## MAS332 Complex Analysis

 Semester 1, 2018/19 10 Credits Lecturer: Dr Mary Hart Reading List Aims Outcomes Teaching Methods Assessment Full Syllabus

It is natural to use complex numbers in algebra, since these are the numbers we need to provide the roots of all polynomials. In fact, it is equally natural to use complex numbers in analysis, and this course introduces the study of complex-valued functions of a complex variable. Complex analysis is a central area of mathematics. It is both widely applicable and very beautiful, with a strong geometrical flavour. This course will consider some of the key theorems in the subject, weaving together complex derivatives and complex line integrals. There will be a strong emphasis on applications. Anyone taking the course will be expected to know the statements of the theorems and be able to use them correctly to solve problems.

Prerequisites: MAS211 (Advanced Calculus and Linear Algebra)

The following modules have this module as a prerequisite:

 MAS430 Analytic Number Theory MAS436 Functional Analysis

## Outline syllabus

• Revision of complex numbers
• Special functions
• Simple integrals of complex-valued functions
• Open sets, neighbourhoods and regions
• Differentiability; Cauchy-Riemann equations, harmonic functions
• Power series and special functions
• Complex line integrals
• Cauchy's Theorem
• Cauchy's integral formula and Cauchy's formula for derivatives
• Taylor's Theorem
• Laurent's Theorem and singularities
• Cauchy's Residue Theorem and applications

## Aims

• To introduce complex functions of a complex variable
• To demonstrate the critical importance of differentiability of complex functions of a complex variable, and its surprising relation with path-independence of line integrals
• To demonstrate the relevance of power series in complex analysis
• To develop the subject of complex analysis rigorously, highlighting its logical structure and proving several of the fundamental theorems
• To discuss some applications of the theory, including to the calculation of real integrals

## Learning outcomes

• Appreciate the stronger implications of differentiability for complex function.
• Be able to use Cauchy's Theorem, and Cauchy's Integral Formulae to solve problems.
• Be able to develop Taylor Series and Laurent Series.
• Be familiar with the computation of residues and be able to evaluate integrals using the Residue Theorem.
• Be able to quote the statement of theorems (including all the conditions which guarantee their validity) and demonstrate that they can use these results correctly to solve problems (including checking conditions before applying the conclusion of the theorem).

## Teaching methods

Lectures, problem solving

There are two lectures per week. All the material for the course is on "MOLE2" so students can spend their time in lectures thinking about the work rather than just copying down notes.
At times extra problems are incorporated into the lecture. These extra problems and their solutions are then added to the relevant folder in "MOLE2".
There are Exercise Sheets on all parts of the course and students are asked to submit solutions to a number of these at about fortnightly intervals. Detailed solutions for any exercises sheet are made available to view after the problems from the exercise sheet have been handed in. Students are also advised that they would be wise to attempt the remaining problems also.
As there are no formal tutorials, several office hours a week are allocated so that students can consult the lecturer about problems on the course.

20 lectures, no tutorials

## Assessment

One formal 2.5 hour written examination. Format: 4 questions from 5.

## Full syllabus

1. Revision of complex numbers
Revision of elementary arithmetic operations with complex numbers, the modulus of a complex number and modulus-argument form for a complex number. A reminder of the triangle inequalities. 2. Special functions
Definition of the exponential function ez and a proof that many of the familiar properties of the real exponential function continue to hold in the complex case. Definition of the trigonometric and hyperbolic functions in terms of the exponential function. A brief introduction to the complex logarithm. 3. Simple line integrals of complex valued-functions
Definition of a path. Evaluation of line integrals integrals along a path. 4. Regions in the complex plane
Definitions of the terms region and simply-connected region. 5. Differentiation.
The definition of a derivative. The definition of an analytic function in a region and at a point. The Cauchy-Riemann equations for differentiable functions. Special functions - the derivative of the exponential functions, the trigonometric functions and the hyperbolic functions. Harmonic functions. 6. Power series
A brief reminder of the first year results on series (see SOM111 notes) and a discussion of how these results can easily be extended to complex series. The concept of the radius of convergence and a useful formula for it (which can be used in many cases). The idea of a disc of convergence and the fact that power series may be treated in much the same way as polynomials inside the disc of convergence. For example power series may be differentiated and integrated term by term in the disc of convergence. 7. More on ∫ ;amma
The introduction of primitives. The ML estimates for | ∫ ;amma|. 8. Cauchy's Theorem
Statement of Cauchy's Theorem and its derivation from Green's Theorem. Independence of path for integrals of analytic functions on simply-connected regions. 9. Cauchy's integral formulae
Cauchy's Theorem is used to derive Cauchy's Integral Formula for a function and for its derivatives. 10. Taylor's Theorem
Cauchy's Integral Formulae are used to derive power series expansions for analytic functions (Taylor series). Taylor series are then used to investigate zeros of analytic functions. 11. Isolated singularities, Laurent's Theorem, Classification of singularities
Definition of an isolated singularity. Series expansion for a function which is analytic in a disc except at its centre (Laurent series) . Classification of the type of singularity by the use of the Laurent series or other methods. The residue at an isolated singularity. Quick ways of calculating residues at poles. 12. Cauchy's Residue Theorem
Cauchy's Residue Theorem and it application to evaluating integrals.