Mathematical Analysis
| Professor | Prof. Dr. Florian Bünger |
| Programm of study | GESBC, CIBC |
| Typus | mandatory course |
| Turnus | 2. Semester |
| Course Format | 6h lectures per week, 2h instructions per week, 2h exercises per week |
| ECTS Credit Points | 9 |
| Prerequisite | High school mathematics |
Period
The lecture will be read in summer semester. The detailed schedule information about time and place can be found in "Vorlesungsankündigungen". The additional contact person at the institute is Michael Dudzinski (office hours).
Contents
Sequences
examples, limits, bounded sequences, sums and products of sequences, increasing and decreasing sequences, Cauchy sequences, subsequences, accumulation points, limes inferior, limes superior, sequences in vector spaces, compactness, completeness
Series
geometric series, harmonic series, convergence, absolute convergence, convergence criterion
( Leibnitz-, comparison-, root-, ratio-)
Continuous functions
epsilon-characterization, sum, difference and ratio of continuous functions, Bolzano theorem, intermediate value theorem, inverse functions, extreme values, limits of functions
Polynomials
Horner scheme, zeros of polynomials, interpolation with polynomials, error of the interpolation
Limits in function spaces
pointwise convergence, uniform convergence, series of functions, approximation, Bernstein polynomials
Power series
radius of convergence, ratio test
Elementary functions
logarithm, exponential, trigonometric and hyperbolic functions
Differentiable real functions
characterization of differentiability sums, products, ratios of differentiable real functions, differentiation of power series, rules for differentiation, Rolle's theorem, mean-value-theorem, l'Hôpital rules, Taylor's theorem
Analysis of functions
domain of definition, symmetries, zeros, poles and limits, asymptotes, monotonicity behavior, extreme values, points of inflection
Numerical solution of equations (Optional)
bisection method, fixed point theorems, Lipschitz continuity, different types of fixed points, Newton's method, iterative methods for linear systems
The Riemann integral
integrability, rules for integration, the indefinite integral, antiderivatives, the fundamental theorem of differential and integral calculus, integration by parts, integration by substitution, integration of rational functions
Improper integrals
unbounded domains, unbounded functions
Numerical integration (Optional)
Newton-Cotes formulas, composite Newton-Cotes formulas, error of quadrature formulas
Periodic functions
Fourier expansions, approximation, uniform convergence of Fourier expansion, complex Fourier expansion, trigonometric interpolation
References
- J. Levin:
Introduction to Mathematical Analysis, Cambridge University Press, 2000 - V.A.Zorich:
Mathematical Analysis I, Springer Verlag, 2004 - W. Mackens, H. Voss:
Mathematik für Studierende der Ingenieurwissenschaften II (Manuscript) - W. Rudin:
Principies of Mathematical Analysis, Mc Grau-Hill, 1976 - W. Walter:
Analysis I, Springer Verlag, 1985 - H. Amann, J. Escher:
Analysis I, Birkhäuser Verlag, 1998
