MAIR22 Complex Analysis and Differential Equations
(for B. Tech Computer Science and Engineering)
OBJECTIVES. The course presents
an introduction to analytic functions and power series
various Cauchy’ theorems and its applications in evaluation of integral
various approach to find general solution of the ordinary differential equations
Laplace transform techniques to find solution of differential equations
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
understand analytic functions discuss its properties
obtain series representation of analytic functions
evaluate various integrals by using Cauchy’s residue theorem
classify singularities and derive Laurent series expansion
find the solutions of first and some higher order ordinary differential equations
apply properties of special functions in discussion the solution of ODE.
Find Laplace transform of a given function and its inverse Laplace transform.
Find solution of first order partial differential equations.
References.
James Ward Brown, Ruel Vance Churchill, Complex Variables and Applications, McGraw-Hill Higher Education, 2004
Dennis Zill, Warren S. Wright, Michael R. Cullen, Advanced Engineering Mathematics, Jones & Bartlett Learning, 2011
Erwin Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons, 2019.
William E. Boyce, Richard C. DiPrima, Douglas B. Meade, Elementary Differential Equations and Boundary Value Problems, Wiley, 2017.
Ian N. Sneddon, Elements of Partial Differential Equations, Courier Corporation, 2013
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