Course Title: Special Functions Course Code:
MTH4421Program: B.Sc. in Mathematics Department: Mathematics
College: Jamoum University Colleg
Institution: Umm Al-Qura University
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Table of Contents
A. Course Identification ... 3
6. Mode of Instruction (mark all that apply) ... 3
B. Course Objectives and Learning Outcomes ... 3
1. Course Description ... 3
2. Course Main Objective ... 3
3. Course Learning Outcomes ... 3
C. Course Content ... 4
D. Teaching and Assessment ... 5
1. Alignment of Course Learning Outcomes with Teaching Strategies and Assessment Methods ... 5
2. Assessment Tasks for Students ... 5
E. Student Academic Counseling and Support ... 6
F. Learning Resources and Facilities ... 6
1.Learning Resources ... 6
2. Facilities Required ... 6
G. Course Quality Evaluation ... 6
H. Specification Approval Data ... 7
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A. Course Identification
1. Credit hours: 3 2. Course type
a. University College Department Others
b. Required Elective
3. Level/year at which this course is offered: Tenth level/Fourth year 4. Pre-requisites for this course (if any):
Calculus + Ordinary Differential Equations
5. Co-requisites for this course (if any):
Not applicable
6. Mode of Instruction (mark all that apply)
No Mode of Instruction Contact Hours Percentage
1 Traditional classroom Three hours/week 100%
2 Blended 0 0
3 E-learning 0 0
4 Distance learning 0 0
5 Other 0 0
7. Contact Hours (based on academic semester)
No Activity Contact Hours
1 Lecture 30
2 Laboratory/Studio 0
3 Tutorial 10
4 Others (specify) 0
Total 40
B. Course Objectives and Learning Outcomes
1. Course Description
This course is flexible enough to be presented to undergraduate students or beginning graduate students majoring in applied mathematics, engineering, chemistry or physics who wishes to use special functions. It is an introductory course which presents the fundamental concepts of various types of polynomials and their properties.
2. Course Main Objective
The purpose of this course is to condense into an introductory text the definitions and techniques arising in special functions. The material is presented to develop a physical understanding of the mathematical concepts associated with different types of functions and develop the recurrence relation of various types of polynomials and solve their differential equations.
3. Course Learning Outcomes
CLOs Aligned
PLOs 1 Knowledge and Understanding: by the end of this course, the
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CLOs Aligned
PLOs student is expected to be able to
1.1 Define the infinite series in the interval using the Euler’s formula.
1.2 Write the Fourier series of even and odd functions,
1.3 Express the polynomials in terms of Fourier series and obtained the necessary deduction.
1.4 Use of Beta and Gamma function in evaluating the complicated integral with ease.
2 Skills: by the end of this course, the student is expected to be able to 2.1 Express the solution of the physical problem in terms of special
functions.
2.2 Apply Laplace transform operator and solve the problems of different nature.
2.3 To express the polynomials in term of Hermite, Laguerre’s, Legendre and Bessel functions.
3 Values: by the end of this course, the student is expected to be able to
3.1 Find the solution of Hermite equation, Legendre, Laguerre and Bessel by general power series and the proof of orthogonality of Hermite polynomials and recurrence relations for Hermite polynomials using the generating function.
3.2 Demonstrate Fourier transforms, their properties and the solution of the initial boundary value problems for PDEs using Fourier transforms 3.3 Understand the applications of the polynomials mentioned in the
content.
C. Course Content
No List of Topics Contact
Hours .
1
Basic definitions of Fourier series, Fourier series of even and odd functions, periodic functions, Dirichlet’s condition, Fourier series expansion of algebraic functions, absolute value function, step-function.
6
2.
Definition and properties of Gamma function, transformations of Gamma functions, use of Gamma function in integrating a function.
Definition and properties of Beta function, evaluation of Beta function in an explicit form, transformations of Beta function, use of Beta function in evaluating the integrals, relation between Gamma and Beta functions and Laplace transformation.
