*Instructor:* Dr. Gergely AMBRUS

*Text:* handouts

*Prerequisite:* Calculus

Course description: The aim of the course is two-fold: we present theorems which belong to the principles of mathematical analysis as well as topics that are of independent interest. We shall especially emphasise on areas of combinatorial and geometric flavour and applications in discrete mathematics.

*Topics:*

1. Basic analysis: continuity, differentiation, mean value theorems, L'Hospital's Rule, derivatives of higher order. Essentials of complex analysis.

2. Infinite series: convergence, summability, power series. The number e. Taylor series.

3. Special functions: Exponential and logarithmic functions. Trigonometric functions. The Gamma functions; Stirling's formula.

4. Extremum problems and polynomials: Lagrange multipliers. Interpolating polynomials; Chebyshev polynomials. Bernstein's inequality. Trigonometric polynomials; Fejér's Theorem.

5. Fourier series: basic notions; Bessel's Inequality, Parseval's formula. Weierstrass approximation theorem. Discrete Fourier transforms; applications.

6. Convex analysis: separation theorems, elementary convexity. Depending on time, Fritz John's theorem about extremal ellipsoids.