- To discuss the complex number system, different types of complex functions, analytic properties of complex numbers, theorems in complex analysis to carryout various mathematical operations in complex plane, roots of a complex equation.
- To discuss limits, continuity, differentiability, contour integrals, analytic functions and harmonic functions.
- Cauchy–Riemann equations in the Cartesian and polar coordinates, Cauchy’s integral formula, Cauchy–Goursat theorem, convergence of sequence and series, Taylor series, Laurents series.
- Integral transforms with a special focus on Laplace integral transform. Fourier transform.
COURSE LEARNING OUTCOMES (CLO)
CLO-1: Define the complex number system, complex functions and integrals of complex functions (C1)
CLO-2: Explain the concept of limit, continuity, differentiability of complex valued functions (C2)
CLO-3: Apply the results/theorems in complex analysis to complex valued functions (C3)
1. Introductory Concepts – Three Lectures
- Introduction to Complex Number System
- Argand diagram
- De Moivre’s theorem and its Application Problem Solving Techniques
2. Analyticity of Functions – Four Lectures
- Complex and Analytical Functions,
- Harmonic Function, Cauchy-Riemann Equations.
- Cauchy’s theorem and Cauchy’s Line Integral.
3. Singularities – Five Lectures
- Singularities, Poles, Residues.
- Contour Integration.
4. Laplace transform – Six Lectures
- 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
5. Fourier series and Transform – Seven Lectures
- Fourier theorem and coefficients in Fourier series,
- Even and odd functions,
- Complex form of Fourier series,
- Fourier transform definition,
- Fourier transforms of simple functions,
- Magnitude and phase spectra,
- Fourier transform theorems,
- Inverse Fourier transform,
6. Solution of Differential Equations– Seven Lectures
- Series solution of differential equations,
- Validity of series solution, Ordinary point,
- Singular point, Forbenius method,
- Indicial equation,
- Bessel’s differential equation, its solution of first kind and recurrence formulae,
- Legendre differential equation and its solution,
- Rodrigues formula