK11 20221
year
type
M2 - Bachelor Degree
subject
keywords
Normal families, Meromorphic functions, Picard, Montel, Marty's theorem, Montel's three value theorem, Picard's great theorem, Omitted values, Spherical derivatives, Chordal metric, Extended complex plane
publication/series
Bachelor's Theses in Mathematical Sciences
report number
LUNFMA-4132-2022
ISSN
1654-6229
other publication id
2022:K5
language
English
id
9087909
date added to LUP
2024-04-15 16:33:01
date last changed
2024-04-15 16:33:01
@misc{9087909,
  abstract     = {{We establish the theory of normal families of meromorphic functions taking values in the extended complex plane, which can be regarded as a metric space equipped with the spherical metric. Using the notion of spherical derivatives, we state and prove Marty's theorem on normal families of meromorphic functions. As a result, we deduce the classical Montel's theorem on relatively compact families of analytic functions. With this knowledge, we obtain the fundamental normality test - Montel's three value theorem and prove it using the lemma of Zalcman. Applying these results, the proof of the celebrated Picard's great theorem easily follows.}},
  author       = {{Jahic, Ena}},
  issn         = {{1654-6229}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Bachelor's Theses in Mathematical Sciences}},
  title        = {{Normal Families and Picard's Great Theorem}},
  year         = {{2022}},
}