Capital University of Science & Technology

Complex Variables and Transforms (MTEE2053)


Applied Differential Equation (MT-1043)

Recommended Book(s)

Complex Variables & Applications, By J. W. Brown & R. V. Churchill

Reference Book(s)

Complex Analysis For Mathematics & Engineering,  By J. H. Mathew & Howells,

Basic Complex Analysis,  by J. E. Marsden

Analytic Function Theory,  By E. Hille 1974

Course Objectives

1.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. 2.To discuss limits, continuity, differentiability, contour integrals, analytic functions and harmonic functions. 3.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. 4.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  (Level: C1)

CLO:2.  Explain  the concept of limit, continuity, differentiability of complex valued functions   (Level: C2)

CLO:3. Apply the results/theorems in complex analysis  to complex valued functions   (Level: C3)

CLO:3. Explain the concept of integral transforms, e.g., Laplace, Fourier transforms and the related inverse transforms by using the following  Partial fractions method, Tables, Convolution theorems and apply these transformation for engineering problems (Level: 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


  1. Analyticity of Functions – Four lectures
  • Complex and Analytical Functions,
  • Harmonic Function, Cauchy-Riemann Equations.
  • Cauchy’s theorem and Cauchy’s Line Integral.
  • Singularities, Poles, Residues.
  • Contour Integration.
  1. Singularities – Five lectures


  1. 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


  1. 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,


  1. 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

Mapping of CLOs to Program Learning Outcomes







(Engineering Knowledge)


(Problem Analysis)






(Design and Development of Solutions)












(Modern Tool Usage)






(The Engineer and Society)






(Environment and Sustainability)












(Individual and Team Work)












(Project Management)






(Life Long Learning)