Department of Applied Mathematics & Theoretical Physics -- (302) 857-7516 (main), -7517 (fax)
Complex Analysis II
AMTP-871 (17600) : W, 4:30–7:15 pm, ETV.137
[Topics][Daily
Schedule][Assignments][Welcome]
Grading recipe (no make-ups are given, except in cases of proven medical emergency):
Component |
Time |
Remark |
% of Grade |
---|---|---|---|
Late HW = 0 credit !!! |
30% |
||
Term-paper |
Last week of semester |
Chosen topic |
70% |
Complex analysis (Math 871) is a sequel to Math-571, which thus is an absolute prerequisite. The course extends upon this, revisiting the maximum and minimum modulus principles, continues with the study of meromorphic functions and analytic continuation. This leads into Conformal mapping, harmonic functions and the Picard Theorems. We end this course with a brief forray into multi-variable complex analysis. It is important to identify the common underlying principles, methods and techniques, as they can then freely be used in a vast variety of applications.
“Success
= 1% inspiration + 99% perspiration”--T.A. Edison
But, learning is still 100% learning!
§6: The Maximum Modulus Principle
§7: Entire and meromorphic Functions
§8: Analytic Continuation
§9: Normal Families
§10: Conformal Mapping
§11: Harmonic Functions
§12. Picard Theorems
01/16: Introductory
Matters
01/23: The Maximum Modulus Principle and
the Schwarz Lemma, Extensions and the Phragmén-Lindelöf
Theorem:
§6.1–6
01/30: Theorems of Mittag-Leffler, Weierstrass and Extensions: §7.1–3 [§6 HW due]
02/06: Ininite Products and Factorization of Entire Functions:
§7.5–6
02/13: The Jensen Formula and Entire Functions of Finite Order and the Runge Approximation Theorem: §7.7–9
02/20: Power Series, Natural Boundaries, Multi-Valued Functions and Riemann Sheets (= Surfaces):
§8.1–4 [§7 HW due]
02/27: The
Schwarz Principle, Monodromy, Permanence, the Gamma Function and its relatives (extra): §8.5–8
03/05: Normal Families, Regions, Riemann Mapping Theorem, Examples, §10.1–2 [§8 HW due]
03/12: Examples, Conformal Mapping of Multiply Connected Regions: §10.4
03/17-21: Spring
Recess
03/26: Harmonic Conjugate, and Max/Min Modulus Principles, The Poisson Integral Formula: §11.1–3 [§10 HW due]
04/02: The Dirichlet Problem and the Harnack Theorem: §11.4–5
04/09: Green’s Functions, the Bloch Theorem: §11.6 and §12.1
04/16: Theorems of Schottky and Picard: §12.2–3 [§11 HW due]
04/23: A Forray into Many-Variable Complex Analysis, Holomorphic Functions and Mappings: extra [§12 HW due]
04/30: Meromorphic Functions, Residues, Rational Maps and Exceptional Sets: extra, Review
Particular problems for each homework set will be announced in class; the due dates are given in the day-to-day schedule. All homework assignments are due in class, on the day indicated and should be either given to the instructor in hand, left in the instructor's mailbox in ETV#116. Late homework will not be accepted, except in cases of proven emergency.
Collaboration policy
Collaboration -- but not blind copying -- on the homework assignments is strongly encouraged; students should use this to learn from each other. All exams and quizzes are open text and open class-notes (including notebooks and class handouts), but no collaboration is allowed; by signing the exams and quizzes, the student implicitly agrees to abide by this policy. Violation of this policy is covered under University regulations on academic dishonesty and cheating.
Presentation and organization
While a neat presentation of home,- quiz- and exam-work is not required for full credit, it certainly makes it easier to assess the quality of the work and give the proper credit due. In all cases, include a simple sketch if it might help conveying the approach or the calculations. Where necessary, include all units and symbols such as the measure of an integral, arrow on a vector, vertical bars for the absolute value of a quantity, for the magnitude of a vector or for the determinant of a matrix, etc.
© Tristan Hübsch, 2007