**Duration:** 22 weeks

**Number of lectures per week:** 2.5

**Assessment:**

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

**Description: **

This course follows on directly from 2E1 and develops the mathematics of engineering and physics. It covers complex analysis, Fourier series, Fourier transforms, Laplace transforms, partial differential equations.

**Complex Analysis (Kreyszig cpt. 12-15)**

Complex function and mappings. Complex Differentiation. Analytic
functions. Cauchy-Riemann equations theorem and its proof.
Exponential, Trigonometric and Hyperbolic complex functions.
Logarithmic and Complex Power functions. Harmonic functions.
Conformal mapping applied to two dimensional Laplace equation.

Complex Integration. Cauchy's integral theorem and its proof.
Cauchy's derivative formula. Laurent's theorem and its application to
representation of functions, classification of singularities and
calculation of residues. Proof of important formulae for calculation
of residue at a pole of order m. Residue theorem and its proof.
Application of residue theorem to calculation of complex integrals
such as Inverse Laplace Transform and to real
integrals.

**Fourier Analysis and Partial Differential Equations (Kreyszig cpt. 10-11)**

Fouriers Theorem. Even and Odd Functions. Half-Range Fourier Series. Least Squares Approximation
property of Fourier Series.

Derivation of Fourier Transform from complex Fourier Series.
Evaluation of some important Fourier transform pairs by contour
integration. Relationship between Fourier and Laplace
transforms. Application to partial differential equations.

Classifications of Partial Differential Equations. Method of
Separation of Variables applied to Solution of the Diffusion Equation,
Poisson's Equation and the Wave equation subject to appropriate
initial and boundary conditions d'Alemberts Solution. Bending vibration of bars and
different boundary conditions. Natural Modes. Application to Heat
Transfer Problems.

**Textbooks:**

*Advanced Engineering Mathematics* (8th edition) - Erwin
Kreyszig - John Wiley.

Oct 8, 1999