Syllabus for Master of Science Mathematics Subject Code: 02MA0555 Subject Name: Special Functions-II M.Sc. Year-II (Sem-4) Objective: This course aims at providing essential ideas of integral transforms. Numerous problems, which are not easily solvable, can be properly dealt with integral transform methods. Credits Earned: 5 Credits Course Outcomes: After completion of this course, student will be able to Understand concept of Fourier series, Fourier Integral and Fourier transform. Analyze the properties of Hankel transform, Mellin Transform and Z-transform. Perform operations with orthogonal polynomials, Legendre’s polynomial and Laguerre polynomial with their differential equations along with the corresponding recurrence formulas. Demonstrate their understanding of how physical phenomena are modeled using special functions. Explain the applications of Integral transforms and solving problems using transform method. Pre-requisite of course: Real Analysis, Complex Analysis, Differential Equations Teaching and Examination Scheme Tutorial/ Practical Teaching Scheme (Hours) Theory Marks Marks Total Credits Mid Term ESE Internal Viva Marks Theory Tutorial Practical Sem work (E) (I) (V) (M) (TW) 4 2 - 5 50 30 20 25 25 150 Contents: Unit Topics Contact Hours 1 Fourier transforms and their applications: 15 Introduction, Fourier series, Fourier integral formulas, Fourier transform of generalized functions, Basic properties of Fourier transform, Applications of Fourier transform. Syllabus for Master of Science Mathematics 2 Hankel Transforms and their applications 9 Hankel transforms and examples, Operational properties of Hankel transform, Application of Hankel transform to PDE 3 Mellin Transform and Z-trasnfrom 12 Introduction to Mellin transform, Operational properties of Mellin transform, Application of Mellin Transforms, Introduction to Z-transform, basic Operational properties of Z-transform, Inverse Z-transform and examples. 4 Orthogonal Polynomials 08 Simple set of polynomials, Orthogonality, Equivalent condition for orthogonality, Zeros of orthogonal polynomials, Recurrence relation, Christoffel-Darboux formula. 5 Legendre’s polynomials 08 Generating function, Recurrence relations, Legendre’s differential equation, Rodrigue’s formula, Bateman Generating function 6 Laguerre polynomials 08 Introduction,Recurrence relations, Laguerre differential equation, Rodrigue’s formula, Orthogonality Total Hours 60 Recommended Textbooks: 1. E. D. Rainville, Special Functions, The Macmillan Company, New York, 1960. 2. Z. X. Wang and D. R. Guo,Special Functions, World Scientific publishing Co., 1989. 3. L. Debnath,Integral transforms and their Applications, CRC Press, New York- London- Tokyo, 1995. 4. M. Ya. Antimirov, A.A. Kolyshkin, R. Valliancourt, Applied Integral Transforms, CRM Monograph Series,American Mathematical Society, 2007. 5. A.D Poularikas, The Transforms and Applications Handbook, CRC Press, 1996. Suggested Theory distribution: The suggested theory distribution as per Bloom’s taxonomy is as per follows. This distribution serves as guidelines for teachers and students to achieve effective teaching- learning process Distribution of Theory for course delivery and evaluation Remember Understand Apply Analyze Evaluate Create 20% 20% 30% 15% 10% 5% Instructional Method: Syllabus for Master of Science Mathematics a. The course delivery method will depend upon the requirement of content and need of students. The teacher in addition to conventional teaching method by black board, may also use any of tools such as demonstration, role play, Quiz, brainstorming, MOOCs etc. b. The internal evaluation will be done on the basis of continuous evaluation of students in the laboratory and class-room. c. Practical examination will be conducted at the end of semester for evaluation of performance of students in laboratory. d. Students will use supplementary resources such as online videos, NPTEL videos, e- courses, Virtual Laboratory Supplementary Resources: 1. http://mathworld.wolfram.com/OrthogonalPolynomials.html 2. https://en.wikipedia.org/wiki/Laguerre_polynomials 3. https://www.youtube.com/watch?v=fnErOqHoPtQ 4. http://nptel.ac.in/courses/106106097/pdf/Lecture10_ZTransForm.pdf 5. https://en.wikibooks.org/wiki/Digital_Signal_Processing/Z_Transform .