Learning objectives
The course teaches the basic facts of complex analysis and ordinary differential equations.
Prerequisites
Mathematical Analysis AB and C and Geometry A.
Course unit content
1) Sequences and series of functions.
Pointwise and uniform convergence of sequences of functions. Uniform convergence and continuity. Integration and differentiation of sequences of functions. Total convergence of series of functions and Weierstrass' M-test. Integration and differentiation of series of functions.
2) Complex numbers.
Algebra and topology of complex numbers. Sequences and series of complex numbers. Limits and continuity of complex functions. Curves and integration. Exponential and trigonometric complex functions. Euler's formula.
3) Holomorphic functions.
Complex derivatives. Cauchy-Riemann equations and application.
4) Power series.
Abel's lemma and radius of convergence. Cauchy-Hadamard formula. Differentiation and integration of power series.
5) Cauchy theorem.
The index of a closed curve. Local Cauchy theorem and Cauchy integral formula. Applications:
power series representation of holomorphic functions, zeros of holomorphic functions,
Cauchy estimates, Liouville theorem and the fundamental theorme of algebra.
6) Singularities.
Classification of singularirties. Laurent series. The theorem of residues.
7) Ordinary differential equations.
Cauchy problem. Local existence and uniqueness. Maximal and global solutions. Solutions of
some first order ode's: linear, separable and Bernoulli's equations.
8) Linear systems of ode's.
Fundamental system of solutions. Lagrange's method of undetermined coefficients. Linear: semisimple and nihilpotent matrices, Jordan canonical form. Exponential of a matrix. Linear equations of higher order.
Bibliography
G. C. Barozzi: Matematica per l'ingegneria dell'informazione, Zanichelli, Bologna, 2001;
J. B. Conway: Functions of one complex variable, Graduate Text in Mathematics n.11, Springer-Verlag, New York, 1978;
M. W. Hirsch - S. Smale: Differential equations, dynamical systems, and linear algebra, Academic Press, New York, 1974;
C. D. Pagani - S. Salsa: Analisi matematica vol.2, Masson, Milano, 1991.
Teaching methods
Written and oral final examination.
