Integral Transforms & Complex Methods - MAT00081H

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Module summary

This module further develops Complex Analytic methods, and provides techniques that can be used to evaluate nontrivial integrals of functions with branch cuts. It also introduces and develops asymptotic methods which give useful estimates of the growth of functions and can also be used to give accurate estimates of various integrals. These techniques are used in various areas of pure and applied mathematics. Some “special functions” such as the Gamma and Beta functions are studied in detail using the methods of the module. Two important integral transforms, due to Fourier and Laplace, are introduced and studied - both have many applications in modern mathematics and its applications.

Related modules

Additional information

For elective students/MSc: a good grounding in contour integration and residue calculus is necessary.

Metric Spaces (from 2024/25 onwards)

Module will run

Occurrence	Teaching period
A	Semester 1 2023-24

Module aims

This module further develops Complex Analytic methods, and provides techniques that can be used to evaluate nontrivial integrals of functions with branch cuts. It also introduces and develops asymptotic methods which give useful estimates of the growth of functions and can also be used to give accurate estimates of various integrals. These techniques are used in various areas of pure and applied mathematics. Some “special functions” such as the Gamma and Beta functions are studied in detail using the methods of the module. Two important integral transforms, due to Fourier and Laplace, are introduced and studied - both have many applications in modern mathematics and its applications.

Module learning outcomes

By the end of the module, students should be able to:

1. Apply tools and techniques of complex analysis in a variety of problems, including evaluation of contour integrals and solving differential equations.

2. Demonstrate and employ various properties of the Gamma and Beta functions.

3. Compute Fourier and Laplace transforms and inverse transforms.

4. Apply Fourier and Laplace methods to solve concrete problems.

5. Find asymptotic expansions for a variety of functions.

Module content

Complex analysis and contour integration are developed further, to include multivalued functions and the evaluation of improper integrals of real functions. The important Gamma and Beta functions are introduced and studied. Fourier and Laplace integral transforms are introduced and several of their main properties are derived; they are then applied to problems including ordinary and partial differential equations. Asymptotic expansions are developed along with techniques for approximately evaluating integrals.

Indicative assessment

Task	% of module mark
Closed/in-person Exam (Centrally scheduled)	100

Special assessment rules

None

Indicative reassessment

Task	% of module mark
Closed/in-person Exam (Centrally scheduled)	100

Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.

Indicative reading

• H A Priestley, Introduction to Complex Analysis (S 7.55 PRI)

• Ian Stewart and David Tall, Complex Analysis: The Hitchhiker's Guide to the Plane (S 7.55 STE)

• M.J. Ablowitz and A.S. Fokas, Complex variables: Introduction and Applications (Cambridge University press)

• G F Simmons, Differential Equations, with Applications and Historical Notes, Tata MacGraw-Hill (paperback) (S7.38 SIM)

• E T Whittaker and G N Watson, A Course of Modern Analysis, (Cambridge University Press)

• A Pinkus and S Zafrany, Fourier Series and Integral Transforms (S 7.39 PIN).