## COURSE OBJECTIVES

• 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)

## COURSE CONTENTS

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