Year of entry: 2022
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Course unit details:
|Unit level||Level 3|
|Teaching period(s)||Semester 1|
|Offered by||Department of Mathematics|
|Available as a free choice unit?||No|
This unit introduces the student to properties of regular (or analytic) functions of a complex variable. It proceeds with a study of singularities of such functions, introducing methods of contour integration and its applications to evaluating some new real integrals (problems unsolvable by other means).
With this material mastered, the course unit proceeds to describe analytic continuation of real analytic functions, and applications to solving various types of differential equation using Fourier and Laplace transforms.
|Unit title||Unit code||Requirement type||Description|
|Real Analysis A||MATH20101||Pre-Requisite||Compulsory|
|Real Analysis B||MATH20111||Pre-Requisite||Compulsory|
|Partial Differential Equations and Vector Calculus A||MATH20401||Pre-Requisite||Compulsory|
|Partial Differential Equations and Vector Calculus B||MATH20411||Pre-Requisite||Compulsory|
The unit aims to expose students to the fascinating and very rich theory of analytic functions of a complex variable. This is a central pillar of pure mathematics, as well as being a domain with many applications, both within mathematics and without. The theory is developed sufficiently far to be able to perform contour integration, and to develop the Fourier and Laplace transforms and illustrate their uses for solving differential equations.
- Use the Cauchy-Riemann Theorem and its converse to decide whether a given function is regular
- Use elementary regular functions including those defined by power series.
- Use Taylor's Theorem and Laurent's Theorem to expand a regular function as a power series on a disc or on an annulus.
- Identify the location and nature of a singularity of a complex function and, in the case of a pole, to calculate the order and the residue
- Evaluate the properties of functions such as ln(z) or za (where a is not an integer) involving Branch Cut(s).
- Use a variety of methods (including the Fundamental Theorem and Cauchy’s Residue Theorem) to calculate the complex integral of a given function.
- Perform contour integration of complex functions around suitable closed contours (including circular and rectangular contours, D-contours, keyhole contours and dumb-bell contours) in order to evaluate certain real, definite integrals.
- Apply techniques from complex analysis to deduce results in other areas of mathematics, such as proving the Fundamental Theorem of Algebra and performing the summation of series
- Apply the process of Analytic Continuation to certain functions.
- State and use the properties of the Gamma Function.
- Use the properties of Fourier and Laplace Transforms to solve certain PDEs
A. Series. Complex series, power series and the radius of convergence.
B. Continuity. Continuity of complex functions
C. The complex plane. The topology of the complex plane, open sets, paths and continuous functions.
D. Differentiation. Differentiable complex functions and the Cauchy-Riemann equations.
E. Integration. Integration along paths, the Fundamental Theorem of Calculus, the Estimation Lemma, statement of Cauchy's Theorem.
F. Taylor and Laurent Series. Cauchy's Integral Formula and Taylor Series, Zeros and Poles, Laurent Series.
G. Residues. Cauchy's Residue Theorem, the evaluation of definite integrals and summation of series. Cauchy’s Integral Formula and Liouville's Theorem. Jordan’s Lemma.
H. ‘Multivalued Functions’: The functions ln(z) or za (where a is not an integer). Branch cuts and branch points. Functions with finite branch cuts.
I. Real Definite Integrals: More evaluation of real definite integrals by contour methods, now involving multivalued functions of z and keyhole and dumbbell contours.
J. Analytic Continuation: Examples of regular functions defined by series or integrals and their analytic continuations. Uniqueness of analytic continuations and applications. Contact continuation theorem and Schwarz's principle.
K. The Gamma Function: Definition of as an integral. The functional relation. Analytic continuation of , its poles and residues. The Reflection Formula. The Digamma function.
L. Fourier and Laplace Transforms: Integral transforms in general. Fourier's integral theorem. The complex Fourier Transform and its inverse. The Fourier Cosine and Sine Transforms and their inverses. Extension to considering the inverse transform as a contour integral. Half-range FTs and their analyticity in the complex plane. The Laplace transform and its relationship to the complex Fourier transform. The Bromwich integral inversion formula. Transforms of derivatives and derivatives of transforms and examples.
M. Applications of Integral Transforms to Partial Differential Equations: A simple linear ODE solved by Laplace transform. Initial value problem for the one-dimensional heat equation for the infinite bar. Same problem for the semi-infinite bar with appropriate end conditions.
Teaching and learning methods
Videos/podcasts; tutorials and review classes.
Coursework (mid-term) - worth 20%
End of Semester examination - worth 80% - 3 hours Duration
Coursework feedback delivered via Blackboard
End of Semester examination - generic feedback to cohort following marking of the exam
E.T. Copson, Functions of a Complex Variable, 1995
|Scheduled activity hours|
|Independent study hours|
|Mike Simon||Unit coordinator|
|Anastasia Kisil||Unit coordinator|