Special Functions

Paper Code: 
24DMAT813
Credits: 
6
Contact Hours: 
90.00
Max. Marks: 
100.00
Objective: 

 This course will enable the students to

  1. Understand the properties of special functions like Gauss hypergeometric, Legendre functions with their integral representations.
  2. Understand the concept of Bessel’s function, Hermite function etc, with its properties like recurrence relations, orthogonal properties, generating functions etc.
  3. Understand how special functions are useful in differential equations.

 

Course Outcomes: 

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

 

 

 

 

 

 

24DMAT

813

 

 

 

 

 

 

 

Special Functions

(Theory)

 

CO192: Explore the concept hypergeometric functions, Including basic properties, Series representations, integral representations and transformations.

CO193: Classify and explain the functions of different types of differential equations.

CO194: Describe Legender function and   properties such as orthogonality, Recurrence relations, Generating functions and Rodrigues' formula.

CO195: Analyse properties such as recurrence relations, Generating functions, Asymptotic expansions and integral representations of Bessel’s function.

CO196: Explore Laguerre and Hermite functions with properties.

CO197: Contribute effectively in course-specific interaction.

Approach in teaching:

Discussion, Demonstration, Team teaching, Presentation

 

Learning activities for the students:

Self-learning, Presentation, Effective questions, Assigned tasks

 

 

 

Quiz,

Power Point Presentations, Individual and group projects,

Open Book Test, Semester End Examination

 

 

Unit I: 
Gauss hypergeometric functions
18.00

Definition and properties, Condition of convergence, Integral representation, Gauss theorem, Vandermonde’s theorem, Kummer’s theorem, Linear transformation, Differentiation formulae, Relations of contiguity.

 

Unit II: 
Special cases of hypergeometric functions and properties
18.00

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.

 

Unit III: 
Legendre polynomials and functions
18.00

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.

 

Unit IV: 
Bessel Functions
18.00

Bessel's equation and its solution, Recurrence relations, Generating function, Integral representations of Bessel’s function, Integrals involving Bessel’s functions.

 

Unit V: 
Hermite polynomials
18.00

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.

 

Essential Readings: 
  • 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.

 

References: 
  • I.N. Sneddon, Special Functions, McGraw Hill, New Delhi, 1956.
  • N.N. Lebedev and 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

JOURNALS

 

Academic Year: 
2024-2025
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