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