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3. Definition of Hermite polynomial, Laguerre polynomial and their 8
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equations, generating function and recurrence relation. Pochhammer symbols, hypergeometric functions and their properties, differential and integral representation of Hypergeometric functions
4.
Definition of Legendre polynomial, Bessel function and their generating function and recurrence relation. Definition of orthogonality, Orthogonal set of functions, Orthogonality of Hermite, Laguerre’s, Legendre and Bessel functions.
8
Total 30
D. Teaching and Assessment
1. Alignment of Course Learning Outcomes with Teaching Strategies and Assessment Methods
Code Course Learning Outcomes Teaching Strategies Assessment Methods 1.0 Knowledge and Understanding
1.1 Identify polynomials and their differential equation, series solution.
Lecture and Tutorials Exams, quizzes 1.2 Knowledge of Laplace operator and
Fourier transformation.
Lecture and Tutorials Exams, quizzes 1.3 Present an account of basic concepts
and definitions of polynomials.
Lecture and Tutorials Exams, quizzes
1.4
Describe the polynomials in term of special functions and able to find its generating function.
Lecture and Tutorials Exams, quizzes
2.0 Skills
2.1 Demonstrate the ability for solving mathematical problems involving polynomials described by differential equations.
Lecture/Individual or group work
Exams, quizzes
2.2 Explain the Fourier series technique. Lecture/Individual or group work
Exams, quizzes 2.3 Apply Laplace transform in the
solution of some physical problems of science and engineering.
Lecture/Individual or group work
Exams, quizzes
3.0 Values
3.1 Recognize the notions of Pochhammer symbols and obtained hypergeometric function from these symbols.
Lecture/ Self-learning through the website
Exams, quizzes
3.2 Interpret graphical and qualitative representations of solutions to problems.
Lecture/ Self-learning through the website
Exams, quizzes 3.3 Generalize mathematical concepts in
problem and their applications in physics and chemistry.
Lecture/Individual or group work
Exams, quizzes
2. Assessment Tasks for Students
# Assessment task* Week Due Percentage of Total
Assessment Score
6
# Assessment task* Week Due Percentage of Total
Assessment Score
1 Midterm Exam 6th %25
2 Quizes and homeworks During semester %25
3 Final exam End of semester %50
*Assessment task (i.e., written test, oral test, oral presentation, group project, essay, etc.)
E. Student Academic Counseling and Support
Arrangements for availability of faculty and teaching staff for individual student consultations and academic advice:
All faculty members are required to give their office hours. Each member allocates at least 3 hours per week to give academic advice to students.
Students are required to complete the home assignments and attend regular lectures
F. Learning Resources and Facilities
1.Learning ResourcesRequired Textbooks
• Special Functions of mathematical physics and Chemistry, by Sneddon, I. N. (1996).
• Special functions: A graduate text by Richard Beals (2010)
• Handbook of special functions: derivatives, integrals, series and other formulas by Yury A. Brychkov (2008).
Essential References
Materials None
Electronic Materials Laptop, smart board, and projector.
Other Learning
Materials None
2. Facilities Required
Item Resources
Accommodation
(Classrooms, laboratories, demonstration rooms/labs, etc.)
Large classrooms that can accommodate at least 30 students.
Technology Resources
(AV, data show, Smart Board, software, etc.)
Data Show, Smart Board Other Resources
(Specify, e.g. if specific laboratory equipment is required, list requirements or
attach a list)
None
G. Course Quality Evaluation
Evaluation
Areas/Issues Evaluators Evaluation Methods
Effectiveness of teaching and Students Direct
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Evaluation
Areas/Issues Evaluators Evaluation Methods
assessment
Quality of learning resources Students Direct
Extent of achievement of course learning outcomes
Faculty Member Direct
Evaluation areas (e.g., Effectiveness of teaching and assessment, Extent of achievement of course learning outcomes, Quality of learning resources, etc.)
Evaluators (Students, Faculty, Program Leaders, Peer Reviewer, Others (specify) Assessment Methods (Direct, Indirect)
H. Specification Approval Data
Council / Committee Council of the Mathematics Department
Reference No.
Date
