2P1: Complex Analysis

MATH29141 2P1: Complex Analysis 10 Semester 1 Undergraduate Level 2 James Montaldi Unit coordinator Michael Coleman Unit coordinator Department of Mathematics 1 FHEQ level (Framework for Higher Education Qualifications) ' Middle part of Bachelors ' European Credit Transfer & Accumulation System Rating : 5.0 <div>This course introduces the analysis of complex functions of a complex variable.  Complex differentiability is a very strong condition, and differentiable (or holomorphic or regular) functions have many strong properties.  Integration is along paths in the complex plane. The central result of this spectacularly beautiful part of mathematics is Cauchy’s Theorem guaranteeing that certain integrals along closed paths are zero.  After studying properties of isolated singularities of functions, Cauchy’s theorem leads to useful techniques for evaluating real integrals based on the ‘calculus of residues’ (problems unsolvable by other means).  Syllabus:A.    Complex functions: Domains and paths in the complex plane. Differentiation and the Cauchy-Riemann equations; holomorphic functions. Path integrals and the Fundamental Theorem of contour integration.  B.    Power series: Power series and their radius of convergence. Derivatives and integrals of power series. Elementary functions such as exp, sin, cos, sinh, cosh and log. C.    Cauchy’s Theorem and Formula: Winding numbers, Cauchy’s theorem, Cauchy’s integral formula and the estimation lemma.  D.    Taylor and Laurent series: The Cauchy-Taylor theorem, Liouville’s theorem, Laurent’s theorem and calculation of Laurent series. E.    Residues: Isolated singularities, poles and their residues; Cauchy’s residue theorem. Applications of the residue theorem to the evaluation of trigonometric integrals, integrals over the real line, and summation of series.</div>  <div>This course introduces the analysis of complex functions of a complex variable.  Complex differentiability is a very strong condition, and differentiable (or holomorphic or regular) functions have many strong properties.  Integration is along paths in the complex plane. The central result of this spectacularly beautiful part of mathematics is Cauchy’s Theorem guaranteeing that certain integrals along closed paths are zero.  After studying properties of isolated singularities of functions, Cauchy’s theorem leads to useful techniques for evaluating real integrals based on the ‘calculus of residues’ (problems unsolvable by other means).  Syllabus:A.    Complex functions: Domains and paths in the complex plane. Differentiation and the Cauchy-Riemann equations; holomorphic functions. Path integrals and the Fundamental Theorem of contour integration.  B.    Power series: Power series and their radius of convergence. Derivatives and integrals of power series. Elementary functions such as exp, sin, cos, sinh, cosh and log. C.    Cauchy’s Theorem and Formula: Winding numbers, Cauchy’s theorem, Cauchy’s integral formula and the estimation lemma.  D.    Taylor and Laurent series: The Cauchy-Taylor theorem, Liouville’s theorem, Laurent’s theorem and calculation of Laurent series. E.    Residues: Isolated singularities, poles and their residues; Cauchy’s residue theorem. Applications of the residue theorem to the evaluation of trigonometric integrals, integrals over the real line, and summation of series.</div>  <div>The unit aims to introduce the basic ideas of complex analysis, with particular emphasis on contour integration, Cauchy’s Theorem and the calculus of residues. The course is only available to students on the BSc/MMathPhys Mathematics & Physics programmes and the BSc Computer Science and Mathematics programme. </div>  On successful completion of the course, students will be able to:<ul><li>Demonstrate properties of holomorphic (regular) functions, such as the Cauchy-Riemann equations, and apply the Cauchy-Riemann Theorem to elementary functions Define, analyse and make use of elementary 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 Countour Integration and the Cauchy Residue Theorem, and apply to the evaluation of some real integrals. </li></ul> <div>This course will be delivered together with the first half of MATH34011 (which is 20 credits). It will involve 3 hours of lectures plus an examples class and a tutorial each week for 6 weeks. The course is only available to students on the BSc/MMathPhys Mathematics & Physics programmes and the BSc Computer Science and Mathematics programme.  </div>  0 Other 20% 1 Written exam 80% Online coursework test (using stack) Test mid-way through semester; Stack feedback available after test is closed 20% WeightingWritten exam General feedback provided after exam is marked 80% Weighting Online coursework test (using stack) Test mid-way through semester; Stack feedback available after test is closed 20% WeightingWritten exam General feedback provided after exam is marked 80% Weighting MATH11121 Mathematical Foundations & Analysis Pre-Requisite Compulsory N Ian Stewart and David Tall, Complex Analysis, Cambridge University Press, 1983. Assessment written exam 6 Lectures 22 Tutorials 2 0 70