MA3000: Complex Function Theory
| School | Cardiff School of Mathematics |
| Department Code | MATHS |
| Module Code | MA3000 |
| External Subject Code | G100 |
| Number of Credits | 10 |
| Level | L6 |
| Language of Delivery | English |
| Module Leader | Professor Marco Marletta |
| Semester | Spring Semester |
| Academic Year | 2016/7 |
Outline Description of Module
A lecture based module covering some advanced topics in complex analysis. The module aims to cover topics which are of particular relevance to spectral theory, differential equations and special functions.
Prerequisite Modules: MA2003 Complex Analysis
Recommended Modules: MA0221 Analysis III
On completion of the module a student should be able to
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Understand concepts of convergence for analytic functions and their consequences;
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State and prove the standard theorems on omitted values for entire and meromorphic functions and use them to prove results;
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Define the concept of order for an entire function, compute it for concrete examples, understand the relationship between this order and the growth of the sequence of zeros, state and prove the Hadamard Factorisation Theorem, and use uniqueness results to compute Hadamard factorisations of certain elementary functions;
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State and prove the Mittag-Leffler Theorem and compute the Mittag-Leffler expansion of certain functions;
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State and prove the Maximum Modulus Theorem and the Phragmen-Lindelöf Theorem and apply them.
How the module will be delivered
27 fifty-minute lectures
Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.
Students are also expected to undertake at least 50 hours private study including preparation of solutions to given exercises.
Skills that will be practised and developed
Skills:
Competence in complex analysis at a level which would allow a start to graduate study.
Transferable Skills:
Logical analysis and abstract reasoning. Cognitive skills.
How the module will be assessed
Formative assessment is carried out by means of regular tutorial exercises. Feedback to students on their solutions and their progress towards learning outcomes is provided during lectures.
The summative assessment is the written examination at the end of the module. This gives students the opportunity to demonstrate their overall achievement of learning outcomes. It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.
The examination paper has a choice of three from four equally weighted questions.
Assessment Breakdown
| Type | % | Title | Duration(hrs) |
|---|---|---|---|
| Exam - Spring Semester | 100 | Complex Function Theory | 2 |
Syllabus content
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Understand concepts of convergence for analytic functions and their consequences;
-
State and prove the standard theorems on omitted values for entire and meromorphic functions and use them to prove results;
-
Define the concept of order for an entire function, compute it for concrete examples, understand the relationship between this order and the growth of the sequence of zeros, state and prove the Hadamard Factorisation Theorem, and use uniqueness results to compute Hadamard factorisations of certain elementary functions;
-
State and prove the Mittag-Leffler Theorem and compute the Mittag-Leffler expansion of certain functions;
-
State and prove the Maximum Modulus Theorem and the Phragmen-Lindelöf Theorem and apply them.
Background Reading and Resource List
E. Hille, Analytic Function Theory. Ginn, 1962.
A.I. Markushevich, Theory of Functions of a Complex Variable. AMS Chelsea Publishing, 2005.