Description:

The course starts from outlining the Jordan curve theorem and the Riemann-Stieltjes integral. Then basic results of complex analysis in one variable are explained in detail: the derivative of a complex function and the Cauchy-Riemann equations, holomorphic and analytic functions, the index of a point with respect to a closed curve, Cauchy's integral theorem, Morera's theorem, roots of a holomorphic function, analytic continuation, isolated singularities, the maximum modulus principle, Liouville's theorem, the Cauchy estimates, Laurent series, residue theorem.

Contents:

1. Connected, path-connected, simply connected spaces, the Jordan curve theorem
2. Variation of a function, length of a curve, the Riemann-Stieltjes integral (survey)
3. Derivative of a complex function, the Cauchy-Riemann equations
4. Holomorphic functions, power series, analytic functions
5. Regular curves, integration of a function along a curve (contour integral), the index of a point with respect to a closed curve
6. Cauchy's integral theorem for triangles
7. Cauchy's integral formula for convex sets, relation between holomorphic and analytic functions, Morera's theorem
8. Roots of a analytic function, analytic continuation
9. Isolated singularities
10. The maximum modulus principle, Liouville's theorem
11. The Cauchy estimates, uniform convergence of analytic functions
12. Cauchy's integral theorem (general version)
13. The residue theorem

Recommended literature:

Key references:
[1] W. Rudin: Real and Complex Analysis, (McGrew-Hill, Inc., New York, 1974)

Recommended references:
[2] J. B. Conway: Functions of One Complex Variable I, Springer-Verlag, New York, 1978

Keywords:

Jordan curve theorem, Riemann-Stieltjes integral, Cauchy-Riemann equations, Morera's theorem, isolated singularity, maximum modulus principle, Liouville's theorem, Cauchy estimates, Laurent series, residue theorem

Abbreviations used:

Semester:

• W ... winter semester (usually October - February)
• S ... spring semester (usually March - June)
• W,S ... both semesters

Mode of completion of the course:

• A ... Assessment (no grade is given to this course but credits are awarded. You will receive only P (Passed) of F (Failed) and number of credits)
• GA ... Graded Assessment (a grade is awarded for this course)
• EX ... Examination (a grade is awarded for this course)
• A, EX ... Examination (the award of Assessment is a precondition for taking the Examination in the given subject, a grade is awarded for this course)

Weekly load (hours per week):

• P ... lecture
• C ... seminar
• L ... laboratory
• R ... proseminar
• S ... seminar