Integral Transforms (Optional Paper)

Laplace Transform: Definition and its properties, Laplace Transform of derivatives and Integrals, Multiplication and Division by Power of x, Evaluation of Integrals by using Laplace Transforms, Periodic functions, Initial-value and Final value Theorem.

Inverse Laplace transform: Definition and its properties, Inverse Laplace Transform of derivatives and Integrals, Multiplication and Division by Power of x. Convolution theorem for Laplace transform, Evaluation of Integrals by using Inverse Laplace transform, use of Partial fractions, The Heaviside Expansion formula, Complex inversion formula.

Fourier transform: Definition and properties of Fourier sine, cosine and complex transforms, Inversion Theorem, Relationship between Fourier transform and Laplace transform, Modulation Theorem, Convolution theorem for sine, cosine and complex transforms, Parseval’s Identity, Fourier transform of derivatives.

Mellin transform– Definition and elementary properties. Mellin transforms of derivatives and integrals. Inversion theorem. Convolution theorem, Inverse Mellin transform of two functions.

Infinite Hankel transform– Definition and elementary properties. Hankel transform of Exponential functions, Inversion formula, Hankel transform of derivatives, Basic operational property of Hankel transform, Parseval’s Theorem.

Solution of ordinary differential equations with constant and variable coefficients by Laplace transform. Application to the simple boundary value problem by Laplace, Fourier and infinite Hankel transforms.