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Bessel functions
SPECIAL FUNCTIONS Using (18.65) and setting x = cos θ immediately gives a rearrangement of the required result (18.69). Similarly, adding the plus and minus cases of result (18.68) gives sin(n + 1)θ + sin(n − 1)θ = 2 sin nθ cos θ. Dividing through on both sides by sin θ and using (18.66) yields (18.70). The recurrence relations (18.69) and (18.70) are extremely useful in the practical computation of Chebyshev polynomials. For example, given the values of T0 (x) and T1 (x) at some point x, the result (18.69) may be used iteratively to obtain the value of any Tn (x) at that point; similarly, (18.70) may be used to calculate the value of any Un (x) at some point x, given the values of U0 (x) and U1 (x) at that point. Further recurrence relations satisfied by the Chebyshev polynomials are Tn (x) = Un (x) − xUn−1 (x), (1 − x2 )Un (x) = xTn+1 (x) − Tn+2 (x), (18.71) (18.72) which establish useful relationships between the two sets of polynomials Tn (x) and Un (x). The relation (18.71) follows immediately from (18.68), whereas (18.72) follows from (18.67), with n replaced by n + 1, on noting that sin2 θ = 1 − x2 . Additional useful results concerning the derivatives of Chebyshev polynomials may be obtained from (18.65) and (18.66), as illustrated in the following example. Show that Tn (x) = nUn−1 (x), (1 − x2 )Un (x) = xUn (x) − (n + 1)Tn+1 (x). These results are most easily derived from the expressions (18.65) and (18.66) by noting that d/dx = (−1/ sin θ) d/dθ. Thus, Tn (x) = − Similarly, we find 1 d(cos nθ) n sin nθ = = nUn−1 (x). sin θ dθ sin θ sin(n + 1)θ cos θ (n + 1) cos(n + 1)θ − sin3 θ sin2 θ x Un (x) (n + 1)Tn+1 (x) = − , 1 − x2 1 − x2 which rearranges immediately to yield the stated result. Un (x) = − 1 d sin θ dθ sin(n + 1)θ sin θ = 18.5 Bessel functions Bessel’s equation has the form x2 y + xy + (x2 − ν 2 )y = 0, (18.73) which has a regular singular point at x = 0 and an essential singularity at x = ∞. The parameter ν is a given number, which we may take as ≥ 0 with no loss of 602 18.5 BESSEL FUNCTIONS generality. The equation arises from physical situations similar to those involving Legendre’s equation but when cylindrical, rather than spherical, polar coordinates are employed. The variable x in Bessel’s equation is usually a multiple of a radial distance and therefore ranges from 0 to ∞. We shall seek solutions to Bessel’s equation in the form of infinite series. Writing (18.73) in the standard form used in chapter 16, we have 1 ν2 (18.74) y + y + 1 − 2 y = 0. x x By inspection, x = 0 is a regular singular point; hence we try a solution of the n form y = xσ ∞ n=0 an x . Substituting this into (18.74) and multiplying the resulting equation by x2−σ , we obtain ∞ ∞ (σ + n)(σ + n − 1) + (σ + n) − ν 2 an xn + an xn+2 = 0, n=0 n=0 which simplifies to ∞ ∞ (σ + n)2 − ν 2 an xn + an xn+2 = 0. n=0 n=0 0 Considering the coefficient of x , we obtain the indicial equation σ 2 − ν 2 = 0, and so σ = ±ν. For coefficients of higher powers of x we find (σ + 1)2 − ν 2 a1 = 0, (σ + n)2 − ν 2 an + an−2 = 0 for n ≥ 2. (18.75) (18.76) Substituting σ = ±ν into (18.75) and (18.76), we obtain the recurrence relations (1 ± 2ν)a1 = 0, n(n ± 2ν)an + an−2 = 0 (18.77) for n ≥ 2. (18.78) We consider now the form of the general solution to Bessel’s equation (18.73) for two cases: the case for which ν is not an integer and that for which it is (including zero). 