MAIR22 Complex Analysis and Differential Equations

(for B. Tech Computer Science and Engineering)

OBJECTIVES. The course presents

1. an introduction to analytic functions and power series

2. various Cauchy’ theorems and its applications in evaluation of integral

3. various approach to find general solution of the ordinary differential equations

4. Laplace transform techniques to find solution of differential equations

5. Partial differential equations and methods to find solution of it.

COURSE CONTENT.

Analytic functions; Cauchy-Riemann equations; Line integral, Cauchy's integral theorem and integral formula (without proof); Taylor's series and Laurent series; Residue theorem (without proof) and its applications.

Higher order linear differential equations with constant coefficients; Second order linear differential equations with variable coefficients; Method of variation of parameters; Cauchy-Euler equation.

Laplace Transform of Standard functions, derivatives and integrals – Inverse Laplace transform – Convolution theorem – Periodic functions – solution of ordinary differential equation and simultaneous equations with constant coefficients and integral equations by Laplace Transform.

Formation of partial differential equations by eliminating arbitrary constants and functions – solution of first order equations – four standard types – Lagrange’s equation. Method of separation of variables.

COURSE OUTCOME. On completion of the course, student will be able to

1. understand analytic functions discuss its properties

2. obtain series representation of analytic functions

3. evaluate various integrals by using Cauchy’s residue theorem

4. classify singularities and derive Laurent series expansion

5. find the solutions of first and some higher order ordinary differential equations

6. apply properties of special functions in discussion the solution of ODE.

7. Find Laplace transform of a given function and its inverse Laplace transform.

8. Find solution of first order partial differential equations.

References.

1. James Ward Brown, Ruel Vance Churchill, Complex Variables and Applications, McGraw-Hill Higher Education, 2004

2. Dennis Zill, Warren S. Wright, Michael R. Cullen, Advanced Engineering Mathematics, Jones & Bartlett Learning, 2011

3. Erwin Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons, 2019.

4. William E. Boyce, Richard C. DiPrima, Douglas B. Meade, Elementary Differential Equations and Boundary Value Problems, Wiley, 2017.

5. Ian N. Sneddon, Elements of Partial Differential Equations, Courier Corporation, 2013