CC BY
1
УДК 53
Rejepov A., student.
Yazdurdyyev M., teacher.
Oguzhan Egineering and Technology University of Turkmenistan.
Ashgabat, Turkmenistan.
SOLVING MATHEMATICAL PROBLEMS USING SPECIAL FUNCTIONS
Abstract
Special functions, including gamma functions, Bessel functions, and Legendre polynomials, are essential tools in solving advanced mathematical problems. They provide solutions to differential equations, integral evaluations, and optimization problems commonly encountered in applied mathematics, physics, and engineering. This article explores the theoretical foundations of special functions and their practical applications, with examples demonstrating their relevance and utility in mathematical problem-solving. Special functions are mathematical constructs that extend the capabilities of elementary functions to solve complex equations and problems. These functions, such as the gamma function (a generalization of factorials), Bessel functions (used in wave mechanics), and Legendre polynomials (applied in potential theory), serve as solutions to specific types of differential equations and boundary-value problems. The significance of special functions lies in their wideranging applications, from quantum mechanics to fluid dynamics and beyond. This paper investigates their properties and demonstrates how they can be used to address challenging mathematical problems.
Key words:
special functions, gamma functions, advanced mathematical problems, problem-solving, mechanics.
Theoretical Background
Special functions are categorized by their unique properties and their origin as solutions to classical differential equations:
Gamma Function (Г(х))
Defined as an extension of the factorial for real and complex numbers:
Applications: Probability theory, combinatorics, and complex analysis.
Bessel Functions
Solutions to Bessel's differential equation:
Applications: Vibrations, heat conduction, and wave equations.
Legendre Polynomials (P_n(x))
Solutions to Legendre's differential equation:
Applications: Potential theory, spherical harmonics, and celestial mechanics.
Applications of Special Functions
Special functions are indispensable for solving mathematical problems in various domains.
Solving Differential Equations
Differential equations such as the wave equation and the heat equation often require solutions involving Bessel or Legendre functions.
Example: Modeling vibrations in circular membranes uses Bessel functions to describe radial motion.
Evaluating Complex Integrals
The gamma function simplifies the evaluation of integrals with factorial-like growth. Optimization Problems
Special functions like the gamma function and beta function are employed in probability distributions for optimization tasks.
Practical Examples Using the Gamma Function
Evaluate the integral:
Solution: Using the gamma function
F
JQ
xne~*dx — Г(п + 1)
рос
/ xne~x dx - Г(п + 1) JO
Here, n=4:
Solving a Boundary Value Problem with Bessel Functions
A circular drum's vibration modes can be modeled using Bessel functions of the first kind, Jn(x), with boundary conditions at the drum's edge.
Conclusion Special functions provide a powerful toolkit for solving mathematical problems that extend beyond the capabilities of elementary functions. Their applications in differential equations, integrals, and optimization demonstrate their importance in theoretical and applied mathematics. Mastery of these functions enables mathematicians and scientists to approach complex problems effectively. References:
1. Abramowitz, M., & Stegun, I. A. (1965). Handbook of Mathematical Functions.
2. Andrews, L. C. (1998). Special Functions of Mathematics for Engineers.
3. Watson, G. N. (1995). A Treatise on the Theory of Bessel Functions.
© Rejepov A., Yazdurdyyev M., 2024
УДК 551.509.617
Будаев А.Х.
младший научный сотрудник, ФГБУ «ВГИ»
г. Нальчик, РФ
ЛАБОРАТОРНЫЕ ИССЛЕДОВАНИЯ КЛАСТЕРОВ ИЗ НАНОТРУБОК ОКСИДА ЦИНКА
Аннотация
В статье представлены результаты исследования кластеров оксида цинка. Исходным материалом является цинк с добавлением порошка графита в качестве катализатора. Приведено описание комплекса лабораторного оборудования и метода проведения экспериментов. Кластеры синтезированы в условиях высокой относительной влажности при отрицательных температурах. Согласно результатам исследования кластеры оксида цинка образуют различные модификации размерами от десятков нанометров до сотен микрометров и преимущественно состоят из нанотрубок.
