**
School of Mathematics
School of Mathematics**

**Course 3E1 ** 2002-03 (JS Engineering, option JS MSISS
)

**Lecturer:** Dr James Drummond and Dr. Richard M. Timoney

**Requirements/prerequisites:** 2E1 and 2E2
(Calculus and elementary ODE. Laplace transforms.
Theory of series.)

**Duration:** 22 weeks

**Number of lectures per week:** 2 lectures plus 1 tutorial

**Assessment:** Weekly tutorial problems.

**End-of-year Examination:** One 3-hour examination

**Description: **
This course follows on directly from 2E1/2E2 and develops the mathematics
of engineering and physics. It covers Fourier
series, Fourier transforms, partial differential
equations, linear programming and optimisation, complex analysis.

#### Fourier Analysis and Partial Differential Equations

This section is based on Kreysig chapters 10-11.

Fouriers Theorem. Even and Odd Functions. Half-Range Fourier
Series.

Derivation of Fourier Transform from complex Fourier Series.
Linearity of the
Fourier transform and the Fourier tranform of a derivative.
Application to partial differential equations.

Partial differential equations. Wave equation with d'Alemberts
solution.
Method of
separation of variables applied to solutions of the diffusion (heat)
equation,
Laplace's equation and the wave equation subject to appropriate
initial and boundary conditions.
Fourier series applied to matching initial conditions.
Natural modes and nodal lines.
Classifications of partial differential equations in two variables.

#### Linear Programming and Optimisation

This section is based on Kreysig Chapter 20 and part of Chapter 21.

It will introduce some basic aspects of linear programming and graph
theory.

#### Complex Analysis

(Kreyszig chapters 12-15)
Complex function and mappings. Complex Differentiation. Analytic
functions. Cauchy-Riemann equations theorem and its proof.
Harmonic functions.
Power series and radius of convergence (without proofs).
Exponential, trigonometric and hyperbolic functions (for complex
arguments).
Logarithm and complex power functions. Mapping properties of some
examples.

Complex Integration. Cauchy's integral theorem and its proof.
Cauchy's integral formula. Independence of path consequence of
Cauchy's theorem and use of the theory to evaluate complex integrals
in simple cases (residue theorem not covered). Power series
representations in discs.

See `http://www.maths.tcd.ie/~richardt/3E1/` for further
information.

#### Textbook:

Erwin Kreyszig, Advanced Engineering Mathematics, (8th
edition) Wiley, 1999.

May 16, 2003

File translated from
T_{E}X
by
T_{T}H,
version 2.70.

On 16 May 2003, 17:43.