18.5.1 Bessel functions for non-integer ν If ν is a non-integer then, in general, the two roots of the indicial equation, σ1 = ν and σ2 = −ν, will not differ by an integer, and we may obtain two linearly independent solutions in the form of Frobenius series. Special considerations do arise, however, when ν = m/2 for m = 1, 3, 5, . . . , and σ1 − σ2 = 2ν = m is an (odd positive) integer. When this happens, we may always obtain a solution in 603 SPECIAL FUNCTIONS the form of a Frobenius series corresponding to the larger root, σ1 = ν = m/2, as described above. However, for the smaller root, σ2 = −ν = −m/2, we must determine whether a second Frobenius series solution is possible by examining the recurrence relation (18.78), which reads n(n − m)an + an−2 = 0 for n ≥ 2. Since m is an odd positive integer in this case, we can use this recurrence relation (starting with a0 = 0) to calculate a2 , a4 , a6 , . . . in the knowledge that all these terms will remain finite. It is possible in this case, therefore, to find a second solution in the form of a Frobenius series, one that corresponds to the smaller root σ2 . Thus, in general, for non-integer ν we have from (18.77) and (18.78) an = − = 0 1 an−2 n(n ± 2ν) for n = 2, 4, 6, . . . , for n = 1, 3, 5, . . . . Setting a0 = 1 in each case, we obtain the two solutions x4 x2 + − ··· . y±ν (x) = x±ν 1 − 2(2 ± 2ν) 2 × 4(2 ± 2ν)(4 ± 2ν) It is customary, however, to set a0 = 1 , 2±ν Γ(1 ± ν) where Γ(x) is the gamma function, described in subsection 18.12.1; it may be regarded as the generalisation of the factorial function to non-integer and/or negative arguments.§ The two solutions of (18.73) are then written as Jν (x) and J−ν (x), where x ν 1 x 4 1 1 1 x 2 Jν (x) = + − ··· 1− Γ(ν + 1) 2 ν+1 2 (ν + 1)(ν + 2) 2! 2 ∞ n ν+2n x (−1) = ; (18.79) n!Γ(ν + n + 1) 2 n=0 replacing ν by −ν gives J−ν (x). The functions Jν (x) and J−ν (x) are called Bessel functions of the first kind, of order ν. Since the first term of each series is a finite non-zero multiple of xν and x−ν , respectively, if ν is not an integer then Jν (x) and J−ν (x) are linearly independent. This may be confirmed by calculating the Wronskian of these two functions. Therefore, for non-integer ν the general solution of Bessel’s equation (18.73) is given by y(x) = c1 Jν (x) + c2 J−ν (x). § In particular, Γ(n + 1) = n! for n = 0, 1, 2,. . ., and Γ(n) is infinite if n is any integer ≤ 0. 604 (18.80) 18.5 BESSEL FUNCTIONS We note that Bessel functions of half-integer order are expressible in closed form in terms of trigonometric functions, as illustrated in the following example. Find the general solution of x2 y + xy + (x2 − 14 )y = 0. This is Bessel’s equation with ν = 1/2, so from (18.80) the general solution is simply y(x) = c1 J1/2 (x) + c2 J−1/2 (x). However, Bessel functions of half-integral order can be expressed in terms of trigonometric functions. To show this, we note from (18.79) that J±1/2 (x) = x±1/2 ∞ n=0 (−1)n x2n . 22n±1/2 n!Γ(1 + n ± 12 ) Using the fact that Γ(x + 1) = xΓ(x) and Γ( 21 ) = J1/2 (x) = = = ( 12 x)1/2 Γ( 23 ) ( 12 x)1/2 √ ( 12 ) π ( 12 x)1/2 √ ( 12 ) π √ π, we find that, for ν = 1/2, ( 1 x)5/2 ( 1 x)9/2 − 2 5 + 2 7 − ··· 1!