10 Bessel FunctionsModified Bessel Functions10.28 Wronskians and Cross-Products10.30 Limiting Forms

§10.29 Recurrence Relations and Derivatives

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Permalink:

http://dlmf.nist.gov/10.29

See also:

Annotations for Ch.10

Contents

1. §10.29(i) Recurrence Relations
2. §10.29(ii) Derivatives

§10.29(i) Recurrence Relations

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Keywords:

modified Bessel functions, recurrence relations

Notes:

See Watson (1944, p. 79).

Referenced by:

§10.43(ii), Erratum (V1.1.3) for Subsections 10.6(i), 10.29(i), Erratum (V1.1.3) for References

Permalink:

http://dlmf.nist.gov/10.29.i

Addition (effective with 1.1.3):

A sentence was added just below (10.29.3) regarding results on modified quotients of the form $z{𝒵}_{\nu \pm 1}\left(z\right)/{𝒵}_{\nu }\left(z\right)$.

See also:

Annotations for §10.29 and Ch.10

With ${𝒵}_{\nu }\left(z\right)$ defined as in §10.25(ii),

10.29.1		${𝒵}_{\nu -1}\left(z\right)-{𝒵}_{\nu +1}\left(z\right)$	$=(2\nu /z){𝒵}_{\nu }\left(z\right),$
		${𝒵}_{\nu -1}\left(z\right)+{𝒵}_{\nu +1}\left(z\right)$	$=2{𝒵}_{\nu }^{\prime }\left(z\right).$
ⓘ Symbols: ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.6.26 Referenced by: §10.29(ii), §10.51(ii), §7.6(ii) Permalink: http://dlmf.nist.gov/10.29.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.29(i), §10.29 and Ch.10

10.29.2		${𝒵}_{\nu }^{\prime }\left(z\right)$	$={𝒵}_{\nu -1}\left(z\right)-(\nu /z){𝒵}_{\nu }\left(z\right),$
		${𝒵}_{\nu }^{\prime }\left(z\right)$	$={𝒵}_{\nu +1}\left(z\right)+(\nu /z){𝒵}_{\nu }\left(z\right).$
ⓘ Symbols: ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.6.26 Referenced by: §10.28, §10.43(i), §10.51(ii) Permalink: http://dlmf.nist.gov/10.29.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.29(i), §10.29 and Ch.10

10.29.3		$I_0^{\prime }\left(z\right)$	$=I_1\left(z\right),$
		$K_0^{\prime }\left(z\right)$	$=-K_1\left(z\right).$
ⓘ Symbols: $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind and $z$: complex variable A&S Ref: 9.6.27 Referenced by: §10.29(i), Erratum (V1.1.3) for Subsections 10.6(i), 10.29(i) Permalink: http://dlmf.nist.gov/10.29.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.29(i), §10.29 and Ch.10

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

§10.29(ii) Derivatives

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Keywords:

derivatives, explicit forms, modified Bessel functions

Notes:

See Watson (1944, p. 79). For (10.29.5) use induction combined with the second of (10.29.1).

Permalink:

http://dlmf.nist.gov/10.29.ii

See also:

Annotations for §10.29 and Ch.10

For $k=0,1,2,\mathrm{\dots }$,

10.29.4		${\left({\displaystyle \frac{1}{z}}{\displaystyle \frac{d}{dz}}\right)}^k(z^{\nu }{𝒵}_{\nu }\left(z\right))$	$=z^{\nu -k}{𝒵}_{\nu -k}\left(z\right),$
		${\left({\displaystyle \frac{1}{z}}{\displaystyle \frac{d}{dz}}\right)}^k(z^{-\nu }{𝒵}_{\nu }\left(z\right))$	$=z^{-\nu -k}{𝒵}_{\nu +k}\left(z\right).$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.6.28 Permalink: http://dlmf.nist.gov/10.29.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.29(ii), §10.29 and Ch.10

10.29.5		$${𝒵}_{\nu }^{(k)}\left(z\right)=\frac{1}{2^k}\left({𝒵}_{\nu -k}\left(z\right)+\left(\genfrac{}{}{0.0pt}{}{k}{1}\right){𝒵}_{\nu -k+2}\left(z\right)+\left(\genfrac{}{}{0.0pt}{}{k}{2}\right){𝒵}_{\nu -k+4}\left(z\right)+\mathrm{\cdots }+{𝒵}_{\nu +k}\left(z\right)\right).$$
ⓘ Symbols: $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.6.29 Referenced by: §10.29(ii) Permalink: http://dlmf.nist.gov/10.29.E5 Encodings: TeX, pMML, png See also: Annotations for §10.29(ii), §10.29 and Ch.10

10.28 Wronskians and Cross-Products10.30 Limiting Forms

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