Teaching Scheme (in Hours)
| Theory |
Tutorial |
Practical |
Total |
| 3 |
2 |
0 |
5 |
Subject Credit : 5
Examination Scheme (in Marks)
Theory
ESE (E)
|
Theory
PA (M)
|
Practical
ESE Viva (V)
|
Practical
PA (I)
|
Total
|
| 70 |
30 |
0 |
0 |
100 |
Syllabus Content
Unit-1: Polar Form of Complex Numbers, Powers and Roots
Complex Variable – Differentiation : Differentiation, Cauchy-Riemann equations, analytic functions, harmonic functions, finding harmonic conjugate; elementary analytic functions (exponential, trigonometric, logarithm) and their properties; Conformal mappings, Mobius transformations and their properties
Unit-2: Complex Variable - Integration
Contour integrals, Cauchy-Goursat theorem (without proof), Cauchy Integral formula (without proof), Liouville’s theorem and Maximum-Modulus theorem (without proof); Sequences, Series, Convergence Tests, Power Series, Functions Given by Power Series, Taylor and Maclaurin Series, Uniform Convergence.
Unit-3: Laurent’s series
Zeros of analytic functions, singularities, Residues, Cauchy Residue theorem (without proof), Residue Integration Method, Residue Integration of Real Integrals
Unit-4: Partial differential equations
First order partial differential equations, solutions of first order linear and nonlinear PDEs, Charpit’s Method
Unit-5: Homogeneous and nonhomogeneous linear partial differential equations
Solution to homogeneous and nonhomogeneous linear partial differential equations second and higher order by complementary function and particular integral method. Separation of variables method to simple problems in Cartesian coordinates, second-order linear equations and their classification, Initial and boundary conditions, Modeling and solution of the Heat, Wave and Laplace equations.
Course Outcome
- Convert complex number in a polar form, plot the roots of a complex number in complex plane, find harmonic conjugate of analytic functions and apply conformal mapping in geometrical transformation
- Evaluate complex integration by using various result, test convergence of complex sequence and series and expand some analytic function in Taylor’s series
- Find Laurent’s series and pole of order, and apply Cauchy Residue theorem in evaluating some real integrals
- Form and solve first order linear and nonlinear partial differential equations
- Apply the various methods to solve higher order partial differential equations, modeling and solve some engineering problems related to Heat flows, Wave equation and Laplace equation