Γ( 2 ) 2!Γ( 2 ) ( 12 x)5/2 ( 12 x)9/2 − √ + √ − ··· 1!( 32 )( 12 ) π 2!( 52 )( 32 )( 21 ) π ( 1 x)1/2 sin x x2 x4 + − · · · = 21 √ = 1− 3! 5! (2) π x 2 sin x, πx whereas for ν = −1/2 we obtain ( 1 x)3/2 ( 1 x)7/2 ( 21 x)−1/2 − 2 3 + 2 5 − ··· 1 Γ( 2 ) 1!Γ( 2 ) 2!Γ( 2 ) ( 21 x)−1/2 x2 2 x4 1− = √ + − ··· = cos x. 2! 4! πx π J−1/2 (x) = Therefore the general solution we require is y(x) = c1 J1/2 (x) + c2 J−1/2 (x) = c1 2 sin x + c2 πx 2 cos x. πx 18.5.2 Bessel functions for integer ν The definition of the Bessel function Jν (x) given in (18.79) is, of course, valid for all values of ν, but, as we shall see, in the case of integer ν the general solution of Bessel’s equation cannot be written in the form (18.80). Firstly, let us consider the case ν = 0, so that the two solutions to the indicial equation are equal, and we clearly obtain only one solution in the form of a Frobenius series. From (18.79), 605 SPECIAL FUNCTIONS 1.5 1 J0 J1 J2 0.5 2 4 6 8 10 x −0.5 Figure 18.5 The first three integer-order Bessel functions of the first kind. this is given by J0 (x) = ∞ n=0 =1− (−1)n x2n 22n n!Γ(1 + n) x4 x6 x2 + 2 2 − 2 2 2 + ··· . 22 2 4 2 46 In general, however, if ν is a positive integer then the solutions of the indicial equation differ by an integer. For the larger root, σ1 = ν, we may find a solution Jν (x), for ν = 1, 2, 3, . . . , in the form of the Frobenius series given by (18.79). Graphs of J0 (x), J1 (x) and J2 (x) are plotted in figure 18.5 for real x. For the smaller root, σ2 = −ν, however, the recurrence relation (18.78) becomes n(n − m)an + an−2 = 0 for n ≥ 2, where m = 2ν is now an even positive integer, i.e. m = 2, 4, 6, . . . . Starting with a0 = 0 we may then calculate a2 , a4 , a6 , . . . , but we see that when n = m the coefficient an is formally infinite, and the method fails to produce a second solution in the form of a Frobenius series. In fact, by replacing ν by −ν in the definition of Jν (x) given in (18.79), it can be shown that, for integer ν, J−ν (x) = (−1)ν Jν (x), 606 18.5 BESSEL FUNCTIONS and hence that Jν (x) and J−ν (x) are linearly dependent. So, in this case, we cannot write the general solution to Bessel’s equation in the form (18.80). One therefore defines the function Yν (x) = Jν (x) cos νπ − J−ν (x) , sin νπ (18.81) which is called a Bessel function of the second kind of order ν (or, occasionally, a Weber or Neumann function). As Bessel’s equation is linear, Yν (x) is clearly a solution, since it is just the weighted sum of Bessel functions of the first kind. Furthermore, for non-integer ν it is clear that Yν (x) is linearly independent of Jν (x). It may also be shown that the Wronskian of Jν (x) and Yν (x) is non-zero for all values of ν. Hence Jν (x) and Yν (x) always constitute a pair of independent solutions. If n is an integer, show that Yn+1/2 (x) = (−1)n+1 J−n−1/2 (x). From (18.81), we have Yn+1/2 (x) = Jn+1/2 (x) cos(n + 12 )π − J−n−1/2 (x) . sin(n + 12 )π If n is an integer, cos(n + 12 )π = 0 and sin(n + 12 )π = (−1)n , and so we immediately obtain Yn+1/2 (x) = (−1)n+1 J−n−1/2 (x), as required. The expression (18.81) becomes an indeterminate form 0/0 when ν is an integer, however. This is so because for integer ν we have cos νπ = (−1)ν and J−ν (x) = (−1)ν Jν (x). Nevertheless, this indeterminate form can be evaluated