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10 Bessel FunctionsSpherical Bessel Functions

§10.49 Explicit Formulas

Contents

§10.49(i) Unmodified Functions

Define ak(ν) as in (10.17.1). Then

10.49.1 ak(n+12)={(n+k)!2kk!(n-k)!,k=0,1,,n,0,k=n+1,n+2,.
10.49.2 jn(z)=sin(z-12nπ)k=0n/2(-1)ka2k(n+12)z2k+1+cos(z-12nπ)k=0(n-1)/2(-1)ka2k+1(n+12)z2k+2.
10.49.3 j0(z) =sinzz,
j1(z) =sinzz2-coszz,
j2(z) =(-1z+3z3)sinz-3z2cosz.
10.49.4 yn(z)=-cos(z-12nπ)k=0n/2(-1)ka2k(n+12)z2k+1+sin(z-12nπ)k=0(n-1)/2(-1)ka2k+1(n+12)z2k+2.
10.49.5 y0(z) =-coszz,
y1(z) =-coszz2-sinzz,
y2(z) =(1z-3z3)cosz-3z2sinz.
10.49.6 hn(1)(z) =eizk=0nik-n-1ak(n+12)zk+1,
10.49.7 hn(2)(z) =e-izk=0n(-i)k-n-1ak(n+12)zk+1.

§10.49(ii) Modified Functions

Again, with ak(n+12) as in (10.49.1),

10.49.8 in(1)(z)=12ezk=0n(-1)kak(n+12)zk+1+(-1)n+112e-zk=0nak(n+12)zk+1.
10.49.9 i0(1)(z) =sinhzz,
i1(1)(z) =-sinhzz2+coshzz,
i2(1)(z) =(1z+3z3)sinhz-3z2coshz.
10.49.10 in(2)(z)=12ezk=0n(-1)kak(n+12)zk+1+(-1)n12e-zk=0nak(n+12)zk+1.
10.49.11 i0(2)(z) =coshzz,
i1(2)(z) =-coshzz2+sinhzz,
i2(2)(z) =(1z+3z3)coshz-3z2sinhz.
10.49.12 kn(z)=12πe-zk=0nak(n+12)zk+1.
10.49.13 k0(z) =12πe-zz,
k1(z) =12πe-z(1z+1z2),
k2(z) =12πe-z(1z+3z2+3z3).

k=0nak(n+12)zn-k is sometimes called the Bessel polynomial of degree n. For a survey of properties of these polynomials and their generalizations see Grosswald (1978). See also §18.34, de Bruin et al. (1981a, b), and Dunster (2001c).

§10.49(iii) Rayleigh’s Formulas

10.49.14 jn(z) =zn(-1zddz)nsinzz,
yn(z) =-zn(-1zddz)ncoszz.
10.49.15 in(1)(z) =zn(1zddz)nsinhzz,
in(2)(z) =zn(1zddz)ncoshzz.
10.49.16 kn(z)=(-1)n12πzn(1zddz)ne-zz.

§10.49(iv) Sums or Differences of Squares

Denote

10.49.17 sk(n+12)=(2k)!(n+k)!22k(k!)2(n-k)!,
k=0,1,,n.

Then

10.49.18 jn2(z)+yn2(z)=k=0nsk(n+12)z2k+2.
10.49.19 j02(z)+y02(z) =z-2,
j12(z)+y12(z) =z-2+z-4,
j22(z)+y22(z) =z-2+3z-4+9z-6.
10.49.20 (in(1)(z))2-(in(2)(z))2=(-1)n+1k=0n(-1)ksk(n+12)z2k+2.
10.49.21 (i0(1)(z))2-(i0(2)(z))2 =-z-2,
(i1(1)(z))2-(i1(2)(z))2 =z-2-z-4,
(i2(1)(z))2-(i2(2)(z))2 =-z-2+3z-4-9z-6.