CBCGS [2017 - current]

CBGS [2013 - 2016]

Old [2000 - 2012]

## Topics with syllabus and resources

1 Module 1

1.01 Laplace Transform (Lt) of Standard Functions

- Definition. unilateral and bilateral Laplace Transform
- LT of sin(at), cos(at), e
^{at},t^{n}, sinh(at), cosh(at), erf(t) - Heavi-side unit step
- dirac-delta function
- LT of periodic function.

1.02 Properties of Laplace Transform

- Linearity,
- first shifting theorem,
- second shifting theorem,
- multiplication by t
^{n}, division by t , - Laplace Transform of derivatives and integrals,
- change of scale,
- convolution theorem,
- initial and final value theorem,
- Parsavel‘s identity.

1.03 Inverse Laplace Transform

- Partial fraction method
- long division method
- residue method.

1.04 Applications of Laplace Transform

- Solution of ordinary differential equations.

2 Module 2

2.01 Introduction

- Definition
- Dirichlet‘s conditions
- Euler‘s formulae.

2.02 Fourier Series of Functions

- Exponential, trigonometric functions, even and odd functions, half range sine and cosine series.
- Complex form of Fourier series, orthogonal and orthonormal set of functions, Fourier integral representation.

3 Module 3

3.01 Solution of Bessel Differential Equation

- Series method, recurrence relation, properties of Bessel function of order +1/2 and -1/2 Generating function, orthogonality property.
- Bessel Fourier series of functions.

4 Module 4

4.01 Scalar and Vector Product

- Scalar and vector product of three and four vectors and their properties.

4.02 Vector Differentiation

- Gradient of scalar point function
- divergence and curl of vector point function.

4.03 Properties

- Solenoidal and irrotational vector fields
- conservative vector field.

4.04 Vector Integral

- Line integral
- Green‘s theorem in a plane
- Gauss‘ divergence theorem
- Stokes‘ theorem.

5 Module 5

5.01 Complex Variable

- Analytic Function: Necessary and sufficient conditions, Cauchy.
- Reiman equation in polar form.
- Harmonic function, orthogonal trajectories.

5.02 Mapping

- Conformal mapping.
- bilinear transformations.
- cross ratio.
- fixed points.
- bilinear transformation of straight lines and circles.