MATHEMATICAL ANALYSIS I

Learning Objectives

• First Module

  1. Knowledge and understanding: Assimilation of definitions and main results concerning basic mathematical analysis and functions of one real variable.

  2. Applying knowledge and understanding: acquisition of an appropriate level of autonomy in theoretical knowledge and in the use of basic analytical tools.

  3. Making judgements: ability to draw conclusions, ability to reflect and calculate. Ability to apply the notions learned to solving problems and exercises.

  4. Communication skills: ability to communicate the notions acquired through an adequate scientific language.

  5. Learning skills: ability to deepen and develop the knowledge acquired. Ability to critically use computer tools of symbolic computation.

• II

  Primary goal of the course is to master the techniques and methods that are peculiar to Mathematical Analysis.

Course Structure

• First Module

  Lectures in classroom. Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programm planned and outlined in the syllabus.

• II

  The Teacher via examples and results uses blackboard to show the rigorous methods of Mathematical Analysis.

  Should teaching be carried out blendedly or remotely necessary changes will be set up. Learning assessments may also be carried out on line, should the conditions require it.

Detailed Course Content

• First Module
  • Sets of numbers. Real numbers. The ordering of real numbers. Completeness of R. Factorials and binomial coefficients. Relations in the plane. Complex numbers. Algebraic operation. Cartesian coordinates. Trigonometric and exponential form. Powers and nth roots. Algebraic equations.
  • Limits. Neighbourhoods. Real functions. Limits of functions. Theorems on limits: uniqueness and sign of the limit, comparison theorems, algebra of limits. Indeterminate forms of algebraic and exponential type. Substitution theorem. Limits of monotone functions. Sequences. Limit of a sequence. Sequential characterization of a limit. Cauchy's criterion for convergent sequences. Infinitesimal and infinite functions. Local comparison of functions. Landau symbols and their applications.
  • Continuity. Continuous functions. Sequential characterization of the continuity. Points of discontinuity. Discontinuities for monotone functions. Properties of continuous functions (Weierstrass's theorem, Intermediate value theorem). Continuity of the composition and the inverse functions.
  • Numerical series. Round-up on sequences. Numerical series. Series with positive terms. Alternating series. The algebra of series. Absolute and Conditional Convergence.
• II

  Differential Calculus. Integral Calculus. Ordinary differential equations.

Textbook Information

• First Module

  [1] Bramanti, C. Pagani, S. Salsa, Analisi Matematica uno, Zanichelli.

  [3] N. Fusco, P. Marcellini, C. Sbordone, Analisi Matematica Uno, Liguori Editore.

  [3] G. Di Fazio, P. Zamboni, Analisi matematica 1, Monduzzi Editoriale

• II
  1. M.Giaquinta - G. Modica Mathematical Analysis: Functions of one variable Birkhäuser (2013)
  2. W. Rudin - Principles of Mathematical Analysis 3 ed Mc Graw Hill