using l’Hôpital’s rule (see chapter 4). Therefore, for integer ν, we set Jµ (x) cos µπ − J−µ (x) Yν (x) = lim , (18.82) µ→ν sin µπ which gives a linearly independent second solution for this case. Thus, we may write the general solution of Bessel’s equation, valid for all ν, as y(x) = c1 Jν (x) + c2 Yν (x). (18.83) The functions Y0 (x), Y1 (x) and Y2 (x) are plotted in figure 18.6 Finally, we note that, in some applications, it is convenient to work with complex linear combinations of Bessel functions of the first and second kinds given by Hν(2) (x) = Jν (x) − iYν (x); Hν(1) (x) = Jν (x) + iYν (x), these are called, respectively, Hankel functions of the first and second kind of order ν. 607 SPECIAL FUNCTIONS 1 Y0 0.5 2 Y1 Y2 4 6 8 10 x −0.5 −1 Figure 18.6 The first three integer-order Bessel functions of the second kind. 18.5.3 Properties of Bessel functions Jν (x) In physical applications, we often require that the solution is regular at x = 0, but, from its definition (18.81) or (18.82), it is clear that Yν (x) is singular at the origin, and so in such physical situations the coefficient c2 in (18.83) must be set to zero; the solution is then simply some multiple of Jν (x). These Bessel functions of the first kind have various useful properties that are worthy of further discussion. Unless otherwise stated, the results presented in this section apply to Bessel functions Jν (x) of integer and non-integer order. Mutual orthogonality In section 17.4, we noted that Bessel’s equation (18.73) could be put into conventional Sturm–Liouville form with p = x, q = −ν 2 /x, λ = α2 and ρ = x, provided αx is the argument of y. From the form of p, we see that there is no natural interval over which one would expect the solutions of Bessel’s equation corresponding to different eigenvalues λ (but fixed ν) to be automatically orthogonal. Nevertheless, provided the Bessel functions satisfied appropriate boundary conditions, we would expect them to obey an orthogonality relationship over some interval [a, b] of the form b xJν (αx)Jν (βx) dx = 0 for α = β. (18.84) a 608 18.5 BESSEL FUNCTIONS To determine the required boundary conditions for this result to hold, let us consider the functions f(x) = Jν (αx) and g(x) = Jν (βx), which, as will be proved below, respectively satisfy the equations x2 f + xf + (α2 x2 − ν 2 )f = 0, (18.85) x2 g + xg + (β 2 x2 − ν 2 )g = 0. (18.86) Show that f(x) = Jν (αx) satisfies (18.85). If f(x) = Jν (αx) and we write w = αx, then df d2 Jν (w) dJν (w) d2 f = α2 . =α and dx dw dx2 dw 2 When these expressions are substituted into (18.85), its LHS becomes dJν (w) d2 Jν (w) + xα +(α2 x2 − ν 2 )Jν (w) dw 2 dw dJν (w) d2 Jν (w) = w2 +w + (w 2 − ν 2 )Jν (w). dw 2 dw But, from Bessel’s equation itself, this final expression is equal to zero, thus verifying that f(x) does satisfy (18.85). x 2 α2 Now multiplying (18.86) by f(x) and (18.85) by g(x) and subtracting them gives d [x(fg − gf )] = (α2 − β 2 )xfg, dx where we have used the fact that d [x(fg − gf )] = x(fg − gf ) + (fg − gf ). dx By integrating (18.87) over any given range x = a to x = b, we obtain b b 1 xf(x)g (x) − xg(x)f (x) , xf(x)g(x) dx = 2 2 a α −β a (18.87) which, on setting f(x) = Jν (αx) and g(x) = Jν (βx), becomes b b 1 βxJ xJν (αx)Jν (βx) dx = 2 (αx)J (βx) − αxJ (βx)J (αx) . ν ν ν ν a α − β2 a (18.88) If α = β, and the interval [a, b] is such that the expression on the RHS of (18.88) equals zero, then we obtain the orthogonality condition (18.84). This happens, for example, if Jν (αx) and Jν (βx) vanish at x = a and x = b, or if Jν (αx) and Jν (βx) vanish at x = a and x = b, or for many more general conditions. It should be noted that the boundary term is automatically zero at the point x = 0, as one might expect from the fact that the Sturm–Liouville form of Bessel’s equation has p(x) = x. If α = β, the RHS of (18.88) takes the indeterminant form 0/0. This may be 609 SPECIAL FUNCTIONS evaluated using l’Hôpital’s rule, or alternatively we may calculate the relevant integral directly. Evaluate the integral b Jν2 (αx)x dx. a Ignoring the integration limits for the moment, 1 Jν2 (αx)x dx = 2 Jν2 (u)u du, α where u = αx. Integrating by parts yields I = Jν2 (u)u du = 12 u2 Jν2 (u) − Jν (u)Jν (u)u2 du. Now Bessel’s equation (18.73) can be rearranged as u2 Jν (u) = ν 2 Jν (u) − uJν (u) − u2 Jν (u), which, on substitution into the expression for I, gives I = 12 u2 Jν2 (u) − Jν (u)[ν 2 Jν (u) − uJν (u) − u2 Jν (u)] du = 12 u2 Jν2 (u) − 12 ν 2 Jν2 (u) + 12 u2 [Jν (u)]2 + c. Since u = αx, the required integral is given by b b 1 ν2 x2 − 2 Jν2 (αx) + x2 [Jν (αx)]2 , Jν2 (αx)x dx = 2 α a a (18.89) which gives the normalisation condition for Bessel functions of the first kind. Since the Bessel functions Jν (x) possess the orthogonality property (18.88), we may expand any reasonable function f(x), i.e. one obeying the Dirichlet conditions discussed in chapter 12, in the interval 0 ≤ x ≤ b as a sum of Bessel functions of a given (non-negative) order ν, f(x) = ∞ cn Jν (αn x), (18.90) n=0 provided that the αn are chosen such that Jν (αn b) = 0. The coefficients cn are then given by b 2 f(x)Jν (αn x)x dx. (18.91) cn = 2 2 b Jν+1 (αn b) 0 The interval is taken to be 0 ≤ x ≤ b, as then one need only ensure that the appropriate boundary condition is satisfied at x = b, since the boundary condition at x = 0 is met automatically. 610 18.5 BESSEL FUNCTIONS Prove the expression (18.91). If we multiply (18.90) by xJν (αm x) and integrate from x = 0 to x = b then we obtain b b ∞ xJν (αm x)f(x) dx = cn xJν (αm x)Jν (αn x) dx 0 0 n=0 b Jν2 (αm x)x dx = cm 0 2 = 12 cm b2 J ν (αm b) = 12 cm b2 Jν+1 (αm b), 2 where in the last two lines we have used (18.88) with αm = α = β = αn , (18.89), the fact that Jν (αm b) = 0 and (18.95), which is proved below. Recurrence relations The recurrence relations enjoyed by Bessel functions of the first kind, Jν (x), can be derived directly from the power series definition (18.79). Prove the recurrence relation d ν [x Jν (x)] = xν Jν−1 (x). dx (18.92) From the power series definition (18.79) of Jν (x) we obtain ∞ (−1)n x2ν+2n d ν d [x Jν (x)] = ν+2n dx dx n=0 2 n!Γ(ν + n + 1) = ∞ n=0 = xν (−1)n x2ν+2n−1 2ν+2n−1 n!Γ(ν + n) ∞ n=0 (−1)n x(ν−1)+2n = xν Jν−1 (x). 2(ν−1)+2n n!Γ((ν − 1) + n + 1) It may similarly be shown that d −ν [x Jν (x)] = −x−ν Jν+1 (x). (18.93) dx From (18.92) and (18.93) the remaining recurrence relations may be derived. Expanding out the derivative on the LHS of (18.92) and dividing through by xν−1 , we obtain the relation xJν (x) + νJν (x) = xJν−1 (x). (18.94) Similarly, by expanding out the derivative on the LHS of (18.93), and multiplying through by xν+1 , we find xJν (x) − νJν (x) = −xJν+1 (x). (18.95) Adding (18.94) and (18.95) and dividing through by x gives Jν−1 (x) − Jν+1 (x) = 2Jν (x). 