CC BY
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2. Hyndman, R. J., & Athanasopoulos, G. (2018). Forecasting: Principles and Practice.
3. Shumway, R. H., & Stoffer, D. S. (2017). Time Series Analysis and Its Applications: With R Examples.
© Dowletgeldiyeva A., Kakyshov E., 2024
yflK 53
Gylyjova D., student.
Oguzhan Egineering and Technology University of Turkmenistan.
Ashgabat, Turkmenistan.
SPECIAL FUNCTIONS AND THEIR PROPERTIES
Abstract
Special functions play a vital role in mathematics, providing solutions to complex problems in fields such as physics, engineering, and computational science. These functions, including gamma functions, Bessel functions, and hypergeometric functions, exhibit unique properties that make them indispensable in mathematical analysis. This paper explores the key properties of special functions, their classifications, and their applications in solving real-world problems.
Keywords:
special functions, gamma function, Bessel functions, Legendre polynomials, hypergeometric functions, orthogonality, recurrence relations, differential equations.
Special functions extend the capabilities of elementary mathematical functions to address complex problems. Originating from the solutions of differential equations, these functions have been studied extensively for their unique properties and applications. For instance, the gamma function generalizes factorials, while Bessel functions describe wave propagation and vibrations.
This article investigates the definitions and fundamental properties of widely used special functions, emphasizing their significance in mathematical modeling and problem-solving.
Overview of Special Functions
Special functions are solutions to particular differential equations and are often defined in terms of integrals or series expansions. Some of the most common special functions include:
Gamma Function (r(x))
Defined as:
Key Properties: Recurrence relation:
Г(х + 1) - яГ(з:)
Connection to factorials:
Г(п) - (n - 1)!
Bessel Functions (Jn (x)
Solutions to Bessel's differential equation:
ai у" 4- xy + (iE — n2)y — (J
Key Properties:
Orthogonality: Bessel functions of different orders are orthogonal. Useful in cylindrical coordinate systems. Legendre Polynomials (Pn (x) Solutions to Legendre's differential equation:
Used in spherical harmonics and potential theory. Hypergeometric Functions (F(a,b;c;x) Generalized series expansions:
Key Properties:
Solutions to hypergeometric differential equations. Widely used in mathematical physics.
Key Properties of Special Functions Continuity and Differentiability Most special functions are continuous and differentiable, making them suitable for solving differential equations.
Orthogonality Functions like Bessel functions and Legendre polynomials exhibit orthogonality, aiding in decomposing functions into series expansions. Recurrence Relations Special functions often satisfy recurrence relations, simplifying computations.
Symmetry Certain special functions, such as Legendre polynomials, have symmetric or antisymmetric properties under transformations. Applications of Special Functions Physics Bessel functions describe heat conduction in cylindrical objects and electromagnetic wave propagation. Engineering Hypergeometric functions are used in control systems and signal processing. Mathematics Gamma functions simplify combinatorial and integral calculations.
Conclusion Special functions are indispensable in solving advanced mathematical and physical problems. Their unique properties, such as orthogonality, recurrence relations, and symmetry, provide powerful tools for tackling differential equations and complex integrals. Understanding these functions enhances problem-solving capabilities in theoretical and applied sciences References
1. Abramowitz, M., & Stegun, I. A. (1965). Handbook of Mathematical Functions.
2. Andrews, G. E., Askey, R., & Roy, R. (1999). Special Functions.
3. Watson, G. N. (1995). A Treatise on the Theory of Bessel Functions.
© Gylyjova D., 2024
УДК 53
Muhammetnyyazova M., student
Pudakov B., teacher.
Oguzhan Egineering and Technology University of Turkmenistan.
Ashgabat, Turkmenistan.
INTERACTIVE METHODS OF TEACHING MATHEMATICS IN PRIMARY SCHOOL
Abstract
Interactive teaching methods have become essential in primary school education, especially in
