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10 Bessel FunctionsBessel and Hankel Functions

§10.6 Recurrence Relations and Derivatives

  1. §10.6(i) Recurrence Relations
  2. §10.6(ii) Derivatives
  3. §10.6(iii) Cross-Products

§10.6(i) Recurrence Relations

With 𝒞ν(z) defined as in §10.2(ii),

10.6.1 𝒞ν1(z)+𝒞ν+1(z) =(2ν/z)𝒞ν(z),
𝒞ν1(z)𝒞ν+1(z) =2𝒞ν(z).
10.6.2 𝒞ν(z) =𝒞ν1(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 fν1(z)+fν+1(z) =(2ν/λ)zqfν(z),
(p+νq)fν1(z)+(pνq)fν+1(z) =(2ν/λ)z1qfν(z).
10.6.5 zfν(z) =λqzqfν1(z)+(pνq)fν(z),
zfν(z) =λqzqfν+1(z)+(p+νq)fν(z).

For results on modified quotients of the form z𝒞ν±1(z)/𝒞ν(z) see Onoe (1955) and Onoe (1956).

§10.6(ii) Derivatives

For k=0,1,2,,

10.6.6 (1zddz)k(zν𝒞ν(z)) =zνk𝒞νk(z),
(1zddz)k(zν𝒞ν(z)) =(1)kzνk𝒞ν+k(z).
10.6.7 𝒞ν(k)(z)=12kn=0k(1)n(kn)𝒞νk+2n(z).

§10.6(iii) Cross-Products


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+1pν1 =2νaqν2νbrν,
qν+1+rν =νapνν+1bpν+1,
rν+1+qν =νbpνν+1apν+1,
sν =12pν+1+12pν1ν2abpν,


10.6.10 pνsνqνrν=4/(π2ab).