FUNCTIONS OF COMPLEX VARIABLES
Compulsory course 4 credits.
Theory + Exercises : 4 credits
I. Course description/overview
This course provides
- Properties of complex numbers and complex functions
- Properties analytic functions, power expansions,
- Properties of line integrals and the theory of residus,
- the theory of conformal mappings.
II. Prerequisites : Analysis 1, 2, 3.
III. Contents
Chapter 1 Complex numbers
1.1 complex numbers and algebraic operations
1.2 Complex plane
1.3 fractional power of complex numbers
1.4 Riemann sphere
Chapter 2. Differentiability
2.1 Limits and continuous
2.2 Differentiability
2.3 Analyticity
2.4 Harmonic functions
Chapter 3 Elementary complex functions
3.1 Exponential functions
3.2. Trigonometric and hyperbolic functions
3.3 Logarithmic functions
3.4 Inverse trigonometric and hyperbolic functions
3.5 Branch points and brach cuts
Chapter 4 Line integration on the complex plane
4.1 Line integration
4.2 Green theorem
4.3 Path independent and indefinite integrals
4.4 Cauchy integral formula
Chapter 5 Power series
5.1 Convergence of complex series
5.2 Uniform convergence of series of complex functions
5.3 Power series and Taylor series
5.4 Techniques for expanding Taylor series
5.5 Zeroes of analytic functions and identity theorem
5.5 Laurence series
Chapter 6 Residue and applications
6.1 Singular points
6.2 Definition of the residue
6.3 Techniques for calculating the residue
6.4 Integration using residue calculus
6.5 Applications for Fourier and Laplace transforms
Chapter 7 Conformal mappings
7.1 The conformal properties
7.2 One-to-one mappings
7.3 Linear fractional transformations
7.4 Conformal mapping and boundary value problems
7.5 Schwarz-Christoffel transformations
References
[1] L.V. Ahlfors, Complex analysis, McGaw-Hill, New York, 1966.
[2] H. Cartan, Theorie elementaire des fonctions analytiques d’une ou plusieurs variables complexes, Hermann, Paris, 1961.
[3] A. David Wunsch, Complex variables with applications, Addison-Wesley, 1994.