## Real and Complex Analysis

 Unit code: MATH20101 Credit Rating: 20 Unit level: Level 2 Teaching period(s): Semester 1 Offered by School of Mathematics Available as a free choice unit?: N

Prerequisite

#### Aims

The course unit unit aims to introduce the basic ideas of real analysis (continuity, differentiability and Riemann integration) and their rigorous treatment, and then to introduce the basic elements of complex analysis, with particular emphasis on Cauchy's Theorem and the calculus of residues.

#### Overview

The first half of the course describes how the basic ideas of the calculus of real functions of a real variable (continuity, differentiation and integration) can be made precise and how the basic properties can be developed from the definitions. It builds on the treatment of sequences and series in MATH10242. Important results are the Mean Value Theorem, leading to the representation of some functions as power series (the Taylor series), and the Fundamental Theorem of Calculus which establishes the relationship between differentiation and integration.

The second half of the course extends these ideas to complex functions of a complex variable. It turns out that complex differentiability is a very strong condition and differentiable functions behave very well. Integration is along paths in the complex plane. The central result of this spectacularly beautiful part of mathematics is Cauchy's Theorem guaranteeing that certain integrals along closed paths are zero. This striking result leads to useful techniques for evaluating real integrals based on the 'calculus of residues'.

#### Learning outcomes

On completion of this unit successful students will be able to:

Real Analysis

• state the definition of limit of a function;  calculate  the  limit  for  simple  functions; prove and apply the Rules for Limits to calculations for more complicated functions,
• state the definition of continuity; prove that simple functions are continuous at given points; prove and apply the Rules for Continuous functions to more complicated functions,
• state the definition of differentiable; prove that simple functions are differentiable and calculate their derivatives at given points; prove and apply the Rules for Derivatives to more complicated functions,
• prove and apply the Intermediate Value Theorem; Inverse function Theorem; various results on the composition of functions; various mean value theorems,
• calculate Taylor polynomials; state Taylors Theorem with an error term; derive bounds on the error terms; prove that Taylor series for a function converge to that function,
• state the definition of the Riemann integral; calculate the Riemann integral for various functions.

Complex Analysis

• prove the Cauchy-Riemann Theorem and its converse and use them to decide whether a given function is holomorphic;
• use power series to define a holomorphic function and calculate its radius of co nvergence;
• define and perform computations with elementary holomorphic functions such as sin, cos, sinh, cosh, exp, log, and functions defined by power series;
• define the complex integral and use a variety of methods (the Fundamental Theorem of Contour Integration, Cauchy’s Theorem, the Generalised Cauchy Theorem and the Cauchy Residue Theorem) to calculate the complex integral of a given function ;
• use Taylor’s Theorem and Laurent’s Theorem to expand a holomorphic function in terms of power series on a disc and Laurent series on an annulus, respectively;
• identify the location and nature of a singularity of a function and, in the case of poles, calculate the order and the residue;
• apply techniques from complex analysis to deduce results in other areas of mathematics, including proving the Fundamental Theorem of Algebra and calculating infinite real integrals, trigonometric integrals, and the summation of series.

#### Assessment methods

• Other - 20%
• Written exam - 80%

#### Assessment Further Information

• Coursework; An in-class test in reading week for Real Analysis, an on-line test for Complex Analysis, each counting 10%.
• 3 hours end of semester examination; Weighting within unit 80%.

#### Syllabus

Real analysis

• Limits. Limits of real-valued functions, sums, products and quotients of limits. [5 lectures]
• Continuity. Continuity of real-valued functions, sums, products and quotients of continuous functions, the composition of continuous functions. Boundedness of continuous functions on a closed interval. The Intermediate Value Theorem. The Inverse Function Theorem. [5]
• Differentiability. Differentiability of real-valued functions, sums, products and quotients of continuous functions, Rolle's Theorem, the Mean Value Theorem, Taylor's Theorem. [5]
• Integration. Definition of the Riemann integral, integrability of monotonic and continuous function, the Fundamental Theorem of Calculus. [5]

Complex analysis

• The complex plane. The topology of the complex plane, open sets, complex sequences and series, power series, and continuous functions. [4]
• Differentiation. Differentiable complex functions and the Cauchy-Riemann equations. [2]
• Integration. Integration along paths, the Fundamental Theorem of Calculus, the Estimation Lemma, Cauchy's Theorem, Argument and Logarithm. [5]
• Taylor and Laurent Series. Cauchy's Integral Formula and Taylor series, Louiville's Theorem and the Fundamental Theorem of Algebra, zeros and poles, Laurent series. [5]
• Residues. Cauchy's Residue Theorem, the evaluation of definite integrals and summation of series. [6]

• Mary Hart, Guide to Analysis, Macmillan Mathematical Guides, Palgrave Macmillan; second edition 2001.
• Rod Haggerty, Fundamentals of Mathematical Analysis, Addison-Wesley, second edition 1993.
• Ian Stewart and David Tall, Complex Analysis, Cambridge University Press, 1983.

#### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

#### Study hours

• Lectures - 44 hours
• Tutorials - 22 hours
• Independent study hours - 134 hours

#### Teaching staff

Mark Coleman - Unit coordinator

Charles Walkden - Unit coordinator