10 Bessel Functions Modified Bessel Functions 10.28 Wronskians and Cross-Products 10.30 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

§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 $\ifrac{z\mathscr{Z}_{\nu\pm 1}\left(z\right)}{\mathscr{Z}_{\nu}\left(z\right)}$ .
See also:: Annotations for §10.29 and Ch.10

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

10.29.1	$\displaystyle\mathscr{Z}_{\nu-1}\left(z\right)-\mathscr{Z}_{\nu+1}\left(z\right)$	$\displaystyle=(2\nu/z)\mathscr{Z}_{\nu}\left(z\right),$
10.29.1	$\displaystyle\mathscr{Z}_{\nu-1}\left(z\right)+\mathscr{Z}_{\nu+1}\left(z\right)$	$\displaystyle=2\mathscr{Z}_{\nu}'\left(z\right).$
ⓘ Symbols: $\mathscr{Z}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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	$\displaystyle\mathscr{Z}_{\nu}'\left(z\right)$	$\displaystyle=\mathscr{Z}_{\nu-1}\left(z\right)-(\nu/z)\mathscr{Z}_{\nu}\left(% z\right),$
10.29.2	$\displaystyle\mathscr{Z}_{\nu}'\left(z\right)$	$\displaystyle=\mathscr{Z}_{\nu+1}\left(z\right)+(\nu/z)\mathscr{Z}_{\nu}\left(% z\right).$
ⓘ Symbols: $\mathscr{Z}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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	$\displaystyle I_{0}'\left(z\right)$	$\displaystyle=I_{1}\left(z\right),$
10.29.3	$\displaystyle K_{0}'\left(z\right)$	$\displaystyle=-K_{1}\left(z\right).$
ⓘ Symbols: $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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 $\ifrac{z\mathscr{Z}_{\nu\pm 1}\left(z\right)}{\mathscr{Z}_{\nu}\left(z\right)}$ 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 $k=0,1,2,\dotsc$ ,

10.29.4	$\displaystyle\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{k}(z^{\nu% }\mathscr{Z}_{\nu}\left(z\right))$	$\displaystyle=z^{\nu-k}\mathscr{Z}_{\nu-k}\left(z\right),$
10.29.4	$\displaystyle\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{k}(z^{-% \nu}\mathscr{Z}_{\nu}\left(z\right))$	$\displaystyle=z^{-\nu-k}\mathscr{Z}_{\nu+k}\left(z\right).$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathscr{Z}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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		${\mathscr{Z}_{\nu}}^{(k)}\left(z\right)=\frac{1}{2^{k}}\left(\mathscr{Z}_{\nu-% k}\left(z\right)+\genfrac{(}{)}{0.0pt}{}{k}{1}\mathscr{Z}_{\nu-k+2}\left(z% \right)+\genfrac{(}{)}{0.0pt}{}{k}{2}\mathscr{Z}_{\nu-k+4}\left(z\right)+% \cdots+\mathscr{Z}_{\nu+k}\left(z\right)\right).$
ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\mathscr{Z}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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-Products 10.30 Limiting Forms

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§10.29 Recurrence Relations and Derivatives

Contents

§10.29(i) Recurrence Relations

§10.29(ii) Derivatives