Special Functions
This course will enable the students to
- Understand the properties of special functions like Gauss hypergeometric, Legendre functions with their integral representations.
- Understand the concept of Bessel’s function, Hermite function with its properties like recurrence relations, orthogonal properties, Generating functions etc.
- Understand how special functions are useful in differential equations.
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Learning outcomes (at course level) |
Learning and teaching strategies |
Assessment Strategies |
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Course Title |
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Special Functions (Theory)
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CO124: Explore the concept of hypergeometric functions, including basic properties, series and integral representations and transformations. CO125: Classify and explain the functions of different types of differential equations. CO126: Describe Legender function and properties such as orthogonality, Recurrence relations, etc. CO127: Analyse properties such as recurrence relations, Generating functions, Asymptotic expansions and integral representations of Bessel’s function. CO128: Explore Laguerre and Hermite functions with properties. CO129: Contribute effectively in course-specific interaction |
Approach in teaching: Discussion, Demonstration, Interactive sessions, Presentation.
Learning activities for the students: Self-learning, Presentation, Effective questions, Assigned tasks |
Quiz, Individual and group projects, Open Book Test, Semester End Examination
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Definition and its properties, Condition of convergence, Integral representation, Gauss theorem, Vandermonde’s theorem, Kummer’s theorem, Linear transformation, Differentiation formulae, Relations of contiguity.
Gauss’s hypergeometric differential equation and its solution, relation between the solutions of hypergeometric equation, Two summation theorems, Kummer’s confluent hypergeometric function: Definition and differential equation, Integral representation, Differentiation, Kummer’s first and second transformations, contiguous relations.
Definition, Solution of Legendre’s equation, Legendre functions of the first and second kind, Generating functions(first formula), Rodrigue formula for Pn(x), Orthogonality of Legendre polynomials, Recurrence relations for Pn(x), Beltrami’s result, Christoffel expansion, Christoffel’s summation formula, Relation between Pn(x) and Qn(x), Laplace first and second integrals for Legendre polynomials.
Bessel's equation and its solution, Recurrence relations, Generating function, Integral representations of Bessel’s function, Integrals involving Bessel’s functions.
Definition, Generating function, Recurrence relations, Orthogonality of Hn(x), Rodrigue formula, Hermite’s differential equation and it’s solution, Laguerre polynomials: Laguerre’s differential equation and it’s solutions, Generating function, Rodrigue formula, Orthogonality of Laguerre polynomials, Recurrence relation.
- R.K. Saxena and D. C. Gokhroo, Special Functions, Jaipur Publishing House, 2014.
- M. A. Pathan, V. B. L. Chaurasia, J. Banerji and S. P. Goyal, Special Functions and Calculus of Variations, RBD, Jaipur, 2004.
- E. D. Rainville, Special Functions, Macmillan, New York, 1989.
SUGGESTED READING
- N. Sneddon, Special Functions, McGraw Hill, New Delhi, 1956.
- N.N. Lebedev, R. Silverman, Special Functions and Their Application, Dover Publications INC, 2003.
- Z.X. Wang and D. R. Guo, Special Functions, World Scientific books, 1989.
- G.E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University, 2000.
e- RESOURCES
● https://www.pdfdrive.com/special-functions-their-applications-d164947812.html
● https://web.mst.edu/~lmhall/SPFNS/spfns.pdf
● https://pkalika.files.wordpress.com/2020/08/special-function-kalika124pages.pdf
JOURNALS
● https://www.tandfonline.com/toc/gitr20/current
● https://www.mdpi.com/journal/mathematics/special_issues/Special_Functions_Applications
