Advanced Studies of Special Functions and Integral Transforms

Paper Code: 
MAT424 B
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, generalized 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 analyze various mathematical problems.

 

Course Outcomes (COs):

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

 

 

 

 

MAT 424B

 

Advanced Studies of Special Functions and Integral Transforms

 (Theory)

 

 

 

The students will be able to –

 

CO176: Know the concept of Z-transforms and its properties..

CO177: Understand and find Solutions Heat, Wave, Laplace equation under initial and boundary conditions.

CO178: Think logically and mathematically and apply the knowledge of integral transform to solve complex problems.

CO179: Learn properties of the generalised hypergeometric function and its convergence.

CO180: Explain the applications and the usefulness of these special functions.

CO181: Apply the concept of Z-transforms and its importance in engineering like Digital signal processing and digital filters.

 

Approach in teaching:

Interactive Lectures, Discussion, Power Point Presentations, Informative videos

Learning activities for the students:

Self learning assignments, Effective questions, presentations, Field trips

Quiz, Poster Presentations,

Power Point Presentations, Individual and group projects,

Open Book Test, 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 and J. K. Goyal, Integral Transforms, Pragati Prakashan, New Delhi,2015
  • Z.X. Wang and D.R. Guo, Special Functions, World Scientific publishing Ltd., 1989.
 
References: 
  • George E. Andrews, Richard Askey and Ranjan Roy, Special Functions, Cambridge University, 2000.
  • N. N. Lebedev and Richard A. SilverMan, Special Functions and Their Application, Dover Publications INC, 2012.
  • 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, 2008.
 
Academic Year: 
2022-2023
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