JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD

Pre-requisites

Mathematics courses of first year of study.

Course Objectives

To learn

• Expressing periodic function by Fourier series and a non-periodic function by Fourier transforms
• Various numerical methods to find roots of polynomial and transcendental equations.
• Concept of finite differences and to estimate the value for the given data using interpolation.
• Evaluation of integrals using numerical techniques
• Solving ordinary differential equations of first order using numerical techniques.
• Differentiation and integration of complex valued functions.
• Evaluation of integrals using Cauchy’s integral formula and Cauchy’s residue theorem.
• Expansion of complex functions using Taylor’s and Laurent’s series. Course outcomes: After learning the contents of this paper the student must be able to
• Express any periodic function in terms of sine and cosine
• Find the root of a given polynomial and transcendental equations.
• Estimate the value for the given data using interpolation
• Find the numerical solutions for a given first order ODE’s
• Analyze the complex function with reference to their analyticity, integration using Cauchy’s integral and residue theorems
• Taylor’s and Laurent’s series expansions in complex function

UNIT - I: Fourier Series & Fourier Transforms

Fourier series - Dirichlet’s Conditions - Half-range Fourier series - Fourier Transforms: Fourier Sine and cosine transforms - Inverse Fourier transforms.

UNIT - II: Numerical Methods-I

Solution of polynomial and transcendental equations: Bisection method, Iteration Method, Newton- Raphson method and Regula-Falsi method. Jacobi and Gauss-Seidal iteration methods for solving linear systems of equations. Finite differences: forward differences, backward differences, central differences, symbolic relations and separation of symbols, Interpolation using Newton’s forward and backward difference formulae. Central difference interpolation: Gauss’s forward and backward formulae, Lagrange’s method of interpolation.

UNIT - III: Numerical Methods-II

rd th Numerical integration: Trapezoidal rule and Simpson’s 1/3 and 3/8 rules. Ordinary differential equations: Taylor’s series, Picard’s method, Euler and modified Euler’s methods, Runge-Kutta method of fourth order for first order ODE

UNIT - IV: Complex Differentiation

Limit, Continuity and Differentiation of Complex functions. Cauchy-Riemann equations (without proof), Milne- Thomson methods, analytic functions, harmonic functions, finding harmonic conjugate, elementary analytic functions (exponential, trigonometric, logarithm) and their properties. (All theorems without Proofs), Conformal mappings, Mobius transformations.

UNIT - V: Complex Integration

Line integrals, Cauchy’s theorem, Cauchy’s Integral formula, zeros of analytic functions, singularities, Taylor’s series, Laurent’s series, Residues, Cauchy Residue theorem.

and their properties. (All theorems without Proofs)

TEXT BOOKS:

• 1. B.S. Grewal, Higher Engineering Mathematics, Khanna Publishers, 36th Edition, 2010.
• 2. S.S. Sastry, Introductory methods of numerical analysis, PHI, 4th Edition, 2005.

REFERENCE BOOKS:

• 1. M. K. Jain, S.R.K. Iyengar, R.K. Jain, Numerical methods for Scientific and Engineering Computations, New Age International publishers.
• 2. Erwin Kreyszig, Advanced Engineering Mathematics, 9th Edition, John Wiley & Sons, 2006.
• 3. J. W. Brown and R. V. Churchill, Complex Variables and Applications, 7th Edition, Mc-Graw Hill, 2004.