Complex Analysis

1MA022

Lecturer: Maciej Klimek

Tutor: Marcus Olofsson

Last update: 2015-05-21 at 13:53 CET

The following list outlines provisional time distribution of the topics covered in this

course, as well as the order of lectures (L) and tutorials (T). This plan may undergo mi-

nor modifications depending on feedback from the participants. During the semester

two home assignments (HA) will be handed out while they will not be obligatory

each will give an opportunity to earn up to 2 bonus points that can be added to the

final exam score. While complex analysis is one of the classical areas of mathemat-

ics, it is still being developed and over the years has resulted in a range of tech-

nological applications (e.g. in aeronautics, in filter design for signal processing, for

mapping of the human brain). Some of the key concepts of complex analysis can

be conveniently visualized with the help of the computer suitable MATLAB code

and references will be posted on Studentportalen in due course.

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Week 4, 2xL: Complex numbers, stereographic projection, basic complex functions,

visualization, subsets of the complex plane, continuity, complex derivatives (I.1

8, II.12).

Week 5, 2xL+T: Complex derivatives (continuation), the Cauchy-Riemann equations,

harmonic functions, conformal mappings (II.36).

Week 6, L+T: Conformal mappings (continuation), geometry of Mobius transforma-

tions (II.67).

Week 7, 2xL+T: Review of line integration and an extension to the complex case,

basic properties of harmonic functions, fundamental theorem of calculus for

analytic functions, existence of antiderivatives (III.15, IV.12).

Week 8, L+T: Cauchys integration theorem, Cauchys integral formula and conse-

quences (IV.37).

Week 9, 2xL+T+1st HW: Continuation from the previous week.

Week 10, L+T: Sequences and series of functions, power series expansions, Abels

lemma, radius of convergence (V.13).

Week 11, 2xL+T: Power series expansions of analytic functions, zeros of analytic func-

tions, the identity principle, Laurents series expansions (V.47, VI.1).

Week 12, L+T: Isolated singularities of analytic functions: removable singularities, poles

and essential singularities; meromorphic functions (VI.2).

Week 13, 2xL+T: The residue calculus and applications to evaluation of line integrals

(VII.17).

Week 17, L+T: Continuation from Week 13.

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Week 18, 2xL+T+2nd HW: The argument principle, Rouches theorem, Hurwitzs theo-

rem, the open mapping theorem, the inverse mapping theorem, winding num-

bers (VIII.14, VIII.6).

Week 19, 2xL+T: Simple-connectedness, the Riemann mapping theorem, Montels

theorem, (VIII.8, XI.2, 56).

Week 20, T: Continuation from Week 19.

Week 21, 3xL+T: A short introduction to fractals and Julia sets (XII.35); review and

repetition of the material covered in this course.

Week 22, T: Review and repetition of the material covered in this course.

Week 23, L: Review and repetition of the material covered in this course.

The chapter and section numbers in green refer to the textbook Complex Analysis

by Theodore W. Gamelin (published by Springer in 2001).

Additional study material will be posted on Studentportalen and/or handed out in

class as needed.

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