10 Bessel Functions Bessel and Hankel Functions 10.5 Wronskians and Cross-Products 10.7 Limiting Forms

§10.6 Recurrence Relations and Derivatives

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Permalink:: http://dlmf.nist.gov/10.6
See also:: Annotations for Ch.10

§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	$\displaystyle\mathscr{C}_{\nu-1}\left(z\right)+\mathscr{C}_{\nu+1}\left(z\right)$	$\displaystyle=(2\nu/z)\mathscr{C}_{\nu}\left(z\right),$
10.6.1	$\displaystyle\mathscr{C}_{\nu-1}\left(z\right)-\mathscr{C}_{\nu+1}\left(z\right)$	$\displaystyle=2\mathscr{C}_{\nu}'\left(z\right).$
ⓘ 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: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.6(i), §10.6 and Ch.10

10.6.2	$\displaystyle\mathscr{C}_{\nu}'\left(z\right)$	$\displaystyle=\mathscr{C}_{\nu-1}\left(z\right)-(\nu/z)\mathscr{C}_{\nu}\left(% z\right),$
10.6.2	$\displaystyle\mathscr{C}_{\nu}'\left(z\right)$	$\displaystyle=-\mathscr{C}_{\nu+1}\left(z\right)+(\nu/z)\mathscr{C}_{\nu}\left% (z\right).$
ⓘ 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: §10.21(ii), §10.22(i), §10.22(ii), §10.5, §10.51(i), §10.6(i), §10.63(i), §10.63(ii), §10.74(vi) Permalink: http://dlmf.nist.gov/10.6.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.6(i), §10.6 and Ch.10

10.6.3	$\displaystyle J_{0}'\left(z\right)$	$\displaystyle=-J_{1}\left(z\right),$	$\displaystyle Y_{0}'\left(z\right)$	$\displaystyle=-Y_{1}\left(z\right),$
10.6.3	$\displaystyle{H^{(1)}_{0}}'\left(z\right)$	$\displaystyle=-{H^{(1)}_{1}}\left(z\right),$	$\displaystyle{H^{(2)}_{0}}'\left(z\right)$	$\displaystyle=-{H^{(2)}_{1}}\left(z\right).$
ⓘ 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, ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function) and $z$ : complex variable A&S Ref: 9.1.28 Referenced by: §10.6(i) Permalink: http://dlmf.nist.gov/10.6.E3 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §10.6(i), §10.6 and Ch.10

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	$\displaystyle f_{\nu-1}(z)+f_{\nu+1}(z)$	$\displaystyle=(2\nu/\lambda)z^{-q}f_{\nu}(z),$
10.6.4	$\displaystyle(p+\nu q)f_{\nu-1}(z)+(p-\nu q)f_{\nu+1}(z)$	$\displaystyle=(2\nu/\lambda)z^{1-q}f_{\nu}^{\prime}(z).$
ⓘ 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: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.6(i), §10.6 and Ch.10

10.6.5	$\displaystyle zf_{\nu}^{\prime}(z)$	$\displaystyle=\lambda qz^{q}f_{\nu-1}(z)+(p-\nu q)f_{\nu}(z),$
10.6.5	$\displaystyle zf_{\nu}^{\prime}(z)$	$\displaystyle=-\lambda qz^{q}f_{\nu+1}(z)+(p+\nu q)f_{\nu}(z).$
ⓘ 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: TeX, TeX, pMML, pMML, png, png 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	$\displaystyle\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{k}\left(z% ^{\nu}\mathscr{C}_{\nu}\left(z\right)\right)$	$\displaystyle=z^{\nu-k}\mathscr{C}_{\nu-k}\left(z\right),$
10.6.6	$\displaystyle\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{k}(z^{-% \nu}\mathscr{C}_{\nu}\left(z\right))$	$\displaystyle=(-1)^{k}z^{-\nu-k}\mathscr{C}_{\nu+k}\left(z\right).$
ⓘ 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: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.6(ii), §10.6 and Ch.10

10.6.7		${\mathscr{C}_{\nu}}^{(k)}\left(z\right)=\frac{1}{2^{k}}\sum_{n=0}^{k}(-1)^{n}% \genfrac{(}{)}{0.0pt}{}{k}{n}\mathscr{C}_{\nu-k+2n}\left(z\right).$
ⓘ 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: TeX, pMML, png 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	$\displaystyle p_{\nu}$	$\displaystyle=J_{\nu}\left(a\right)Y_{\nu}\left(b\right)-J_{\nu}\left(b\right)% Y_{\nu}\left(a\right),$
	$\displaystyle q_{\nu}$	$\displaystyle=J_{\nu}\left(a\right)Y_{\nu}'\left(b\right)-J_{\nu}'\left(b% \right)Y_{\nu}\left(a\right),$
	$\displaystyle r_{\nu}$	$\displaystyle=J_{\nu}'\left(a\right)Y_{\nu}\left(b\right)-J_{\nu}\left(b\right% )Y_{\nu}'\left(a\right),$
	$\displaystyle s_{\nu}$	$\displaystyle=J_{\nu}'\left(a\right)Y_{\nu}'\left(b\right)-J_{\nu}'\left(b% \right)Y_{\nu}'\left(a\right),$
ⓘ 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: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §10.6(iii), §10.6 and Ch.10

where $a$ and $b$ are independent of $\nu$ . Then

10.6.9	$\displaystyle p_{\nu+1}-p_{\nu-1}$	$\displaystyle=-\frac{2\nu}{a}q_{\nu}-\frac{2\nu}{b}r_{\nu},$
	$\displaystyle q_{\nu+1}+r_{\nu}$	$\displaystyle=\frac{\nu}{a}p_{\nu}-\frac{\nu+1}{b}p_{\nu+1},$
	$\displaystyle r_{\nu+1}+q_{\nu}$	$\displaystyle=\frac{\nu}{b}p_{\nu}-\frac{\nu+1}{a}p_{\nu+1},$
	$\displaystyle s_{\nu}$	$\displaystyle=\tfrac{1}{2}p_{\nu+1}+\tfrac{1}{2}p_{\nu-1}-\frac{\nu^{2}}{ab}p_% {\nu},$
ⓘ 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: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §10.6(iii), §10.6 and Ch.10

and

10.6.10		$p_{\nu}s_{\nu}-q_{\nu}r_{\nu}=4/(\pi^{2}ab).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\nu$ : complex parameter, $p_{\nu}$ : cross-product, $q_{\nu}$ : cross-product, $r_{\nu}$ : cross-product and $s_{\nu}$ : cross-product A&S Ref: 9.1.34 Permalink: http://dlmf.nist.gov/10.6.E10 Encodings: TeX, pMML, png See also: Annotations for §10.6(iii), §10.6 and Ch.10

10.5 Wronskians and Cross-Products 10.7 Limiting Forms

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§10.6 Recurrence Relations and Derivatives

Contents

§10.6(i) Recurrence Relations

§10.6(ii) Derivatives

§10.6(iii) Cross-Products