Bessel Functions and Their Applications Jennifer Niedziela∗ University of Tennessee - Knoxville (Dated: October 29, 2008) Bessel functions are a series of solutions to a second order differential equation that arise in many diverse situations. This paper derives the Bessel functions through use of a series solution to a differential equation, develops the different kinds of Bessel functions, and explores the topic of zeroes. Finally, Bessel functions are found to be the solution to the Schroedinger equation in a situation with cylindrical symmetry. 1 INTRODUCTION 0 0 X 2 n+s−1 (xy ) = an(n + s) x n=0 While special types of what would later be known as 1 0 0 X 2 n+s Bessel functions were studied by Euler, Lagrange, and (xy ) = an(n + s) x the Bernoullis, the Bessel functions were first used by F. n=0 W. Bessel to describe three body motion, with the Bessel functions appearing in the series expansion on planetary When the coefficients of the powers of x are organized, we perturbation [1]. This paper presents the Bessel func- find that the coefficient on xs gives the indicial equation tions as arising from the solution of a differential equa- s2 − ν2 = 0; =) s = ±ν, and we develop the general tion; an equation which appears frequently in applica- formula for the coefficient on the xs+n term: tions and solutions to physical situations [2] [3]. Fre- quently, the key to solving such problems is to recognize an−2 the form of this equation, thus allowing employment of an = − (3) (n + s)2 − ν2 the Bessel functions as solutions. The subject of Bessel Functions and applications is a very rich subject; never- In the case s = ν: theless, due to space and time restrictions and in the in- terest of studying applications, the Bessel function shall an−2 an = − (4) be presented as a series solution to a second order dif- n(n + 2ν) ferential equation, and then applied to a situation with cylindrical symmetry. Appropriate development of ze- and since a1 = 0, an = 0 for all n = odd integers. Coeffi- roes, modified Bessel functions, and the application of cients for even powers of n are found: boundary conditions will be briefly discussed. a a = − 2n−2 (5) 2n 22n(n + ν) THE BESSEL EQUATION Recalling that for the gamma function: Bessel's equation is a second order differential equation Γ(ν + 2) = (ν + 1)Γ(ν + 1); of the form Γ(ν + 3) = (ν + 2)Γ(ν + 2) = (ν + 2)(ν + 1)Γ(ν + 1); x2y00 + xy0 + (x2 − ν2)y = 0 (1) we can write the coefficients: By re-writing this equation as: a0 Γ(1 + ν) a2 = − 2 = − 2 x(xy0)0 + (x2 − ν2)y = 0 (2) 2 (1 + ν) 2 Γ(2 + ν) a0Γ(1 + ν) and employing the use of a generalized power series, we a2n = − n!22nΓ(n + 1 + ν) re-write the terms of (2) in terms of the series: 1 Which allows us to write the terms of the series : X n+s y = anx 1 n n=0 X (−1) x2n+ν y = J (x) = (6) 1 ν Γ(n + 1)Γ(n + ν + 1) 2 0 X n+s−1 n=0 y = an(n + s)x n=0 1 Where Jν (x) is the Bessel function of the first kind, order 0 X n+s ν. The first five Bessel functions of this kind are shown xy = an(n + s)x n=0 in figure 1. 2 g the gravitational constant of acceleration. Eq. (9) is a form of eq. (1), and solution is: 1 1 u = AJ0(2kz 2 ) + BY0(2kz 2 ); (10) where the A and B are determined by the boundary con- ditions. Modified Bessel Functions Modified Bessel functions are found as solutions to the modified Bessel equation FIG. 1: The Bessel Functions of orders ν = 0 to ν = 5 x2y00 + xy0 − (x2 − ν2)y = 0 (11) DIFFERENT ORDERS OF BESSEL FUNCTIONS which transforms into eq. (1) when x is replaced with ix. However, this leaves the general solution of eq. (1) In the preceding section, the form of Bessel functions a complex function of x. To avoid dealing with complex were obtained are known as "Bessel functions of the first solutions in practical applications [2], the solutions to kind."