Advanced Mathmatics (English)

Unit 1. Numerical Methods: Finite differences, Relation between operators, Interpolation using Newton’s forward and backward difference formulae. Gauss’s forward and backward interpolation formulae. Stirling’s Formulae. Interpolation with unequal intervals: Newton’s divided difference and Lagrange’sformulae. Numerical Differentiation, Numerical integration: Trapezoidal rule and Simpson’s 1/3rd and 3/8 rules. Solution of polynomial and transcendental equations-Bisection method, Newton-Raphson method and Regula-Falsi method.

Unit 2. Transform Calculus: Laplace Transform: Definition and existence of Laplace transform, Properties of Laplace Transform and formulae, Unit Step function, Dirac Delta function, Heaviside function, Laplace transform of periodic functions. Finding inverse Laplace transform by different methods, convolution theorem. Fourier Transform: Fourier Complex, Sine and Cosine transform, properties and formulae, inverse Fourier transforms, Convolution theorem. ZTransform: Definition, properties and formulae, Convolution theorem, inverse Z-transform, application of Z-transform to difference equation.

Unit 3. Complex Variable: Differentiation, Cauchy-Riemann equations, analytic functions, harmonic functions, finding harmonic conjugate; elementary analytic functions (exponential, trigonometric, logarithm) and their properties; Conformal mappings, Mobius transformations and their properties