# Special Functions of Applied Mathematics (MATH483) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
Special Functions of Applied Mathematics MATH483 3 0 0 3 6
Pre-requisite Course(s)
Math 262 or Math 276 or Consent of the instructor
Course Language English N/A Bachelor’s Degree (First Cycle) Face To Face Lecture, Question and Answer, Team/Group. This course is intended primarily for the student of mathematics, physics or engineering who wishes to study the “special” functions in connection with the use of “hypergeometric functions”. The students who succeeded in this course; understand and apply the special functions understand the basic properties of the Hypergeometric series and hypergeometric differential equations understand the Gamma and Beta functions know the infitinite products understand the Bessel function and Bessel differential equation. Gamma and Beta functions; Pochhammer`s symbol; hypergeometric series; hypergeometric differential equation; generalized hypergeometric functions; Bessel function; the functional relationships, Bessel`s differential equation; orthogonality of Bessel functions.

### Weekly Subjects and Releated Preparation Studies

Week Subjects Preparation
1 Infinite Products: Introduction, Definition, Necessary condition for convergence, Absolute Convergence, Uniform Convergence pp. 1-5
2 The Gamma and Beta Functions: The Euler Constant, The Gamma function, The order symbols, Evaluation of certain infinite products, Euler's integral for the Gamma function pp. 8-15
3 The Beta Function, The factorial function (Pochhammer's symbol), Legendre's duplication formula, A summation formula due to Euler pp. 16-29
4 Asymptotic Series: Definition of an asymptotic expansion, Asymptotic expansions about infinity, Algebraic properties, Term-by-term integration, Uniqueness, Watson's Lemma pp. 33-41
5 The Hypergeometric Function (HGF) : The Function F(a, b; c; z), A simple integral form, Evaluation of F(a, b; c; 1), The contiguous function relations, The HG differential equation, Logarithmic solutions of the HG equation, pp. 45-65
6 F(a, b; c; z) as a function of its parameters, Elementary series manipulations, Simple transformations, Relation between functions of z and 1-z, A quadratic transformation, A theorem due to Kummer, Additional properties pp. 55-68
7 Midterm
8 Generalized HGF pp. 73. 83
9 Generalized HGF (continued) pp. 83-93
10 Generalized HGF (continued) pp. 93-102
11 Bessel Functions: Remarks, Definition, Bessel's differential equation, Differential recurrence relations pp. 108-111
12 A pure recurrence relation, A generating function, Bessel's Integral, Index Half an odd integer pp. 111-114
13 Modified Bessel functions, Neumann polynomials, Neumann series pp. 116-119
14 The Confluent HGF: Basic properties, Kummer's first formula, Kummer's second formula. pp. 123-125
15 Review
16 Final

### Sources

Course Book 1. Earl D. Rainville, Special Functions, MacMillan, New York, 1960. 2. Z. X. Wang, D. R. Guo, Special Functions, World Scientific, 1989 3. N. N. Lebedev, Special Functions and Their Aslications, Prentice-Hall, 1965

### Evaluation System

Attendance/Participation - -
Laboratory - -
Application - -
Field Work - -
Special Course Internship - -
Quizzes/Studio Critics - -
Homework Assignments 5 10
Presentation - -
Project - -
Report - -
Seminar - -
Midterms Exams/Midterms Jury 2 50
Final Exam/Final Jury 1 40
Toplam 8 100
 Percentage of Semester Work 60 40 100

### Course Category

Core Courses X

### The Relation Between Course Learning Competencies and Program Qualifications

# Program Qualifications / Competencies Level of Contribution
1 2 3 4 5
1 Has the ability to apply scientific knowledge gained in the undergraduate education and to expand and extend knowledge in the same or in a different area
2 Can apply gained knowledge and problem solving abilities in inter-disciplinary research
3 Has the ability to work independently within research area, to state the problem, to develop solution techniques, to solve the problem, to evaluate the obtained results and to apply them when necessary
4 Takes responsibility individually and as a team member to improve systematic approaches to produce solutions in unexpected complicated situations related to the area of study
5 Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework
6 Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility
7 Has the ability to follow recent developments within the area of research, to support research with scientific arguments and data, to communicate the information on the area of expertise in a systematically by means of written report and oral/visual presentation
8 To have an oral and written communication ability in at least one of the common foreign languages ("European Language Portfolio Global Scale", Level B2)
9 Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge
10 Follows scientific, cultural, and ethical criteria in collecting, interpreting and announcing data in the research area and has the ability to teach.
11 Has professional ethical consciousness and responsibility which takes into account the universal and social dimensions in the process of data collection, interpretation, implementation and declaration of results in mathematics and its applications.

Activities Number Duration (Hours) Total Workload
Course Hours (Including Exam Week: 16 x Total Hours)
Laboratory
Application
Special Course Internship
Field Work
Study Hours Out of Class 16 3 48
Presentation/Seminar Prepration
Project
Report
Homework Assignments 5 8 40
Quizzes/Studio Critics
Prepration of Midterm Exams/Midterm Jury 2 12 24
Prepration of Final Exams/Final Jury 1 18 18