# Course Title: Real and Complex Analysis

## Part A: Course Overview

Course Title: Real and Complex Analysis

Credit Points: 12.00

## Terms

### Teaching Period(s)

MATH2150

City Campus

145H Mathematical & Geospatial Sciences

Face-to-Face

Sem 2 2006,
Sem 2 2007,
Sem 2 2008,
Sem 2 2009,
Sem 1 2010,
Sem 1 2011,
Sem 1 2012,
Sem 1 2013,
Sem 1 2014,
Sem 1 2015,
Sem 2 2016

MATH2150

City Campus

171H School of Science

Face-to-Face

Sem 2 2017

Course Coordinator: Dr. Vera Roshchina

Course Coordinator Phone: +61 3 9925 3161

Course Coordinator Email: vera.roshchina@rmit.edu.au

Course Coordinator Location: 8-9-77

Pre-requisite Courses and Assumed Knowledge and Capabilities

You are assumed to have successfully completed the following courses or their equivalent:

MATH1142 Calculus and Analysis 1

MATH1144 Calculus and Analysis 2

MATH2140 Linear Algebra and Vector Calculus

Course Description

This course provides essential mathematical background for many subsequent courses in modern applied mathematics, pure mathematics, numerical analysis, statistics and operations research. Its aim is threefold. Firstly it extends the analysis of real functions of one variable to the analysis of functions of two or more real variables in a systematic manner. Secondly it provides sufficient background in the analysis of functions of a complex variable for you to study advanced engineering mathematics or aspects of probability theory. Thirdly it gives a brief introduction to some concepts that are essential background for the area of modern functional analysis. You will also become more familiar and comfortable with the language, abstract formalisms and proof techniques that are present in all modern texts on analysis and its applications.

Objectives/Learning Outcomes/Capability Development

This course contributes to the following Program Learning Outcomes for BP083 Bachelor of Science (Mathematics):

Knowledge and Technical Competence

• the ability to use the appropriate and relevant, fundamental and applied mathematical and statistical knowledge, methodologies and modern computational tools.

Problem-solving

• the ability to bring together and flexibly apply knowledge to characterise, analyse and solve a wide range of problems
• an understanding of the balance between the complexity / accuracy of the mathematical / statistical models used and the timeliness of the delivery of the solution.

On completion of this course you should be able to:

1. Apply the mathematical concept of convergence and its epsilon delta definition to establish the existence of limits and devise proofs of mathematical statements via the definition of convergence
2. Use fundamental mathematical concepts and theorems, such as mean value and implicit function theorems, to establish inequalities and estimates, to establish if a function of two variables is continuous and\or differentiable at a given point and explain how its partial derivatives relate to this property.
3. Manipulate calculus of functions of two or more variables and in particular make a change of variables using the Jacobian matrix in a multiple integral.
4. Elaborate the special character of functions of a complex variable and their properties and gain practical skills in analysing and manipulating functions of complex variables (including the evaluation of a line integral of a function of a complex variable using Cauchy’s integral formula, evaluation of real integrals using complex integration, and evaluation of Laurent Series and residues).
5. Operate with the concepts of Functional analysis and Hilbert spaces and  in order to obtain an orthonormal basis and approximate functions via the elements of that basis, apply the Gram-Schmidt orthogonalisation process.
6. Communicate a mathematical argument and construct some simple mathematical proofs.

Overview of Learning Activities

The basic theory will be delivered in face-to-face in lectures. Examples provided within lectures will further be developed in weekly tutorials. Tutorial exercises will provide timely and continuous feedback so you may gauge your progress during the semester.

Overview of Learning Resources

Lecture notes should provide sufficient knowledge in order to perform successfully in this course. Weekly tutorial sheets will provide examples of the type of problems that you are expected to master. The RMIT library contains a number of relevant texts that you may use to obtain alternative presentations of the topics given in lectures.

The lectures will be recorded whenever feasible, and additional material will be available on blackboard.

A Library Guide is available at http://rmit.libguides.com/mathstats

Overview of Assessment

Assessment Task 1  Mid semester test

Weighting 26%

This assessment task supports CLOs 1, 2, 3, 6

Assignments are done in class, with typical solutions handed out beforehand for homework. The particular questions will be aligned with the pace of the lectures.

Weighting 24%

These assessment tasks support CLOs 1, 2, ,3, 4, 5, 6

The assignments will be marked in timely fashion with feedback provided as: