Special Function MATH-434

bs-mathematics new 8th

The main aim of this course is the study of the properties and relations of special functions such as gamma, beta, hypergeometric, confluent hypergeometric functions, Bessel functions and generalized hypergeometric functions. Furthermore, the properties, relations and applications of these special functions are also discussed.

**Learning Outcomes:**

1. Learn basic concepts of special functions.

2. Define gamma, beta and hypergeometric functions and prove some properties involving these functions.

3. Discuss some applications of the gamma, beta function and hypergeometric functions.

4. Define normalization, Bessel's inequality, generating functions, differential equations, recurrence relations.

**Course Outlines:**

1. The gamma function: definition, relations satisfied by gamma function,

2. Euler's constant, the order symbol o and O, asymptotic representations of the gamma function for large O(z),

3. The beta function: properties, relations,

4. Hypergeometric functions F(a,b:c:z), F(a,b:c:1), the hgypergeometric differential equation,

5. Simple transformations, a Theorem due Kummer,

6. Orthogonal polynomials: simple set of polynomials, orthogonality, the term recurrence relation, the Christofell-Darboux formula, normalization,

7. Bessel's inequality, Legendere polynomials: generating function, differential equation,the Rodrigues formula, recurrence relations, hypergeometric form of Pn (x), orthogonality.

8. Hermite Polynomials: generating function, differential equation, the Rodrigues formula, recurrence relations, orthoganality, definition of Hn(x).

9. Laguerre Polynomials: The polynomials Ln(x), generating functions, Rodrigues formula, the differential equation, orthogonality.

**Recommended Books**

- Rainville, E. D., Special Functions, The Macmillan Company, New York, 3rd Edition, 1965.
- Singh, U. P. and Denis, R. Y., Special Functions and Their Applications, Dominant Publishers and Distributors, 2001.

**Suggested Books**

- Andrews, G. E., Richard, A. and Roy, R., Special Functions, Cambridge University Press, 1st Edition, 2000.
- Mathai, A. M. and Houbold, H. J., Special Functions for Applied Scientists, Springer Science and Business Media, LLC, New York, 2008.

**Assessment Criteria**

Sessional: 20 (Presentation / Assignment 04, Attendance 08, Result Mid-Term 04, Quiz 04

Mid-Term Exam (Term Paper): 30

Final-Term Exam: 50