## MA302-6-AU-CO:

Complex Variables

## Key module for

BSC G102 Mathematics (Including Year Abroad),

BSC G103 Mathematics (Including Placement Year),

BSC G104 Mathematics (Including Foundation Year),

MMATG198 Mathematics,

BSC G1G4 Mathematics with Computing (Including Year Abroad),

BSC G1G8 Mathematics with Computing (Including Foundation Year),

BSC G1GK Mathematics with Computing,

BSC G1IK Mathematics with Computing (Including Placement Year),

BSC G1F3 Mathematics with Physics,

BSC G1F4 Mathematics with Physics (Including Placement Year),

BSC G1F5 Mathematics with Physics (Including Foundation Year),

BSC GCF3 Mathematics with Physics (Including Year Abroad),

MSCIG199 Mathematics and Data Science

## Module description

Complex numbers and functions appear in Physics (Quantum Mechanics, especially to solve Schrödinger equation to find the wave function), in Engineering (Control theory and Signal Analysis as the Fourier transform requires integrating complex valued functions).

This module covers in details complex functions and variables and expands on complex derivatives and integration, and more advanced topics.

## Module aims

The aims of this module are:

- to introduce functions of a complex variable and techniques for complex integration including Cauchy’s theorem, integral formula, residue formula, and Jordan’s Lemma.

## Module learning outcomes

By the end of the module, students will be expected to:

- Express complex numbers in both Cartesian and polar forms;
- Parametrize curves and plot regions in the complex plane.
- Carry out calculations of limits, continuity, and differentiability of complex functions.
- Determine whether and where a function is holomorphic / analytic;
- Carry out complex integration via line integrals, Cauchy’s Theorem, Cauchy’s integral formula and Cauchy’s differentiation formula
- Obtain appropriate series expansions of functions;
- Evaluate residues at pole singularities;
- Apply the Residue Theorem to the calculation of real integrals.

## Module information

*Indicative syllabus*

Complex numbers

- Cartesian and polar forms
- Lines, circles and regions in the complex plane

Functions of a complex variable

- limits
- continuity
- derivatives
- holomorphic/analytic functions
- Cauchy-Riemann Equations

Elementary functions

- The exponential function
- The logarithmic function
- Branche and Branch cuts
- Complex Exponents
- Trigonometric and hyperbolic functions

Complex Integration

- Line integrals
- Cauchy's theorem
- Cauchy's integral formula
- Derivatives of an analytic function (Cauchy's differentiation formula)
- Consequences of Cauchy’s Integral formula (Liouville's theorem, Maximum Modulus Principle, Fundamental theorem of algebra)
- Morera’s Theorem

Sequences and Series of Complex Numbers

- Taylor series
- Analytic functions and their relationship to holomorphic functions
- Laurent's theorem

Residue and Poles

- Calculation of residues
- Cauchy's residue theorem

Argument principle

- Rouch\'{e}'s theorem.
- Jordan's lemma
- Calculation of definite integrals using residue theory.

## Learning and teaching methods

Teaching in the School will be delivered using a range of face-to-face lectures, classes, and lab sessions as appropriate for each module. Modules may also include online only sessions where it is advantageous, for example for pedagogical reasons, to do so.

## Bibliography*

## Assessment items, weightings and deadlines

Coursework / exam | Description | Deadline | Coursework weighting |
---|---|---|---|

Coursework | Assignment 1 | ||

Coursework | Assignment 2 | ||

Exam | Main exam: In-Person, Open Book (Restricted), 120 minutes during Summer (Main Period) | ||

Exam | Reassessment Main exam: In-Person, Open Book (Restricted), 120 minutes during September (Reassessment Period) |

### Exam format definitions

**Remote, open book:**Your exam will take place remotely via an online learning platform. You may refer to any physical or electronic materials during the exam.**In-person, open book:**Your exam will take place on campus under invigilation. You may refer to any physical materials such as paper study notes or a textbook during the exam. Electronic devices may not be used in the exam.**In-person, open book (restricted):**The exam will take place on campus under invigilation. You may refer only to specific physical materials such as a named textbook during the exam. Permitted materials will be specified by your department. Electronic devices may not be used in the exam.**In-person, closed book:**The exam will take place on campus under invigilation. You may not refer to any physical materials or electronic devices during the exam. There may be times when a paper dictionary, for example, may be permitted in an otherwise closed book exam. Any exceptions will be specified by your department.

Your department will provide further guidance before your exams.

### Overall assessment

Coursework | Exam |
---|---|

20% | 80% |

### Reassessment

Coursework | Exam |
---|---|

20% | 80% |

## External examiner

**30**hours,

**30 (100%)**hours available to students:

**0**hours not recorded due to service coverage or fault;

**0**hours not recorded due to opt-out by lecturer(s), module, or event type.

*** Please note:** due to differing publication schedules, items marked with an asterisk (*) base their information upon the previous academic year.

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