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.

∞ 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 ∞ 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: ∞ Which allows us to write the terms of the series : X n+s y = anx ∞ n n=0 X (−1) x2n+ν y = J (x) = (6) ∞ ν Γ(n + 1)Γ(n + ν + 1) 2 0 X n+s−1 n=0 y = an(n + s)x n=0 ∞ 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)|x=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 ∇2Ψ = 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 ∞. 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. The carrier 2 2 or sideband frequencies disappear when the modulation (19) by r , and setting equal to m one obtains: index (the peak frequency deviation divided by the mod- ulation frequency) is equal to the zero crossing of the r2 d2R r dR th + + k2r2 = m2 (22) function for the n sideband. R dr2 R dr 4 which when rearranged: Mathematica). Thus, we can express the full solution for the m = 2 scenario: d2R dR r2 + r + (k2r2 − m2) = 0 (23) dr2 dr r r 1 1 α2,1r imφ Ψ(r, φ) = J2( )e (29) which is of the same form as eq. (1), Bessel’s differential 0.510377 2π rb equation. The general solution to eq. (23) is of the form And since we’ve effectively set αm,n = rb, of eq. (8), and we write that general solution: r 1 r 1 Ψ(r, φ) = J (r)eimφ (30) 0.510377 2π 2 R(r) = AJm(kr) + BNm(kr), (24) where Jm(kr) and Nm(kr) are respectively the Bessel Admittedly, this solution is somewhat contrived, but it and Neumann functions of order m, and A and B are shows the importance of working with the zeroes of the constants to be determined via application of the bound- Bessel function to generate the particular solution using ary conditions. As the solution must be finite at x = 0, the boundary conditions. and as Nm(kr) → ∞ as x → 0, this means that the coef- ficient of Nm(kr) = B = 0, leaving R(r) to be expressed:

CONCLUSION R(r) = AJm(kr) (25)

Using the boundary condition that Ψ = 0 at the radius of The Bessel functions appear in many diverse scenar- the disk, we have the condition that Jm(krb) = 0, which ios, particularly situations involving cylindrical symme- implicitly requires the argument of Jm to be a zero of the try. The most difficult aspect of working with the Bessel Bessel function. As noted earlier, these zeroes must be function is first determining that they can be applied calculated individually in numerical fashion. Requiring through reduction of the system equation to Bessel’s dif- th that krb = αm,n, which is the nth zero of the m or- ferential or modified equation, and then manipulating der Bessel function[6], the energy of the system is solved boundary conditions with appropriate application of ze- q roes, and the coefficient values on the argument of the by expressing k in terms of α in (eq. k = 2mE ), m,n h¯2 Bessel function. This topic can be greatly expanded arriving at: upon, and the reader is highly encouraged to review the 2 2 applications and development presented in [2]. αm,n¯h Em,n = 2 h (26) 2mrb The full solution for Ψ is thus: REFERENCES

αm,nr imφ Ψm(r, φ) = AJm( )e (27) rb Because the Bessel function zeroes cannot be determined apriori, it is difficult to find a closed solution to express ∗ the normalization constant A. We select an order for m [email protected] [1] J. J. O’ Connor and R. E. F., Friedrich Wilhelm Bessel to continue with the determination of the normalization (School of Mathematics and Statistics University of St An- constant, and arbitrarily choose m = 2, which has a zero drews Scotland, 1997). at r = 5.13562, which we will set to be the radius of the [2] F. E. Relton, Applied Bessel Functions (Blackie and Son circle. Given the preceding, the normalization can be for Limited, 1946). the m = 2, n = 1 case can be found: [3] H. J. Arfken, G. B., Weber, Mathematical Methods for Physicists (Elsevier Academic Press, 2005). Z rboundary α r α r [4] E. W. Weissten, Modified Bessel Function of the Second A2J ( 2,1 )J ( 2,1 ) dr = 1 (28) 2 r 2 r Kind (Eric Weisstein’s World of Physics, 2008). 0 b b [5] M. Boas, Mathematical Methods for the Physical Sciences (Wiley, 1983). q 1 which for rboundary = 5.13562 =⇒ A = 0.510377 , (nu- [6] P. F. Newhouse and K. C. McGill, Journal of Chemical merical values obtained using numerical integration in Education 81, 424 (2004).