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10 Bessel FunctionsBessel and Hankel Functions

§10.6 Recurrence Relations and Derivatives

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Permalink:
http://dlmf.nist.gov/10.6
See also:
Annotations for Ch.10
Contents
  1. §10.6(i) Recurrence Relations
  2. §10.6(ii) Derivatives
  3. §10.6(iii) Cross-Products

§10.6(i) Recurrence Relations

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Keywords:
Bessel functions, Hankel functions, cylinder functions, recurrence relations
Notes:
For (10.6.1) and (10.6.2) see Olver (1997b, pp. 58–59, 240–242) or Watson (1944, pp. 45, 66, 73–74). (10.6.3) are special cases, and (10.6.4), (10.6.5) follow by straightforward substitution.
Referenced by:
Erratum (V1.1.3) for Subsections 10.6(i), 10.29(i), Erratum (V1.1.3) for References
Permalink:
http://dlmf.nist.gov/10.6.i
Addition (effective with 1.1.3):
A sentence was added just below (10.6.5) regarding results on modified quotients of the form \ifrac{z\mathscr{C}_{\nu\pm 1}\left(z\right)}{\mathscr{C}_{\nu}\left(z\right)}.
See also:
Annotations for §10.6 and Ch.10

With \mathscr{C}_{\nu}\left(z\right) defined as in §10.2(ii),

10.6.1
\mathscr{C}_{\nu-1}\left(z\right)+\mathscr{C}_{\nu+1}\left(z\right)=(2\nu/z)% \mathscr{C}_{\nu}\left(z\right),
\mathscr{C}_{\nu-1}\left(z\right)-\mathscr{C}_{\nu+1}\left(z\right)=2\mathscr{% C}_{\nu}'\left(z\right).
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Symbols:
\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right): cylinder function, z: complex variable and \nu: complex parameter
A&S Ref:
9.1.27
Referenced by:
§10.51(i), §10.6(i), §10.6(ii), §10.63(i), §10.74(iv)
Permalink:
http://dlmf.nist.gov/10.6.E1
Encodings:
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See also:
Annotations for §10.6(i), §10.6 and Ch.10
10.6.2
\mathscr{C}_{\nu}'\left(z\right)=\mathscr{C}_{\nu-1}\left(z\right)-(\nu/z)% \mathscr{C}_{\nu}\left(z\right),
\mathscr{C}_{\nu}'\left(z\right)=-\mathscr{C}_{\nu+1}\left(z\right)+(\nu/z)% \mathscr{C}_{\nu}\left(z\right).

If f_{\nu}(z)=z^{p}\mathscr{C}_{\nu}\left(\lambda z^{q}\right), where p,q, and \lambda (\neq 0) are real or complex constants, then

10.6.4
f_{\nu-1}(z)+f_{\nu+1}(z)=(2\nu/\lambda)z^{-q}f_{\nu}(z),
(p+\nu q)f_{\nu-1}(z)+(p-\nu q)f_{\nu+1}(z)=(2\nu/\lambda)z^{1-q}f_{\nu}^{% \prime}(z).
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Symbols:
z: complex variable, \nu: complex parameter and f_{\nu}(z)
A&S Ref:
9.1.29
Referenced by:
§10.6(i)
Permalink:
http://dlmf.nist.gov/10.6.E4
Encodings:
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See also:
Annotations for §10.6(i), §10.6 and Ch.10
10.6.5
zf_{\nu}^{\prime}(z)=\lambda qz^{q}f_{\nu-1}(z)+(p-\nu q)f_{\nu}(z),
zf_{\nu}^{\prime}(z)=-\lambda qz^{q}f_{\nu+1}(z)+(p+\nu q)f_{\nu}(z).
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Symbols:
z: complex variable, \nu: complex parameter and f_{\nu}(z)
A&S Ref:
9.1.29
Referenced by:
§10.6(i), Erratum (V1.1.3) for Subsections 10.6(i), 10.29(i)
Permalink:
http://dlmf.nist.gov/10.6.E5
Encodings:
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See also:
Annotations for §10.6(i), §10.6 and Ch.10

For results on modified quotients of the form \ifrac{z\mathscr{C}_{\nu\pm 1}\left(z\right)}{\mathscr{C}_{\nu}\left(z\right)} see Onoe (1955) and Onoe (1956).

