
Course Outline /Schedule (Weekly) Planned Learning Activities 
 Week  Subject  Student's Preliminary Work  Learning Activities and Teaching Methods 

1 
Topology of C and the Riemann Sphere 
Reading the relevant sections in the textbook and solving problems 
Lecturing 

2 
Möbius transformations and their properties 
Reading the relevant sections in the textbook and solving problems 
Lecturing 

3 
Differentiable and analytic functions. Cauchy Riemann conditions. Harmonic functions. 
Reading the relevant sections in the textbook and solving problems 
Lecturing 

4 
Line integrals, closed curves. Jordan curve theorem. 
Reading the relevant sections in the textbook and solving problems 
Lecturing 

5 
CauchyGoursat Theorem, Cauchy integral formulas. Fundamental Theorem of Algebra. 
Reading the relevant sections in the textbook and solving problems 
Lecturing 

6 
Open mapping property of analytic functions. Conformal mapping 
Reading the relevant sections in the textbook and solving problems 
Lecturing 

7 
Schwartz Lemma and corollaries. Three circle theorem. 
Reading the relevant sections in the textbook and solving problems 
Lecturing 

8 
Isolated singularities. Poles and essential singularities. Riemann s removable singularity theorem. 
Reading the relevant sections in the textbook and solving problems 
Lecturing 

9 
Midterm Exam 
Review of the material and problem solving 
Written Exam 

10 
Generalized Cauchy integral formula, residue theorem. Applications of the residue theorem. 
Reading the relevant sections in the textbook and solving problems 
Lecturing 

11 
Meromorphic functions. Laurent series. 
Reading the relevant sections in the textbook and solving problems 
Lecturing 

12 
MittagLeffler Theorem, topological properties of Meromorphic functions Weierstrass Theorem. 
Reading the relevant sections in the textbook and solving problems 
Lecturing 

13 
Lattices and doubly periodic functions. Properties of doubly periodic functions. 
Reading the relevant sections in the textbook and solving problems 
Lecturing 

14 
Weierstrass s P function, its properties, derivative and differential equation. Structure of the field of meromorphic functions. 
Reading the relevant sections in the textbook and solving problems 
Lecturing 

15 
Eliptic integralelliptic function relation. Picard s theorem. 
Reading the relevant sections in the textbook and solving problems 
Lecturing 

16/17 
Final Exam 
Review of the material and problem solving 
Written Exam 

