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# Complex Variable and Transforms (Engineering BS 212): Course Outline

The objective of the course is to teach them basic manipulations on complex numbers also enable them to identify the singular points of an analytic function with the construction of series and explaining usefulness of Laurent series in the computation of

## Course Outline (BS-212)

Complex Numbers and Functions;

Complex Numbers and Their Geometric Representation, Polar Form of Complex Numbers, Powers and Roots, Exponential Function, Trigonometric and Hyperbolic Functions. Euler’s Formula, Analytic Function, Cauchy–Riemann Equations, Harmonic Function, Construction of Analytic Function (Milne’s Method),

Complex Integration;

Cauchy’s Integral Theorem, Cauchy’s Integral Formula, Derivatives of Analytic Functions, Power Series, Taylor Series, Laurent Series, Power Series and its Radius of Convergence, Maclaurin Series Expansion, Taylor Series Expansion, Laurent Series Expansion

Residue Integration;

Singularities, Poles, and Zeroes, Residue Integration Method,

Laplace transform;

Laplace transform definition, Laplace transforms of elementary functions, Properties of Laplace transform, Periodic functions and their Laplace transforms, Inverse Laplace transform and its properties, Convolution theorem, Inverse Laplace transform by integral and partial fraction methods, Heaviside expansion formula, Solutions of ordinary differential equations by Laplace transform, Applications of Laplace transforms,

Fourier Analysis;

Fourier Series and coefficients in Fourier series, Even and odd functions, Fourier transform definition, Fourier transforms of simple functions, Convolution theorem, Inverse Fourier transform, Partial Differential Equations (PDEs); Basic Concepts of PDEs, Modeling, Solution of PDEs by Laplace Transforms.