Mathematical Methods for Physics and Engineering - Matematica.NET

18.5 BESSEL FUNCTIONSgenerality. The equation arises from physical situations similar to those involvingLegendre’s equation but when cylindrical, rather than spherical, polar coordinatesare employed. The variable x in Bessel’s equation is usually a multiple of a radialdistance 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 havey ′′ + 1 )x y′ +(1 − ν2x 2 y =0. (18.74)By inspection, x = 0 is a regular singular point; hence we try a solution of theform y = x σ ∑ ∞n=0 a nx n . Substituting this into (18.74) and multiplying the resultingequation by x 2−σ ,weobtain∞∑ [(σ + n)(σ + n − 1) + (σ + n) − ν2 ] ∞∑a n x n + a n x n+2 =0,n=0which simplifies ton=0n=0∞∑ [(σ + n) 2 − ν 2] ∞∑a n x n + a n x n+2 =0.Considering the coefficient of x 0 , we obtain the indicial equationσ 2 − ν 2 =0,and so σ = ±ν. For coefficients of higher powers of x we find[(σ +1) 2 − ν 2] a 1 =0, (18.75)[(σ + n) 2 − ν 2] a n + a n−2 =0 forn ≥ 2. (18.76)n=0Substituting σ = ±ν into (18.75) and (18.76), we obtain the recurrence relations(1 ± 2ν)a 1 =0, (18.77)n(n ± 2ν)a n + a n−2 =0 forn ≥ 2. (18.78)We consider now the form of the general solution to Bessel’s equation (18.73) fortwo cases: the case for which ν is not an integer and that for which it is (includingzero).18.5.1 Bessel functions for non-integer νIf ν is a non-integer then, in general, the two roots of the indicial equation,σ 1 = ν and σ 2 = −ν, will not differ by an integer, and we may obtain two linearlyindependent solutions in the form of Frobenius series. Special considerations doarise, however, when ν = m/2 form =1, 3, 5,...,andσ 1 − σ 2 =2ν = m is an(odd positive) integer. When this happens, we may always obtain a solution in603