Course

Unit 1

Complex Variables – Revision of complex numbers. Definitions of continuity, differentiability, analyticity. Cauchy-Riemann equations

Unit 2

Integration along a smooth curve; integration along a contour; Cauchy’s theorem. Cauchy’s integral formula, Laurent’s theorem, Taylor’s theorem. Calculus of residues. Contour integration.

Unit 3

Calculus of Variations: Maxima and minima – The simplest case – Illustrative examples – Natural boundary conditions and transition conditions – Concept of functional with simple example – Variation of a functional (only necessary conditions) – Simple variational problem – Euler equation – The more general case of variational problems – Constraints and Lagrange multipliers – Variable end points – Sturm-Liouville problems – Hamilton’s principle – Lagrange’s equations – Generalized dynamical entities – Constraints in dynamical systems.

Course Objectives

• To perform calculus for complex
• To understand the concepts of Taylor and Laurent
• To understand complex integrations and
• To Familiarize the concepts of calculus of variations and its

Course Outcomes

CO1: To learn differentiation for complex functions.

CO2: To Understand the basic concepts of complex integrations and residues.

CO3: To Understand the maximum and minimum principle, and variations of functionals.

CO4: To understand the Lagrange multiplier method and problems related to Sturm-Liouville and Hamilton’s principle.

CO-PO Mapping

Evaluation Pattern

• CA – Can be Quizzes, Assignment, Lab Practice, Projects, and Reports

Text Books

1. Advanced Engineering Mathematics, E Kreyszig, John Wiley and Sons, Tenth Edition,
2. S. Gupta, calculus of Variations with Applications, Prentice Hall of India, 1997

Reference Books

1. Advanced Engineering Mathematics, Ray Wylie and Louis Barrett, McGraw Hill, Sixth Edition,
2. Engineering Mathematics, Srimanta Pal and Subodh c Bhunia, Oxford press,
3. Gelfand and S. V. Fomin, Calculus of Variations, Dover Publications, 2000