Complex Analysis II (Winter 2014/15)

This is a course of the Berlin Mathematical School held in English.

Contents

Further topics in analysis of one complex variable: conformal mappings, in particular reflection principles and Schwarz-Christoffel formula; introduction to the theory of elliptic functions and some applications; analytic theory of differential equations, in particular Fuchsian differential equations and hypergeometric differential equations.

News

[11.2.15]

The certificates (Übungsscheine) can be found at Mrs Downes' office in MA 701 (Mon,Tue,Thu,Fri 9.30-11.30 am). During the semester break office hours of Yuri Suris and Ulrike Bücking are by appointment (please write an email).

[3.2.15]

Please note that the last tutorial takes place on Wednesday, February 11th.

[14.1.15]

Please note the corrections on homework sheet 10 in exercise 1+2 g).

[16.12.14, 5.1.15]

Please note the corrections on homework sheet 9 in exercise 1 a). Now all signs and exponents are correct.

[15.12.14]

There will be no office hour of Ulrike Bücking on Friday, December 19th. Please make an appointment or write an email.

[27.11.14]

Handwritten lecture notes of Prof. Suris can be found below. Updates will follow as the lecture advances.

[24.11.14]

There will be no office hour of Ulrike Bücking on Friday, November 28th. You can ask questions on Thursday or Monday instead (8.30 am to 3.30 pm) or write an email.

[14.11.14]

Please note the correction on homework sheet 5 in exercise 1. All exponents were missing in the formula for g.

[21.10.14]

There will be no office hour of Ulrike Bücking on Friday, November 7th. You can ask questions on Thursday or Monday instead (8.30 am to 3.30 pm) or write an email.

[21.10.14]

Please note the new room for the lectures on Thursday: MA 376. Also note that there will be no lecture on Thursday, 23.10.14.

[20.10.14]

The tutorial on Monday is very small. You may consider to switch your tutorial from Wednesday to Monday.

[15.10.14]

Please note the correction in homework exercise 3 of the first sheet and the adjustment of the due date.

[14.10.14]

There will be a second tutorial on Monday, 12-14, in MA 545 covering the same topics as the tutorial on Wednesday before.

[1.10.14]

The lecture starts on Tuesday 14.10.
The first tutorial is on October 15. There will possibly be a second tutorial.

Homework policy

To get a certificate for the tutorial, you need to satisfactorily complete 60% of the homework assignments. Homeworks shall be prepared individually or in groups of two students. Homework should be handed in at the beginning of the tutorial.

Exercise sheets

• Exercise Sheet 1
• Exercise Sheet 2
• Exercise Sheet 3
• Exercise Sheet 4
• Exercise Sheet 5
• Exercise Sheet 6
• Exercise Sheet 7
• Exercise Sheet 8
• Exercise Sheet 9
• Exercise Sheet 10
• Exercise Sheet 11
• Exercise Sheet 12
• Exercise Sheet 13
• Exercise Sheet 14
• Exercise Sheet 15 (no homework)
• Exercise Sheet 16 (no homework)

Lecture notes

• Chapter 1 (Conformal maps)
• Chapter 2 (Schwarz-Christoffel)
• Chapter 3 (Circular arc polygons)
• Chapter 4 (Triangular groups)
• Chapter 5 (Elliptic functions)
• Chapter 6 (Weierstraß functions)
• Chapter 7 (Jacobi functions)
• Chapter 8 (Theta functions)
• Chapter 9 (Applications in number theory and differential equations)
• Chapter 10 (Modular properties of theta functions)
• Chapter 11 (Applications to geometry)
• Chapter 12 (Ordinary complex differential equations: existence and uniqueness)
• Chapter 13 (Local behaviour of solutions to linear ODEs with meromorphic coefficients)
• Chapter 14 (Fuchsian systems of ODEs)
• Chapter 15 (Analytic continuation and monodromy)
• Chapter 16 (Hypergeometric differential equation)

Literature

There are many textbooks on complex analysis covering some part of the material of this cours, including all languages and some old volumes. The following choice is not ordered by any aspect.

• Fischer and Lieb. Ausgewaehlte Kapitel aus der Funktionentheorie. Vieweg, 1988, in German.
• Hurwitz. Funktionentheorie. Birkhäuser, 1932, in German.
• Hurwitz. Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen. Springer, 2000, in German.
• Golubew. Vorlesungen über Differentialgleichungen im Komplexen. Deutscher Verlag der Wissenschaften, 1958, in German.
• Ince. Ordinary differential equations. Dover publications, 1926.
• Whittaker and Watson. A course of modern analysis. Cambridge University Press, 1927.
• Lang. Complex Analysis. Springer, several editions.
• Kyrala. Applied Functions of a complex variable. Wiley, 1972.
• Lorenz. Funktionentheorie. Spectrum Verlag, 1997, in German.
• Moretti. Functions of a complex variable. Prentice-Hall, 1964.
• Priwalow. Einführung in die Funktionentheorie. Teil III. Teubner, 1959, in German.
• Tutschke and Vasudeva. An introduction to complex analysis. Chapman and Hall, 2005.
• Hille. Ordinary differential equations in the complex domain. Wiley, 1976.
• Plemelj. Problems in the sense of Riemann and Klein. Interscience, 1964.
• Gray. Linear differential equations and group theory from Riemann to Poincaré. Birkhäuser, 1986.
• Klein. Vorlesungen über die hypergeometrische Funktion. Springer, several editions, in German.
• Kratzer and Franz. Transzendente Funktionen. AVG, 1960, in German.

Office hours