§10.6(ii) Derivatives

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Keywords:
Bessel functions, Hankel functions, cylinder functions, derivatives, explicit forms
Notes:
For (10.6.6) see Watson (1944, pp. 46). For (10.6.7) use induction combined with the second of (10.6.1).
Permalink:
http://dlmf.nist.gov/10.6.ii
See also:
Annotations for §10.6 and Ch.10

For k=0,1,2,\dotsc,

10.6.6
\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{k}\left(z^{\nu}% \mathscr{C}_{\nu}\left(z\right)\right)=z^{\nu-k}\mathscr{C}_{\nu-k}\left(z% \right),
\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{k}(z^{-\nu}\mathscr{C}% _{\nu}\left(z\right))=(-1)^{k}z^{-\nu-k}\mathscr{C}_{\nu+k}\left(z\right).
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Symbols:
\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right): cylinder function, \frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}: derivative of f with respect to x, k: nonnegative integer, z: complex variable and \nu: complex parameter
A&S Ref:
9.1.30
Referenced by:
§10.6(ii)
Permalink:
http://dlmf.nist.gov/10.6.E6
Encodings:
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See also:
Annotations for §10.6(ii), §10.6 and Ch.10
10.6.7 {\mathscr{C}_{\nu}}^{(k)}\left(z\right)=\frac{1}{2^{k}}\sum_{n=0}^{k}(-1)^{n}% \genfrac{(}{)}{0.0pt}{}{k}{n}\mathscr{C}_{\nu-k+2n}\left(z\right).
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Symbols:
\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}: binomial coefficient, \mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right): cylinder function, n: integer, k: nonnegative integer, z: complex variable and \nu: complex parameter
A&S Ref:
9.1.31
Referenced by:
§10.6(ii)
Permalink:
http://dlmf.nist.gov/10.6.E7
Encodings:
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See also:
Annotations for §10.6(ii), §10.6 and Ch.10

§10.6(iii) Cross-Products

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Keywords:
Bessel functions, cross-products, recurrence relations
Notes:
See Goodwin (1949a).
Permalink:
http://dlmf.nist.gov/10.6.iii
See also:
Annotations for §10.6 and Ch.10

Let

10.6.8
p_{\nu}=J_{\nu}\left(a\right)Y_{\nu}\left(b\right)-J_{\nu}\left(b\right)Y_{\nu% }\left(a\right),
q_{\nu}=J_{\nu}\left(a\right)Y_{\nu}'\left(b\right)-J_{\nu}'\left(b\right)Y_{% \nu}\left(a\right),
r_{\nu}=J_{\nu}'\left(a\right)Y_{\nu}\left(b\right)-J_{\nu}\left(b\right)Y_{% \nu}'\left(a\right),
s_{\nu}=J_{\nu}'\left(a\right)Y_{\nu}'\left(b\right)-J_{\nu}'\left(b\right)Y_{% \nu}'\left(a\right),
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Defines:
p_{\nu}: cross-product (locally), q_{\nu}: cross-product (locally), r_{\nu}: cross-product (locally) and s_{\nu}: cross-product (locally)
Symbols:
J_{\NVar{\nu}}\left(\NVar{z}\right): Bessel function of the first kind, Y_{\NVar{\nu}}\left(\NVar{z}\right): Bessel function of the second kind and \nu: complex parameter
A&S Ref:
9.1.32
Permalink:
http://dlmf.nist.gov/10.6.E8
Encodings:
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See also:
Annotations for §10.6(iii), §10.6 and Ch.10

where a and b are independent of \nu. Then

10.6.9
p_{\nu+1}-p_{\nu-1}=-\frac{2\nu}{a}q_{\nu}-\frac{2\nu}{b}r_{\nu},
q_{\nu+1}+r_{\nu}=\frac{\nu}{a}p_{\nu}-\frac{\nu+1}{b}p_{\nu+1},
r_{\nu+1}+q_{\nu}=\frac{\nu}{b}p_{\nu}-\frac{\nu+1}{a}p_{\nu+1},
s_{\nu}=\tfrac{1}{2}p_{\nu+1}+\tfrac{1}{2}p_{\nu-1}-\frac{\nu^{2}}{ab}p_{\nu},
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Symbols:
\nu: complex parameter, p_{\nu}: cross-product, q_{\nu}: cross-product, r_{\nu}: cross-product and s_{\nu}: cross-product
A&S Ref:
9.1.33
Permalink:
http://dlmf.nist.gov/10.6.E9
Encodings:
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See also:
Annotations for §10.6(iii), §10.6 and Ch.10

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