Special Functions

Paper Code: 
DMAT 813
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 (COs):

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

 

 

 

 

DMAT813

 

 

 

 

 

 

Special Functions

(Theory)

 

The students will be able to –

CO156: Explain the applications and the usefulness of these special functions.

CO157: Classify and explain the functions of different types of differential equations.

CO158: Describe concept of hypergeometric, Bessel's, Legender's functions etc.

CO159: Analyse properties of special functions by their integral representations and symmetries.

CO160: Identified the application of some basic mathematical methods via all these special functions.

CO161: Apply these techniques to solve and analyse various mathematical problems.

Approach in teaching:

Discussion, Demonstration, Team teaching, Presentation

 

Learning activities for the students:

Self-learning, Presentation, Effective questions, Giving tasks

Quiz, Poster Presentations,

Power Point Presentations, Individual and group projects,

Open Book Test, Semester End Examination

 

 

Unit I: 
I
18.00
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.
 
Unit II: 
II
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: 
III
18.00
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.
 
Unit IV: 
IV
18.00
Bessel Functions: Bessel's equation and its solution, Recurrence relations, Generating function, Integral representations of Bessel function, Integrals involving Bessel’s functions.
 
Unit V: 
V
18.00
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.
 
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, 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.
Academic Year: 
2023-2024
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