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10 Bessel FunctionsBessel and Hankel Functions

§10.6 Recurrence Relations and Derivatives

Contents

§10.6(i) Recurrence Relations

With 𝒞ν(z) defined as in §10.2(ii),

10.6.1 𝒞ν-1(z)+𝒞ν+1(z) =(2ν/z)𝒞ν(z),
𝒞ν-1(z)-𝒞ν+1(z) =2𝒞ν(z).
10.6.2 𝒞ν(z) =𝒞ν-1(z)-(ν/z)𝒞ν(z),
𝒞ν(z) =-𝒞ν+1(z)+(ν/z)𝒞ν(z).
10.6.3 J0(z) =-J1(z), Y0(z) =-Y1(z),
H0(1)(z) =-H1(1)(z), H0(2)(z) =-H1(2)(z).

If fν(z)=zp𝒞ν(λzq), where p,q, and λ (0) are real or complex constants, then

10.6.4 fν-1(z)+fν+1(z) =(2ν/λ)z-qfν(z),
(p+νq)fν-1(z)+(p-νq)fν+1(z) =(2ν/λ)z1-qfν(z).
10.6.5 zfν(z) =λqzqfν-1(z)+(p-νq)fν(z),
zfν(z) =-λqzqfν+1(z)+(p+νq)fν(z).

§10.6(ii) Derivatives

For k=0,1,2,,

10.6.6 (1zz)k(zν𝒞ν(z)) =zν-k𝒞ν-k(z),
(1zz)k(z-ν𝒞ν(z)) =(-1)kz-ν-k𝒞ν+k(z).
10.6.7 𝒞ν(k)(z)=12kn=0k(-1)n(kn)𝒞ν-k+2n(z).

§10.6(iii) Cross-Products

Let

10.6.8 pν =Jν(a)Yν(b)-Jν(b)Yν(a),
qν =Jν(a)Yν(b)-Jν(b)Yν(a),
rν =Jν(a)Yν(b)-Jν(b)Yν(a),
sν =Jν(a)Yν(b)-Jν(b)Yν(a),

where a and b are independent of ν. Then

10.6.9 pν+1-pν-1 =-2νaqν-2νbrν,
qν+1+rν =νapν-ν+1bpν+1,
rν+1+qν =νbpν-ν+1apν+1,
sν =12pν+1+12pν-1-ν2abpν,

and

10.6.10 pνsν-qνrν=4/(π2ab).