UNIT - I Laplace Transforms - Laplace Transforms; Laplace Transform of standard functions; first shifting theorem; Laplace transforms of functions when they are multiplied and divided by ‘t’. Laplace transforms of derivatives and integrals of function; Evaluation of integrals by Laplace transforms; Laplace transforms of Special functions; Laplace transform of periodic functions. Inverse Laplace transform by different methods, convolution theorem (without Proof), solving ODEs by Laplace Transform method. (Chapter - 1) UNIT - II Numerical Methods - I - Solution of polynomial and transcendental equations – Bisection method, Iteration Method, Newton- Raphson method and Regula-Falsi method. 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 (Chapter - 2) UNIT - III Numerical Methods - II - Numerical integration : Trapezoidal rule and Simpson’s 1/3rd 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. (Chapter - 3) UNIT - IV Complex Variables (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. (Chapter - 4) UNIT - V Complex Variables (Integration) - Line integrals, Cauchy’s theorem, Cauchy’s Integral formula, Liouville’s theorem, Maximum-Modulus theorem (All theorems without proof); zeros of analytic functions, singularities, Taylor’s series, Laurent’s series; Residues, Cauchy Residue theorem (without proof) (Chapter - 5)
