Normal Families and Picard's Great Theorem
(2022) In Bachelor's Theses in Mathematical Sciences MATK11 20221Mathematics (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:
http://lup.lub.lu.se/student-papers/record/9087909
- author
- Jahic, Ena LU
- supervisor
-
- Yacin Ameur LU
- organization
- course
- MATK11 20221
- year
- 2022
- 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}}, }