[3]. Different kinds of Bessel functions are obtained (11) are expressed in the form: with negative values of ν, or with complex arguments. This section briefly explores these different kinds of func- νπi iπ I (x) = e 2 J (xe 2 ) (12) tions ν ν The Iν (x) are a set of functions known as the modified Neumann Functions Bessel functions of the first kind. The general solution of the modified Bessel function is expressed as a combina- tion of I (x) and a function I (x): Bessel functions of the second kind are known as Neu- ν −ν mann functions, and are developed as a linear combina- y = AI−ν (x) − BIν (x) (13) tion of Bessel functions of the first order described: where again A and B are determined from the boundary cosνπJ (x) − J (x) conditions. N (x) = ν −ν (7) ν sinνπ A solution for non-integer orders of ν is found: For integral values of ν, the expression of Nν (x) has an π I−ν (x) − Iν (x) Kν (x) = (14) indeterminate form, and Nν (x)jx=0 = ±∞. Nevertheless 2 sinνπ the limit of this function for x 6= 0, the expression for Nν is valid for any value of ν, allowing the general solution The functions Kν (x) are known as modified Bessel func- to Bessel's equation to be written: tions of the second kind. A plot of the Neumann Func- tions (Nν (x)) and Modified Bessel functions (Iν (x))is y = AJν (x) + BNν (x) (8) shown in figure (2). A plot of the Modified Second Kind functions (Kn(x)) is shown in fig. (3). with A and B as arbitrary constants determined from Modified Bessel functions appear less frequently in ap- boundary conditions. plications, but can be found in transmission line studies, Bessel functions of the first and second kind are the non-uniform beams, and the statistical treatment of a most commonly found forms of the Bessel function in ap- relativistic gas in statistical mechanics. plications. Many applications in hydrodynamics, elastic- ity, and oscillatory systems have solutions that are based on the Bessel functions. One such example is that of a Zeroes of Bessel Functions uniform density chain fixed at one end undergoing small oscillations. The differential equation of this situation is: The zeroes of Bessel functions are of great importance d2u 1 du k2u in applications [5]. The zeroes, or roots, of the Bessel + + = 0 (9) dz2 z dz z functions are the values of x where value of the Bessel function goes to zero (Jν (x) = 0). Frequently, the ze- 2 p2 where z references a point on the chain, k = g , with p roes are found in tabulated formats, as they must the be as the frequency of small oscillations at that point, and numerically evaluated [5]. Bessel function's of the first 3 APPLICATION - SOLUTION TO SCHROEDINGER'S EQUATION IN A CYLINDRICAL WELL Consider a particle of mass m placed into a two- dimensional potential well, where the potential is zero inside of the radius of the disk, infinite outside of the radius of the disk. In polar coordinates using r, φ as representatives of the system, the Laplacian is written: 1 @ @Ψ 1 @2Ψ r2Ψ = r + : (15) r @r @r r2 @φ2 Which in the Schroedinger equation presents: FIG. 2: The Neumann Functions (black) and the Modified Bessel Functions (blue) for integer orders ν = 0 to ν = 5 ¯h2 1 @ @Ψ 1 @2Ψ − r + = EΨ: (16) 2m r @r @r r2 @φ2 Using the method of separation of variables with a pro- posed solution Ψ = R(r)T (φ) in (16), produces ¯h2 1 @ @R(r) 1 @2T − T (φ) r + R(r) = ER(r)T (φ) 2m R(r) @r @r r2 @φ2 (17) and then dividing by Ψ: 1 1 @ @R 1 1 @2T −2mE r + = (18) R r @r @r r2 T @φ2 ¯h2 2mE 2 2 setting h¯2 = k and multiplying through by r pro- duces 2 r d dR 2 2 1 d T FIG. 3: The Modified Bessel Functions of the second kind for r + k r + 2 = 0 (19) orders ν = 0 to ν = 5 [4] R dr dr T dφ which is fully separated in r and φ. To solve, the φ dependent portion is set to −m2, yielding the harmonic oscillator equation in T (φ), which presents the solution: and second kind have an infinite number of zeros as the T (φ) = Aeimφ (20) value of x goes to 1. The zeroes of the functions can be seen in the crossing points of the graphs in figure ( 1), Where A is a constant determined via proper normaliza- and figure ( 2). The modified Bessel functions of the first tion in φ: kind (Iν (x)) have only one zero at the point x = 0, and the modified Bessel equations of the second kind (Kν (x)) functions do not have zeroes. Z 2π r 1 A2T (φ)T (φ) dφ = 1 =) A = (21) 0 2π Bessel function zeros are exploited in frequency modu- lated (FM) radio transmission. FM transmission is math- q Leaving the φ dependent portion T (φ) = 1 eimφ ematically represented by a harmonic distribution of a 2π sine wave carrier modulated by a sine wave signal which Working now with the r dependent portion of the sep- arated equation, multiplying the r dependent portion of can be represented with Bessel Functions.