ADVANCED STUDIES OF SPECIAL FUNCTIONS AND INTEGRAL TRANSFORMS (Optional Paper)

Paper Code: 
MAT424B
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Objective: 
This course will enable the students to -
  1. Aware about properties of special functions ,generalised hypergeometric, Legendre  Functions etc. by their integral representations and symmetries.
  2. Understand  Laplace transform, Z transforms. 
  3. Learn about its applications in partial differential equations of mathematical physics. 
  4. Apply these techniques to solve and analyse various mathematical problems

Learning Outcomes

Learning and teaching strategies

Assessment

After the completion of the course the students will be able to:

CLO133- To able to know the concept of Z-transforms and Properties and its importance in engineering like Digital signal processing and digital filters.

 

CLO134- To able to understand and find Solutions Heat ,Wave, Laplace equation under initial and boundary conditions.

 

CLO135-  Think logically and mathematically and apply the knowledge of integral transform to solve complex problems.

CLO136- Learn properties of  the  generalised hypergeometric function and its convergence.

 

CLO137- Explain the applications and the usefulness of these special functions.

 

Approach in teaching:

Interactive Lectures, Discussion, Tutorials, Reading assignments, Demonstration, Team teaching

Learning activities for the students:

Self learning assignments, Effective questions, Simulation, Seminar presentation, Giving tasks, Field practical

 

 

 

 

Presentations by Individual  Students

Class Tests at end of each unit.

 

Written assignment(s)

Semester  End Examination

 

Unit I: 
I
15.00

Associated Legendre polynomials of first and second kind: Differential equation, Relation between solutions of associated Legendre equation, Recurrence relation, Orthogonal properties, Hyper geometric forms.

 

Unit II: 
II
15.00

Chebyshev polynomials: Chebyshev equation and its solutions, Expansions, Generating relations and orthogonal property

Unit III: 
III
15.00

Generalized Hypergeometric Function: Definition, Special cases, Series, integral and contour representations, Convergence conditions of these representations, Saalssutz, Whipple theorems, Contiguous function relations, Differentiation and integral formulas.

Unit IV: 
IV
15.00

Laplace Transforms: Complex inversion formula, Use of residue theorem in calculation of inverse Laplace transform including the functions with branch points and infinitely many singularities, Solution of Heat conduction and Wave problems by using complex inversion formula for Laplace transform.

Unit V: 
V
15.00

Z-Transforms: Definition, Inverse, Images of elementary functions, Basic operational properties, Partial derivatives, Initial and Final value theorems and applications.

Essential Readings: 
  • E.D.Rainville, Special functions, Macmillan, New York,1989.
  • K P Gupta, J K Goyal, Integral Transforms, Pragati Prakashan, New Delhi,2015
  • Z.X. Wang, D.R.  Guo, Special Functions, World Scientific publishing Ltd., 1989.
References: 
  • George E. Andrews, Richard Askey, Ranjan Roy, Special Functions, Cambridge University, 2000.
  • N.N.Lebedev, Richard A.SilverMan, Special functions and their application, Dover Publications INC,1972.
  • I.N.Sneddon, Special Functions, TMH, New Delhi,1956.
  • Mohamed F. EL-Hewie, Laplace Transform,Create space Independent Publication, 2013.
  • Joel L. Schiff, The Laplace transform: Theory and application, Springer Science & Business Media, 1999.
  • John Miles, Integral Transform in Applied Mathematics, Cambridge University Press, 1971.
Academic Year: 
2019-2020
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