Complex Analysis&Applications

MATH34011 Complex Analysis&Applications 20 Semester 1 Undergraduate Level 3 James Montaldi Unit coordinator Mike Simon Unit coordinator Department of Mathematics 1 FHEQ level (Framework for Higher Education Qualifications) ' Last part of a Bachelors ' European Credit Transfer & Accumulation System Rating : 10.0 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. 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. 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. <ol><li>Demonstrate properties of holomorphic (regular) functions, such as the Cauchy-Riemann equations, and apply the Cauchy-Riemann Theorem to elementary functions </li><li>Use elementary regular functions including those defined by power series.</li><li>Use the Cauchy-Taylor Theorem and Laurent's Theorem to expand a holomorphic function as a power series on a disc or on an annulus</li><li>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, and justify properties of the residue</li><li>Define and calculate complex integrals using a variety of methods, such as the Fundamental Theorem of Contour Integration and the Cauchy Residue Theorem, and apply to the evaluation of some real integrals. </li><li>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</li><li>Evaluate the properties of functions such as ln(z) or za (where a is not an integer) involving Branch Cut(s).</li><li>Use a variety of methods (including the Fundamental Theorem and Cauchy’s Residue Theorem) to calculate the complex integral of a given function.</li><li>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.</li><li>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</li><li>Apply the process of Analytic Continuation to certain functions.</li><li>State and use the properties of the Gamma Function.</li><li>Use the properties of Fourier and Laplace Transforms to solve certain PDEs</li></ol> A. Series. Complex series, power series and the radius of convergence.B. Continuity. Continuity of complex functionsC. 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. Videos/podcasts; tutorials and review classes. 1 Written exam 100% Coursework feedback delivered via Blackboard End of Semester examination - generic feedback to cohort following marking of the exam MATH29141 2P1: Complex Analysis Anti-requisite Compulsory MATH11121 Mathematical Foundations & Analysis Pre-Requisite Compulsory MATH24420 Partial Differential Equations & Vector Calculus Pre-Requisite Compulsory PHYS20171 Mathematics of Waves and Fields Pre-Requisite Compulsory MATH34011 Requisites: (A) Pre-Requisites: MATH11121 and MATH24420/PHYS20171; (B) Anti-Requisites - Students who took MATH29141 are not permitted to take MATH34011PHYS20171 is an acceptable alternative for those Maths-Physics students who took that unit instead of MATH24420. N Stewart, Ian. (1983). Complex analysis: the hitchhiker's guide to the plane. Cambridge University Press. ISBN: 9781139171632Lang, Serge. (1998). Complex analysis. Springer. ISBN: 0387985921Copson, E. T. (1935). An introduction to the theory of functions of a complex variable. Clarendon. ISBN: 0198531451Ahlfors, Lars V. (1979). Complex analysis: an introduction to the theory of analytic functions of one complex variable. McGraw-Hill. ISBN: 0070006571  Lectures 44 Tutorials 11 0 167