Unit of study_

# MATH2023: Analysis

Analysis grew out of calculus, which leads to the study of limits of functions, sequences and series. It is one of the fundamental topics underlying much of mathematics including differential equations, dynamical systems, differential geometry, topology and Fourier analysis. This unit introduces the field of mathematical analysis both with a careful theoretical framework as well as selected applications. It shows the utility of abstract concepts and teaches an understanding and construction of proofs in mathematics. This unit will be useful to students of mathematics, science and engineering and in particular to future school mathematics teachers, because we shall explain why common practices in the use of calculus are correct, and understanding this is important for correct applications and explanations. The unit starts with the foundations of calculus and the real numbers system. It goes on to study the limiting behaviour of sequences and series of real and complex numbers. This leads naturally to the study of functions defined as limits and to the notion of uniform convergence. Returning to the beginnings of calculus and power series expansions leads to complex variable theory: elementary functions of complex variable, the Cauchy integral theorem, Cauchy integral formula, residues and related topics with applications to real integrals.

Code MATH2023 Mathematics and Statistics Academic Operations 6
 Prerequisites: ? (MATH1X21 or MATH1931 or MATH1X01 or MATH1906) and (MATH1X23 or MATH1933 or MATH1X03 or MATH1907) and (MATH1XX2 or a mark of 65 or above in MATH1014) None MATH2923 or MATH3068 or MATH2962

At the completion of this unit, you should be able to:

• LO1. demonstrate a conceptual understanding of limit, continuity, differentiation, and integration as well as a thorough background in variety of techniques and applications of mathematical analysis
• LO2. assess problems in the framework of mathematical analysis, to choose among several potentially appropriate mathematical methods of solution, and persist in the face of difficulty
• LO3. present complete and mathematically rigorous solutions for problems in mathematical analysis that include appropriate justification for their reasoning
• LO4. recognise problems in mathematics, science, engineering and real life that are amenable to mathematical analysis, and to formulate models for such problems and apply the techniques of mathematical analysis in solving them.

## Unit outlines

Unit outlines will be available 2 weeks before the first day of teaching for the relevant session.