10 Bessel FunctionsBessel and Hankel Functions10.5 Wronskians and Cross-Products10.7 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

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 $z{𝒞}_{\nu \pm 1}\left(z\right)/{𝒞}_{\nu }\left(z\right)$.

See also:

Annotations for §10.6 and Ch.10

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

10.6.1		${𝒞}_{\nu -1}\left(z\right)+{𝒞}_{\nu +1}\left(z\right)$	$=(2\nu /z){𝒞}_{\nu }\left(z\right),$
		${𝒞}_{\nu -1}\left(z\right)-{𝒞}_{\nu +1}\left(z\right)$	$=2{𝒞}_{\nu }^{\prime }\left(z\right).$
ⓘ Symbols: ${𝒞}_{\nu }\left(z\right)$: 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		${𝒞}_{\nu }^{\prime }\left(z\right)$	$={𝒞}_{\nu -1}\left(z\right)-(\nu /z){𝒞}_{\nu }\left(z\right),$
		${𝒞}_{\nu }^{\prime }\left(z\right)$	$=-{𝒞}_{\nu +1}\left(z\right)+(\nu /z){𝒞}_{\nu }\left(z\right).$
ⓘ Symbols: ${𝒞}_{\nu }\left(z\right)$: 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		$J_0^{\prime }\left(z\right)$	$=-J_1\left(z\right),$	$Y_0^{\prime }\left(z\right)$	$=-Y_1\left(z\right),$
		$H_0^{(1)}{}_{}{}^{\prime }\left(z\right)$	$=-H_1^{(1)}\left(z\right),$	$H_0^{(2)}{}_{}{}^{\prime }\left(z\right)$	$=-H_1^{(2)}\left(z\right).$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: 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 $f_{\nu }(z)=z^p{𝒞}_{\nu }\left(\lambda z^q\right)$, where $p,q$, and $\lambda$ ($\ne 0$) are real or complex constants, then

10.6.4		$f_{\nu -1}(z)+f_{\nu +1}(z)$	$=(2\nu /\lambda )z^{-q}{f}_{\nu }(z),$
		$(p+\nu q)f_{\nu -1}(z)+(p-\nu q)f_{\nu +1}(z)$	$=(2\nu /\lambda )z^{1-q}{f}_{\nu }^{\prime }(z).$
ⓘ 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		$z{f}_{\nu }^{\prime }(z)$	$=\lambda q{z}^q{f}_{\nu -1}(z)+(p-\nu q)f_{\nu }(z),$
		$z{f}_{\nu }^{\prime }(z)$	$=-\lambda q{z}^q{f}_{\nu +1}(z)+(p+\nu q)f_{\nu }(z).$
ⓘ 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 $z{𝒞}_{\nu \pm 1}\left(z\right)/{𝒞}_{\nu }\left(z\right)$ 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 $k=0,1,2,\mathrm{\dots }$,

10.6.6		${\left({\displaystyle \frac{1}{z}}{\displaystyle \frac{d}{dz}}\right)}^k\left(z^{\nu }{𝒞}_{\nu }\left(z\right)\right)$	$=z^{\nu -k}{𝒞}_{\nu -k}\left(z\right),$
		${\left({\displaystyle \frac{1}{z}}{\displaystyle \frac{d}{dz}}\right)}^k(z^{-\nu }{𝒞}_{\nu }\left(z\right))$	$={(-1)}^k{z}^{-\nu -k}{𝒞}_{\nu +k}\left(z\right).$
ⓘ Symbols: ${𝒞}_{\nu }\left(z\right)$: cylinder function, $\frac{df}{dx}$: 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		$${𝒞}_{\nu }^{(k)}\left(z\right)=\frac{1}{2^k}\sum _{n=0}^k{(-1)}^n\left(\genfrac{}{}{0.0pt}{}{k}{n}\right){𝒞}_{\nu -k+2n}\left(z\right).$$
ⓘ Symbols: $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${𝒞}_{\nu }\left(z\right)$: 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		$p_{\nu }$	$=J_{\nu }\left(a\right)Y_{\nu }\left(b\right)-J_{\nu }\left(b\right)Y_{\nu }\left(a\right),$
		$q_{\nu }$	$=J_{\nu }\left(a\right)Y_{\nu }^{\prime }\left(b\right)-J_{\nu }^{\prime }\left(b\right)Y_{\nu }\left(a\right),$
		$r_{\nu }$	$=J_{\nu }^{\prime }\left(a\right)Y_{\nu }\left(b\right)-J_{\nu }\left(b\right)Y_{\nu }^{\prime }\left(a\right),$
		$s_{\nu }$	$=J_{\nu }^{\prime }\left(a\right)Y_{\nu }^{\prime }\left(b\right)-J_{\nu }^{\prime }\left(b\right)Y_{\nu }^{\prime }\left(a\right),$
ⓘ Defines: $p_{\nu }$: cross-product (locally), $q_{\nu }$: cross-product (locally), $r_{\nu }$: cross-product (locally) and $s_{\nu }$: cross-product (locally) Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: 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		$p_{\nu +1}-p_{\nu -1}$	$=-{\displaystyle \frac{2\nu }{a}}{q}_{\nu }-{\displaystyle \frac{2\nu }{b}}{r}_{\nu },$
		$q_{\nu +1}+r_{\nu }$	$={\displaystyle \frac{\nu }{a}}{p}_{\nu }-{\displaystyle \frac{\nu +1}{b}}{p}_{\nu +1},$
		$r_{\nu +1}+q_{\nu }$	$={\displaystyle \frac{\nu }{b}}{p}_{\nu }-{\displaystyle \frac{\nu +1}{a}}{p}_{\nu +1},$
		$s_{\nu }$	$=\frac{1}{2}{p}_{\nu +1}+\frac{1}{2}{p}_{\nu -1}-{\displaystyle \frac{{\nu }^2}{ab}}{p}_{\nu },$
ⓘ 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-Products10.7 Limiting Forms

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