Math20101 Complex Analysis Coursework Testing

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




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.


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:

  • understand the concept of limit for real functions and be able to calculate limits of standard functions and construct simple proofs involving this concept;
  • understand the concept of continuity and be familiar with the statements and proofs of the standard results about continuous real functions;
  • understand the concept of the differentiability of a real valued function and be familiar with the statements and proofs of the standard results about differentiable real functions;
  • appreciate the definition of the Riemann integral, and be familiar with the statements and proofs of the standard results about the Riemann integral including the Fundamental Theorem of Calculus;
  • understand the significance of differentiability for complex functions and be familiar with the Cauchy-Riemann equations;
  • evaluate integrals along a path in the complex plane and understand the statement of Cauchy's Theorem;
  • compute the Taylor and Laurent expansions of simple functions, determining the nature of the singularities and calculating residues;
  • use the Cauchy Residue Theorem to evaluate integrals and sum 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%.


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]

Recommended reading

  • 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 coordinatorCharles Walkden - Unit coordinator

Exam Feedback 2014/2015 - Semester 1 (January 2015)

Exam Feedback sessions are organised each semester by the Teaching & Learning Office and are an opportunity for students to see their marked examination scripts.

Copies of the examination paper and generic feedback (where available) are listed below. ‌

We are in the process of uploading the exam papers and feedback.

Level 1

Course UnitExam PaperExam Feedback
MATH10001, Mathematical Workshop


MATH10101, Sets, Numbers and Functions AMATH10101MATH10101
MATH10121, Calculus and Vectors AMATH10121MATH10121
MATH10111, Sets, Numbers and Functions BMATH10111MATH10111
MATH10131, Calculus and Vectors BMATH10131MATH10131
MATH10141, Probability 1MATH10141 (corrected)MATH10141
MATH10951, Financial Mathematics for Actuarial Science 1MATH10951 (corrected)MATH10951

Level 2

Level 3

Course UnitExam PaperExam Feedback
MATH31011, Fourier Analysis and Lebesgue IntegrationMATH31011MATH31011
MATH31051, Introduction to TopologyMATH31051MATH31051
MATH31061, Differentiable ManifoldsMATH31061MATH31061
MATH32001, Group TheoryMATH32001MATH32001
MATH32031, Coding TheoryMATH32031MATH32031
MATH32051, Hyperbolic GeometryMATH32051MATH32051
MATH32071, Introduction to Number TheoryMATH32071MATH32071
MATH33001, Predicate LogicMATH33001MATH33001
MATH34001, Applied Complex AnalysisMATH34001MATH34001
MATH34011, Asymptotic Expansions and Perturbation MethodsMATH34011MATH34011
MATH35001, Viscous Fluid FlowMATH35001MATH35001
MATH35021, ElasticityMATH35021MATH35021
MATH36001, Matrix AnalysisMATH36001MATH36001
MATH36041, Essential Partial Differential EquationsMATH36041MATH36041
MATH37001, Martingales with Applications to FinanceMATH37001MATH37001
MATH38001, Statistical InferenceMATH38001MATH38001
MATH38141, Regression AnalysisMATH38141MATH38141
MATH38061, Multivariate StatisticsMATH38061MATH38061
MATH38071, Medical StatisticsMATH38071MATH38071
MATH38091, Statistical ComputingMATH38091MATH38091
MATH38181, Extreme Values and Financial RiskMATH38181 (corrected)MATH38181
MATH38191, Statistical Modelling in FinanceMATH38191 (corrected)MATH38191
MATH39001, Combinatorics and Graph TheoryMATH39001MATH39001
MATH39511, Actuarial ModelsMATH39511MATH39511

Level 4

Course UnitExam PaperExam Feedback
MATH41011, Fourier Analysis and Lebesgue IntegrationMATH41011MATH41011
MATH41051, Introduction to TopologyMATH41051MATH41051
MATH41061, Differentiable ManifoldsMATH41061MATH41061
MATH42001, Group TheoryMATH42001MATH42001
MATH42041, NonCommutative AlgebraMATH42041MATH42041
MATH42051, Hyperbolic GeometryMATH42051MATH42051
MATH42061, Representation TheoryMATH42061MATH42061
MATH42071, Introduction to Number TheoryMATH42071MATH42071
MATH43001, Predicate LogicMATH43001MATH43001
MATH43051, Model TheoryMATH43051MATH43051
MATH44041, Applied Dynamical SystemsMATH44041MATH44041
MATH45051, Singularities, Bifurcations and CatastrophesMATH45051MATH45051
MATH45061, Continuum MechanicsMATH45061MATH45061
MATH46101, Numerical Linear AlgebraMATH46101MATH46101
MATH47101, Stochastic CalculusMATH47101MATH47101
MATH48001, Statistical InferenceMATH48001MATH48001
MATH48011, Linear Models with Nonparametric RegressionMATH48011MATH48011
MATH48061, Multivariate StatisticsMATH48061MATH48061
MATH48091, Statistical ComputingMATH48091MATH48091
MATH48181, Extreme Values and Financial RiskMATH48181 (corrected)MATH48181
MATH48191, Statistical Modelling in FinanceMATH48191 (corrected)MATH48191
MATH49111, Scientific ComputingMATH49111MATH49111

Level 6

Course UnitExam PaperExam Feedback
MATH61011, Fourier Analysis and Lebesgue Integration


MATH61051, Introduction to TopologyMATH61051MATH61051
MATH61061, Differentiable ManifoldsMATH61061MATH61061
MATH62001, Group TheoryMATH62001MATH62001
MATH62041, NonCommutative AlgebraMATH62041MATH62041
MATH62051, Hyperbolic GeometryMATH62051MATH62051
MATH62061, Representation TheoryMATH62061MATH62061
MATH62071, Introduction to Number TheoryMATH62071MATH62071
MATH63001, Predicate LogicMATH63001MATH63001
MATH63051, Model TheoryMATH63051MATH63051
MATH64041, Applied Dynamical SystemsMATH64041MATH64041
MATH64051, Mathematical Methods (as MAGIC022)MATH64051MATH64051
MATH65051, Singularities, Bifurcations and CatastrophesMATH65051MATH65051
MATH65061, Continuum MechanicsMATH65061MATH65061
MATH65740, Transferable Skills for Applied MathematiciansMATH65740MATH65740
MATH66101, Numerical Linear AlgebraMATH66101MATH66101
MATH67001, Martingales with Applications to FinanceMATH67001MATH67001
MATH67101, Stochastic CalculusMATH67101MATH67101
MATH68001, Statistical InferenceMATH68001MATH68001
MATH68011, Linear Models with Nonparametric RegressionMATH68011MATH68011
MATH68061, Multivariate StatisticsMATH68061MATH68061
MATH68091, Statistical ComputingMATH68091MATH68091
MATH68181, Extreme Values and Financial RiskMATH68181 (corrected)MATH68181
MATH68191, Statistical Modelling in FinanceMATH68191 (corrected)MATH68191
MATH69111, Scientific ComputingMATH69111MATH69111
MATH69511, Actuarial ModelsMATH69511MATH69511
MATH69531, General InsuranceMATH69531MATH69531

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