611 (18.96) SPECIAL FUNCTIONS Finally, subtracting (18.95) from (18.94) and dividing by x gives Jν−1 (x) + Jν+1 (x) = 2ν Jν (x). x (18.97) Given that J1/2 (x) = (2/πx)1/2 sin x and that J−1/2 (x) = (2/πx)1/2 cos x, express J3/2 (x) and J−3/2 (x) in terms of trigonometric functions. From (18.95) we have 1 (x) J1/2 (x) − J1/2 2x 1/2 1/2 1/2 1 1 2 2 2 sin x − cos x + sin x = 2x πx πx 2x πx 1/2 1 2 = sin x − cos x . πx x J3/2 (x) = Similarly, from (18.94) we have 1 (x) J−1/2 (x) + J−1/2 2x 1/2 1/2 1/2 1 1 2 2 2 cos x − sin x − cos x =− 2x πx πx 2x πx 1/2 1 2 − cos x − sin x . = πx x J−3/2 (x) = − We see that, by repeated use of these recurrence relations, all Bessel functions Jν (x) of halfinteger order may be expressed in terms of trigonometric functions. From their definition (18.81), Bessel functions of the second kind, Yν (x), of half-integer order can be similarly expressed. Finally, we note that the relations (18.92) and (18.93) may be rewritten in integral form as xν Jν−1 (x) dx = xν Jν (x), x−ν Jν+1 (x) dx = −x−ν Jν (x). If ν is an integer, the recurrence relations of this section may be proved using the generating function for Bessel functions discussed below. It may be shown that Bessel functions of the second kind, Yν (x), also satisfy the recurrence relations derived above. Generating function The Bessel functions Jν (x), where ν = n is an integer, can be described by a generating function in a way similar to that discussed for Legendre polynomials 612 18.5 BESSEL FUNCTIONS in subsection 18.1.2. The generating function for Bessel functions of integer order is given by ∞ x 1 Jn (x)hn . (18.98) h− = G(x, h) = exp 2 h n=−∞ By expanding the exponential as a power series, it is straightfoward to verify that the functions Jn (x) defined by (18.98) are indeed Bessel functions of the first kind, as given by (18.79). The generating function (18.98) is useful for finding, for Bessel functions of integer order, properties that can often be extended to the non-integer case. In particular, the Bessel function recurrence relations may be derived. Use the generating function to prove, for integer ν, the recurrence relation (18.97), i.e. Jν−1 (x) + Jν+1 (x) = 2ν Jν (x). x Differentiating G(x, h) with respect to h we obtain ∞ 1 ∂G(x, h) x 1 + 2 G(x, h) = nJn (x)hn−1 , = ∂h 2 h n=−∞ which can be written using (18.98) again as ∞ ∞ 1 x 1+ 2 Jn (x)hn = nJn (x)hn−1 . 2 h n=−∞ n=−∞ Equating coefficients of hn we obtain x [Jn (x) + Jn+2 (x)] = (n + 1)Jn+1 (x), 2 which, on replacing n by ν − 1, gives the required recurrence relation. Integral representations The generating function (18.98) is also useful for deriving integral representations of Bessel functions of integer order. Show that for integer n the Bessel function Jn (x) is given by 1 π cos(nθ − x sin θ) dθ. Jn (x) = π 0 (18.99) By expanding out the cosine term in the integrand in (18.99) we obtain the integral 1 π [cos(x sin θ) cos nθ + sin(x sin θ) sin nθ] dθ. I= (18.100) π 0 Now, we may express cos(x sin θ) and sin(x sin θ) in terms of Bessel functions by setting h = exp iθ in (18.98) to give ∞ x exp (exp iθ − exp(−iθ)) = exp (ix sin θ) = Jm (x) exp imθ. 2 m=−∞ 613