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 etc, with its properties like recurrence relations, orthogonal properties, generating functions etc.
- Understand how special functions are useful in differential equations.
Course Outcomes (COs):
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Course |
Learning outcomes (at course level) |
Learning and teaching strategies |
Assessment Strategies |
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Course Code |
Course Title |
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MAT 223
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Special Functions (Theory)
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The students will be able to –
CO48: Explain the applications and the usefulness of these special functions. CO49: Classify and explain the functions of different types of differential equations. CO50: Describe concept of hypergeometric, Bessel's, Legender's functions etc. CO51: Analyse properties of special functions by their integral representations and symmetries. CO52: Identified the application of some basic mathematical methods via all these special functions. CO53: Apply these techniques to solve and analyse various mathematical problems.
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Approach in teaching: Discussion, Demonstration, Team teaching, Presentation
Learning activities for the students: Self learning, Presentation, Effective questions, Giving tasks |
Observation, Presentation, Report writing, |
Gauss hypergeometric functions: 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.
Legendre polynomials and functions: 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 Functions: Bessel's equation and its solution, Recurrence relations, Generating function, Integral representations of Bessel function, Integrals involving Bessel’s functions.
Hermite polynomials: 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.
- I. 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.
