1.  Introduction:  This second semester unit follows on from Advanced Calculus I which is a prerequisite. It is in the mainstream of methods teaching for degree programmes in mathematics. The unit divides into two parts namely Complex Analysis, and Ordinary Differential Equations and Special Functions.

2.  Timetable Hours, Credits, Assessments:  This unit is of 20 UCU and is taught in the Spring Semester by means of 20 lectures (Complex Analysis) and 20 lectures (Ordinary Differential Equations and Special Functions). Each of the two parts of the unit is supported by 3 seminars. Assessment is by set regular coursework (20%) and a three-hour examination (80%).

3.  Overview:  Complex Analysis and Differential Equations provide mathematical tools with which the edifice of applied mathematics is constructed.

This unit applies the complex variable theory established in module 2 of Advanced Calculus I. The Complex Analysis module of this unit is centred on Cauchy's residue theorem and its application to the evaluation of contour integrals. Laplace transforms are introduced and the inversion integral quoted to provide further examples of contour integration.

The unit in parallel resumes the development of differential equations started in module 3 of Advanced Calculus I. Some basic theory of linear ordinary differential equations on the real line precedes methods which use complex series. These methods for solution of linear ordinary differential equations in the complex plane are applied to the solution of Legendre's and Bessel's equations and determine associated functions. These are used to complete the solution of Laplace's equation by separation of variables.

4.  Recommended Reading:  The notes taken in lectures are intended to be complete and self-contained in themselves. A particularly useful book that provides coverage of all aspects of the unit is "Advanced Engineering Mathematics" by E Kreyszig (Wiley). Another useful book is "Introduction to Complex Analysis" by H A Priestley (Oxford) which covers all the complex analysis in the unit.

Logarithm and its principal value, argument. Riemann surface examples log, z, z. Integrals involving branch cuts. Cauchy's Theorem for a contour and its proof. The deformation theorem. Cauchy's Integral formula. Taylor's theorem. Cauchy's formula for derivatives. The Identity theorem. The Maximum-Modulus theorem. (10 Lectures)

Contour integral in C . Integration of zm , for integer m , round simple closed curve. Residues and Cauchy's residue theorem. Techniques for finding residues. Summation of series. Contour integrals with poles off and on contour. Integrals along real line, and technique of closing contour at infinity. Integrals of trigonometric functions. Number of zeros, minus number of poles, formula. (10 Lectures)

5B.  Differential Equations, Special Functions and Laplace Transforms:

Laplace transforms. Definition, elementary functions. The Gamma function. Derivatives, integrals and the convolution theorem. Application to differential equations, systems and integral equations. Differentiation and integration of transforms. Particular inversion methods. (5 Lectures)

Linear ordinary differential equations on the real line. Existence and uniqueness for the nth-order equation. Linear dependence; the Wronskian, Liouville/Abel formula. The complete solution, method of variation of parameters. Second-order equations. (4 Lectures)

Second-order linear equations on the real line. Series solutions. Singular points and the method of Frobenius. The indicial equation; equal roots and roots differing by an integer. (5 Lectures)

Legendre polynomials and Bessel functions. Motivation from separable solutions of Laplace's equation. Application to the solution of the heat conduction equation. (6 Lectures)