Pre-requisite modules
Co-requisite modules
- None
Prohibited combinations
- None
Occurrence | Teaching cycle |
---|---|
A | Spring Term 2017-18 |
The Core modules in the second year of mathematics programmes cover material which is essential for accessing a wide range of topics later on. Such material underpins the development of mathematics across all three of the major streams we offer through our single-subject degrees, and it is also very important for many combined programmes. As well as covering fundamental material, these modules address applications and techniques which all students will subsequently be able to draw on in various contexts.
As part of the broad aims outlined above, this module considers two important developments in calculus and analysis. First, differentiable functions of a complex variable have very many properties beyond those of the differentiable functions encountered in Real Analysis. This has applications to the expansion of functions in power series and also to the important techniques of contour integration. Second, a powerful tool both in applied mathematics and analysis is the Fourier transform, which has applications in many areas.
This module will extend the ideas developed in Real Analysis from real to complex numbers, develop the theory of holomorphic functions, and apply this theory to understand problems arising in real analysis or calculus and describe the idea and basic properties of the Fourier transform, indicating their applications in fields such as digital signal processing and differential equations.
Subject content
Review of complex numbers.
Limits and continuity in C. Algebra of limits. Convergence of absolutely convergent series.
Open sets, complex differentiability, linearity, product rule, quotient rule, chain rule. Holomorphic functions, Cauchy-Riemann equations.
Integration of complex-valued functions on a real interval: linearity, triangle inequality, Fundamental Theorem.
Paths and path integrals. Estimation Lemma. Fundamental Theorem.
Cauchy’s Theorem and Morera’s Theorem from Green’s Theorem.
The winding number defined as an integral.
Extension of Cauchy’s Theorem to allow finitely many points of non-differentiability inside the curve, provided the integrand is bounded.
Cauchy Integral Formulae for the function and its derivatives; the derivative of a holomorphic function is holomorphic.
Taylor’s Theorem from the integral formula, with Cauchy’s formula for the coefficients. Liouville’s Theorem. Fundamental Theorem of Algebra.
Taylor’s formula for the Taylor coefficients.
Term-by-term differentiation of power series.
Isolated singularities, removable singularities and poles. Laurent series about a pole.
The Residue Theorem.
Example of residue calculations, to include integrals of (cos(kx))/(1+x^2) and sin(kx)/x
Fourier Transforms. Definition and examples of the Fourier transform, including examples calculated by residue methods. Statement of the inversion theorem.
Properties of the Fourier transform: linearity, translation, dilation, derivatives, convolutions, etc. Square-integrable functions and Plancherel’s Identity.
Applications of Fourier transforms to (ordinary and/or partial) differential equations.
Laplace Transform. Developed only on example sheets in parallel to the Fourier transform.
Academic and graduate skills
Academic Skills: our second year Core modules continue to develop themes which start in the first year, such as the application of rigorous mathematical techniques and ideas to the development of mathematics; the power of abstraction as a way of solving many similar problems at the same time; the development and consolidation of essential skills which a mathematician needs in their toolkit and needs to be able to use without pausing for thought.
Graduate skills: The techniques of Fourier analysis developed in this course are widely used in real-world applications (some of which the students will encounter during the module). These techniques are used by many graduates in their day-to-day work.
Task | Length | % of module mark |
---|---|---|
University - closed examination Functions of a Complex Variable |
1.5 hours | 100 |
None
Task | Length | % of module mark |
---|---|---|
University - closed examination Functions of a Complex Variable |
1.5 hours | 100 |
Current Department policy on feedback is available in the undergraduate student handbook. Coursework and examinations will be marked and returned in accordance with this policy.
H A Priestley "Introduction to Complex Analysis" (2 ed) OUP 2003, JBM S7
A Pinkus and S Zafrany, Fourier Series and Integral Transforms (S 7.39 PIN)