Bessel functions of the first kind

Аннотация

This paper discusses the derivation of Bessel functions of the first kind using power series method and their properties. Additionally, the practical applications of these functions, their graphical analysis, and relationships with other special functions are examined. The research results serve to expand the theoretical and practical significance of Bessel functions. Keywords: Bessel functions, power series, mathematical analysis, physical phenomena, engineering problems, special functions. of the first kind using power series method and related Introduction: Bessel functions are widely used in mathematical properties. Furthermore, it analyzes problems with cylindrical symmetry, particularly in the these functions' applications in physical problems, their analysis of vibrations, heat transfer, and practical significance, and relationships with other electromagnetic waves. These functions have found special functions. The research results contribute to the their significant place in numerous physical broader application of Bessel functions and expand phenomena and engineering problems, especially in their importance in scientific research, engineering, the analysis of oscillations, heat conduction, and and physics. electromagnetic wave propagation. Moreover, their graphical analysis and application through Bessel functions are typically defined through mathematical formulas provide more effective differential equations and are derived using power solutions to physical problems. series. The Bessel function of the first kind is written as: The paper illustrates the derivation of Bessel functions 2n+v (−1)n  x n    2   J (x)= n  2 n!Г(v+n+1) n=0 where J (x)− is the Bessel function, x− is the argument, v− is the order index, and Г − is the Gamma n function. The derivation of Bessel functions is accomplished through the Bessel equation: ( ) x2y'' + xy' + x2 −v2 y =0 (1) Or American Journal of Applied Science and Technology 26 https://theusajournals.com/index.php/ajast American Journal of Applied Science and Technology (ISSN: 2771-2745) y'  v2  y'' + +  1−  y =0 (2) x x this equation is called the Bessel equation, where the constant v− is called the index of the equation. v0 y = xvz In the next step, we introduce the substitution into the equation . ( )' y'= xv z =vxv−1z + xv z' ( )' y''= vxv−1z + xv z' =v(v−1)xv−2 z +2vxv−1z'+ xv z'' y'  v2  y'' + + y =0 x x vxv−1z + xv z'  v2  v(v−1)xv−2 z +2vxv−1z'+ xv z''+ +  1−  xvz =0 Simplifying x x z this expression and to simplify the function, we obtain the equation : (2v+1) z'' + z'+ z =0 (2) x z = c xn We seek the solution of this equation in the form of: . n n=0 z'=c +2c x+3c x2 +...+(n+2)c xn+1 +... 1 2 3 x+2 z' c = 1 +2c +3c x+...+(n+2)c xn +... x x 2 3 x+2 z''= 2c +23c x2 +34c x2 +...+(n+1)(n+2)c xn +... 2 3 4 x+2 Substituting the resulting series into the equation, we obtain the following equality: 2v+1 c +[2c +(2v+1)2c +c ]+[23c +(2v+1)3c +c ]x+[34c + x 1 2 2 0 3 3 1 4 (2v+1)4c +c ]x2 +...+[(n+1)(n+2)c +(2v+1)(n+2)c +c ]xn +...=0 4 2 n+2 n+2 n According to the method of undetermined coefficients, we set all coefficients of powers of x equal to zero. This is because for the series sum to be zero, each coefficient must be zero. x0 for c = 0 (3) 1 (n+1)(n+2)c +(2v+1)(n+2)c +c =0 n=1,2,... n+2 n+2 n From this, we obtain the following recurrence formula: c c =− n n=0, 1, 2,... (4) n+2 (n+2)(n+2v+2) Based on(3) and (4): American Journal of Applied Science and Technology 27 https://theusajournals.com/index.php/ajast American Journal of Applied Science and Technology (ISSN: 2771-2745) c =c =c =...=c =...=0 1 3 5 2n−1 c c = − 0 2 2(2v+2) c c c =− 2 = 0 4 4(2v+4) 24(2v+2)(2v+4) ...................................................... c c = (−1)n 0 = 2n 24...2n(2v + 2)(2v + 4)...(2v + 2n) c = (−1)n 0 22n 12...n(v +1)(v + 2)...(v + n) Thus, the solution of the equation is represented by the series:   (−1)nx2n  z =c 1+ (5) 0 22n 12...n(v+1)(v+2)...(v+n) n=1 Usually, the constant c is chosen as: 0 1 c = 0 2vГ(n+1) The Gamma function is considered a generalization of the factorial: 12...n=n!= Г(n+1) (v+1)(v+ 2)...(v+ n)Г(v+1) =(v+ 2)(v+3)...(v+ n)Г(v+2) = =(v+ n)Г(v+ n) = Г(v+ n+1) We can write the function in the following form: 1  (−1)nx2n z = +  = 2v Г(n +1) 22n+v 12...n(v +1)(v + 2)...(v + n)Г(v +1) n=1  (−1)nx2n =  22n+v Г(n +1)Г(v + n +1) n=0 The solution of the equation consists of the function. We denote this function as J (x) . Therefore, v  (−1)n(x 2)2n+v J (x) =  v Г(n+1)Г(v+n+1) n=0 This function is called the Bessel function of the first kind J (x) has periodic oscillations similar to sine and v of index v or order v In some literature, these functions cosine, but these oscillations gradually increase and are also referred to as cylindrical functions. decrease in amplitude. The Bessel functions exhibit sinusoidal-like behavior. American Journal of Applied Science and Tec