Multivariable Calculus and Transforms: Course Outline (BS-321)

The complex number and complex variables.

Complex differentiation and contents: integration. Laplace Transformation and its applications. Series solution of the DEs.

Complex number systems and Complex Variable theory: Introduction to complex number systems. Argands diagram, modulus and argument of a complex number, polar form of a complex number, De Moivres theorem and its applications. Complex functions, analytical function, harmonic and conjugate harmonic functions, Cauchy Remann equations (in Cartesian and polar coordinates). Line integral, Greens theorem, Cauchys theorem, Cauchys integral formula, singularities, poles, residue and contour integration and application.

Laplace Transforms: Definition, Laplace transforms of elementary functions. Properties of Laplace transform, Laplace transform of derivatives, integrals. Multiplication by t and division by t properties. Periodic functions and their Laplace transforms. Inverse Laplace transforms and their properties. Convolution theorem. Inverse Laplace transforms by integral and partial fraction methods. Heavisides expansion formula. Solution of ordinary differential equations by Laplace transform. Applications of Laplace transformation on various fields of engineering.

Series Solution of Differential Equations: Introduction. The solution of po(x)y + p1
(x) y + p2 (x) y=0, when p(0)=0. Validity of series solution. Ordinary point, singular point. Forbenius method, indicial equation. Bessels differential equation, its solution of first kind and its recurrence formulae. Logendre differential equation and its solution. Rodriguez formula.

Fourier Transform: Definition, Fourier transform of simple functions, magnitude and phase spectra, Fourier transform theorems, Inverse Fourier transform, Solution of differential equation using Fourier transform.