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## MTH2001 - Analysis (2018)

MODULE TITLE | Analysis | CREDIT VALUE | 30 |
---|---|---|---|

MODULE CODE | MTH2001 | MODULE CONVENER | Dr Jimmy Tseng (Coordinator) |

DURATION: TERM | 1 | 2 | 3 |
---|---|---|---|

DURATION: WEEKS | 11 | 11 | 0 |

Number of Students Taking Module (anticipated) | 186 |
---|

Real and complex analysis represent two of the most useful and beautiful theories in mathematics. Developed in response to the crisis in mathematics during the 18th and 19th centuries by Gauss, Cauchy and Riemann, analysis is a powerful tool that makes precise the notions of limit, convergence, continuity and differentiability – and studies the real and complex topologies that permit a rigorous approach to studying these concepts. In this module, you will develop your understanding of these powerful techniques.

Prerequisite modules: MTH1001; MTH1002.

Analysis is the theory that underpins all continuous mathematics. The objective of this module is to provide you with a logically based introduction to real and complex analysis. The primary objective is to define all the basic concepts clearly and to develop them sufficiently to provide proofs of useful theorems. This enables you to see the reason for studying analysis, and develops the subject to a stage where you can use it in a wide range of applications.

On successful completion of this module, **you should be able to**:

**Module Specific Skills and Knowledge:**

1 state and prove key theorems in analysis using a rigorous approach;

2 develop proofs related to basic topological concepts like connectedness;

3 compare and contrast the theory of analytic functions over the real and complex numbers;

4 compute contour integrals and to apply this to real analysis.

**Discipline Specific Skills and Knowledge:**

5 apply fundamental mathematical concepts, manipulations and results in analysis;

6 formulate rigorous arguments as part of your mathematical development.

**Personal and Key Transferable/ Employment Skills and Knowledge:**

7 think analytically and use logical argument and deduction;

8 communicate your ideas effectively in writing and verbally;

9 manage your time and resources effectively.

Review of epsilon-N sequence limits; montone convergence; Bolzano-Weierstrass; Cauchy sequences and completeness.

Epsilon-delta function limits; continuity; differentiability (all in R then R^n).

Function classes: C^k, C^infinity etc; Lipschitz continuity.

Complex analysis; Cauchy-Riemann equations; contrast to real analytic functions.

Inverse and implicit function theorems.

Formal theory of Riemann integration; integrability of monotonic functions and continuous functions; problems interchanging limits in general; pointer towards Lebesgue integration and monotone/dominated convergence theorems.

Contour integrals; poles and singularities (isolated, removable, essential); residues; Cauchy's Theorem; Cauchy integral formulae; Taylor series and Laurent series;

Maximum modulus principle, Liouville's theorem, fundamental theorem of algebra, meromorphic functions, residue theorem;

Rouche's theorem, principle of the argument;

Applications to definite integrals, summation of series and the location of zeros.

Scheduled Learning & Teaching Activities | 76.00 | Guided Independent Study | 224.00 | Placement / Study Abroad |
---|

Category | Hours of study time | Description |

Scheduled learning and teaching activities | 66 | Lectures including example classes |

Scheduled learning and teaching activities | 10 | Tutorials |

Guided independent study | 224 | Lecture and assessment preparation; wider reading |

Form of Assessment | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
---|---|---|---|

Exercise sheets | 10 x 10 hours | All | Discussion at tutorials; tutor feedback on submitted answers. |

Coursework | 0 | Written Exams | 100 | Practical Exams |
---|

Form of Assessment | % of Credit | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
---|---|---|---|---|

Written exam – closed book (Jan) | 30 | 2 hours | All | Via SRS |

Written exam – closed book (May) | 70 | 2 hours | All | Via SRS |

Original Form of Assessment | Form of Re-assessment | ILOs Re-assessed | Time Scale for Re-reassessment |
---|---|---|---|

All above | Written exam (100%) | All | August Ref/Def period |

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

information that you are expected to consult. Further guidance will be provided by the Module Convener

**Basic reading:**

**ELE: http://vle.exeter.ac.uk**

**Web based and Electronic Resources:**

William F. Trench, Introduction to Real Analysis, freely downloadable here: https://digitalcommons.trinity.edu/mono/7/

**Other Resources:**

Reading list for this module:

Type | Author | Title | Edition | Publisher | Year | ISBN | Search |
---|---|---|---|---|---|---|---|

Set | DuChateau P.C. | Advanced Calculus | Harper Collins | 1992 | 000-0-064-67139-9 | [Library] | |

Set | McGregor C., Nimmo J. & Stothers W. | Fundamentals of University Mathematics | 2nd | Horwood, Chichester | 2000 | 000-1-898-56310-1 | [Library] |

Set | Gaughan E. | Introduction to Analysis | 5th | Thompson | 1998 | 000-0-534-35177-8 | [Library] |

Set | Burn R.P. | Numbers and Functions: Steps to Analysis | Electronic | Cambridge University Press | 2005 | 000-0-521-41086-X | [Library] |

Set | Bryant V. | Yet another Introduction to Analysis | Cambridge University Press | 1990 | 978-0521388351 | [Library] | |

Set | Howie, John M | Complex Analysis | Springer | 2003 | 000-1-852-33733-8 | [Library] | |

Set | Priestley H.A. | Introduction to Complex Analysis | Oxford University Press | 2003 | 000-0-198-53428-0 | [Library] | |

Set | Spiegel M.R. | Schaum's outline of theory and problems of complex variables : with an introduction to conformal mapping and its appreciation. | McGraw-Hill | 1981 | 000-0-070-84382-1 | [Library] | |

Set | Stewart I. & Tall D. | Complex Analysis (the hitchhiker's guide to the plane) | Cambridge University Press | 1983 | 000-0-521-28763-4 | [Library] | |

Set | Krantz, Steven G. | Real analysis and foundations | 4th | CRC Press, Boca Raton, FL | 2017 | [Library] | |

Set | Abbott, Stephen | Understanding analysis | 2nd | Springer, New York, | 2015 | [Library] |

CREDIT VALUE | 30 | ECTS VALUE | 15 |
---|---|---|---|

PRE-REQUISITE MODULES | MTH1001, MTH1002 |
---|---|

CO-REQUISITE MODULES |

NQF LEVEL (FHEQ) | 5 | AVAILABLE AS DISTANCE LEARNING | No |
---|---|---|---|

ORIGIN DATE | Wednesday 11 January 2017 | LAST REVISION DATE | Thursday 28 February 2019 |

KEY WORDS SEARCH | Analysis; supremum; infimum; series; functions; limits; continuity; derivatives; integration.; residue; contour integral. |
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