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10 Bessel FunctionsSpherical Bessel Functions

§10.47 Definitions and Basic Properties

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§10.47(i) Differential Equations

10.47.1 z2d2wdz2+2zdwdz+(z2-n(n+1))w=0,
10.47.2 z2d2wdz2+2zdwdz-(z2+n(n+1))w=0.

Here, and throughout the remainder of §§10.4710.60, n is a nonnegative integer. (This is in contrast to other treatments of spherical Bessel functions, including Abramowitz and Stegun (1964, Chapter 10), in which n can be any integer. However, there is a gain in symmetry, without any loss of generality in applications, on restricting n0.)

Equations (10.47.1) and (10.47.2) each have a regular singularity at z=0 with indices n, -n-1, and an irregular singularity at z= of rank 1; compare §§2.7(i)2.7(ii).

§10.47(ii) Standard Solutions

Equation (10.47.1)

10.47.3 jn(z)=12π/zJn+12(z)=(-1)n12π/zY-n-12(z),
10.47.4 yn(z)=12π/zYn+12(z)=(-1)n+112π/zJ-n-12(z),
10.47.5 hn(1)(z)=12π/zHn+12(1)(z)=(-1)n+1i12π/zH-n-12(1)(z),
10.47.6 hn(2)(z)=12π/zHn+12(2)(z)=(-1)ni12π/zH-n-12(2)(z).

jn(z) and yn(z) are the spherical Bessel functions of the first and second kinds, respectively; hn(1)(z) and hn(2)(z) are the spherical Bessel functions of the third kind.

Equation (10.47.2)

10.47.7 in(1)(z) =12π/zIn+12(z)
10.47.8 in(2)(z) =12π/zI-n-12(z)
10.47.9 kn(z)=12π/zKn+12(z)=12π/zK-n-12(z).

in(1)(z), in(2)(z), and kn(z) are the modified spherical Bessel functions.

Many properties of jn(z), yn(z), hn(1)(z), hn(2)(z), in(1)(z), in(2)(z), and kn(z) follow straightforwardly from the above definitions and results given in preceding sections of this chapter. For example, z-njn(z), zn+1yn(z), zn+1hn(1)(z), zn+1hn(2)(z), z-nin(1)(z), zn+1in(2)(z), and zn+1kn(z) are all entire functions of z.

§10.47(iii) Numerically Satisfactory Pairs of Solutions

For (10.47.1) numerically satisfactory pairs of solutions are given by Table 10.2.1 with the symbols J, Y, H, and ν replaced by j, y, h, and n, respectively.

For (10.47.2) numerically satisfactory pairs of solutions are in(1)(z) and kn(z) in the right half of the z-plane, and in(1)(z) and kn(-z) in the left half of the z-plane.

§10.47(iv) Interrelations

10.47.10 hn(1)(z) =jn(z)+iyn(z),
hn(2)(z) =jn(z)-iyn(z).
10.47.11 kn(z)=(-1)n+112π(in(1)(z)-in(2)(z)).
10.47.12 in(1)(z) =i-njn(iz),
in(2)(z) =i-n-1yn(iz).
10.47.13 kn(z)=-12πinhn(1)(iz)=-12πi-nhn(2)(-iz).

§10.47(v) Reflection Formulas

10.47.14 jn(-z) =(-1)njn(z), yn(-z) =(-1)n+1yn(z),
10.47.15 hn(1)(-z) =(-1)nhn(2)(z), hn(2)(-z) =(-1)nhn(1)(z).
10.47.16 in(1)(-z) =(-1)nin(1)(z), in(2)(-z) =(-1)n+1in(2)(z),
10.47.17 kn(-z)=-12π(in(1)(z)+in(2)(z)).