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### Title- Special Functions & Integral Transforms (Code-09010402)

 Unit Content Learning objects Teaching Guidelines Methodology Time (Hrs) one Series solution of differential equations – Power series method,   Definitions of Beta and Gamma functions. Bessel equation and its solution: Bessel functions and their properties-Convergence, recurrence, Relations and generating functions, Orthogonality of Bessel functions Students will be able to   1.  Understand Series solution of differential equations – Power series method,   2. Bessel equation and its solution: Bessel functions and their properties-Convergence, recurrence 3. Orthogonality of Bessel functions Lecture should be effective so that student can grasp the topics easily. 1.  Lecture. 2.  Seminar. 3. Discussion/Interaction with Students. 4. Assignment. 5.  Discussion on Assignment.         6. Evaluation of Assignment. 15 Two Legendre and Hermite differentials equations and their solutions:   Legendre and Hermite functions and their properties-Recurrence   Relations and generating functions. Orhogonality of Legendre and Hermite polynomials.   Rodrigues’ Formula for Legendre & Hermite Polynomials,Laplace   Integral Representation of Legendre polynomial. Students will be able to 1.Legendre and Hermite differentials equations and their solutions:   2.  Relations and generating functions. Orhogonality of Legendre and Hermite polynomials.   3.Rodrigues’ Formula for Legendre & Hermite Polynomials,Laplace Lecture should be effective so that student can grasp the topics easily. 1.     Lecture. 2.     Seminar. 3.    Discussion/Interaction with Students. 4.    Assignment. 5.    Discussion on Assignment.         6.    Evaluation of Assignment 15 Three Laplace Transforms – Existence theorem for Laplace transforms, Linearity of the Laplace transforms, Shifting theorems,   Laplace transforms of derivatives and integrals, Differentiation and integration of Laplace transforms,   Convolution theorem, Inverse Laplace transforms,convolution theorem, Inverse Laplace transforms of derivatives and integrals, solution of ordinary differential equations using Laplace transform.   Fourier transforms: Linearity property, Shifting, Modulation, Convolution Theorem,   Fourier Transform of Derivatives, Relations between Fourier transform and Laplacetransform,   Parseval’s identity for Fourier transforms, solution of differential Equationsusing Fourier Transforms. To make students familiar with Laplace Transforms – Existence theorem for Laplace transforms, Linearity of the Laplace transforms, Shifting theorems, Convolution theorem, Inverse Laplace transforms,convolution theorem, Inverse Laplace transforms of derivatives and integrals, solution of ordinary differential equations using Laplace transform. Fourier Transform of Derivatives, Relations between Fourier transform and Laplacetransform,   Parseval’s identity for Fourier transforms, solution of differential Equationsusing Fourier Transforms. Lecture should be effective sothat student can grasp the topics easily. 1.     Lecture. 2.    Seminar. 3.    Discussion/Interaction with Students. 4.    Assignment. 5.    Discussion on Assignment.        6.    Evaluation of Assignment 15