Complex Analysis I

News:

• Added 3rd supplementary exercises (related to complex integration and winding numbers).

• NO lecture on March 3rd (but exercises as usual). 

• There will be a lecture on March 2nd, but if I run out of material, the lecture on March 3rd *may* be cancelled. Final information here latest on March 2nd. We will also decide on March 2nd if there will be an extra exercise class on March 10 (this would consist of some kind of rehearsal exercises). 

• Seventh exercises online. This will be the last "official" sheet, but there may be further voluntary exercises to support rehearsing.  If there are, we may also have an additional exercise session to discuss those. More information will follow here, and I'll also be happy to hear your wishes during classes. 

• Sixth exercises online.

• Added Section 3.5 to the lecture notes (after having realised that lecturing about "paths" is unsatisfactory without ever mentioning that they are "compact").

• Fifth exercises online. Second "Supplementary exercise" set online.

• New section below: "Errors in lecture notes". These are fixed in the current version of the notes, but if you read the notes a while ago, please check again. Also, if you find something suspicious, don't hesitate to contact me: the notes are new, and very likely there are lots of small(?) mistakes remaining. 

• Fourth exercises online.

• Third exercises online.

• Second exercises online.

• We decided in class that the breaks between lectures are only 5 min. 

• First exercises online.

• First lecture: January 12. First exercise session: January 20. 

Errors in lecture notes:

• The proof of Theorem 8.1 in the lecture notes was initially wrong (or at least seriously incomplete). It has now been fixed. 

• Earlier in the notes, open and connected sets were called "domains", but in fact the right word is "region". Suomeksi "alue".

• In the definition of "accumulation point" (Definition 3.26), it was earlier assumed that "z in X". This should be "z in C".

• Below (3.44) (definition of connected sets) it has been added that U,V are disjoint. 

Lecture times and exercise sessions:

• Lectures Thursdays and Fridays 10.15 - 12.00 (both in MaD302)

• Exercise sessions: Fridays at 8.15 (MaD380) and 12.15 (MaD355) 

(Non-Finnish speaking students: please choose the Friday morning exercise session is possible.)

Course description

This is an introductory course on complex analysis. Our topics include (under construction):

• Definition of complex numbers and basic their operations (product, inverse, modulus, conjugate, argument)  

• Review of metric and topological propertis of the plane (limits and continuity, open and closed sets, connectedness) 

• Analytic functions: defininition, basic properties, and examples (complex exponential, roots, logarithms, trigonometric functions)

• Cauchy-Riemann equations

• Complex integration (paths and path integrals)

• Cauchy's theorem in convex domains

• The Cauchy integral formula (local version) and its corollaries

• Maximum modulus principle, Liouville's theorem, and the fundamental theorem of algebra

Materials

There are lecture notes available in both Finnish and English. The Finnish notes are by Tero Kilpeläinen. These English notes contain the same topics (although details are different), and the lectures will follow the ordering of the English version. 

Exercises

Exercises will be published here. You will need to complete at least 30% of the exercises to take the course exam. 

• First exercises

• Second exercises

• Third exercises

• Fourth exercises

• Fifth exercises

• Sixth exercises

• Seventh exercises

Assistant: Tuomas Oikari

Supplementary exercises

These exercises give you a voluntary chance to improve your "complex calculus" skills, if you feel that you need such improvement. These exercises will not be discussed in the sessions, and they don't yield "points" of any sort, but of course you can ask questions about them if something isn't clear. 

• Supplement I

• Supplement II

• Supplement III