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

§10.29 Recurrence Relations and Derivatives

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

§10.29(i) Recurrence Relations
§10.29(ii) Derivatives

§10.29(i) Recurrence Relations

Notes:: See Watson (1944, p. 79).
Keywords:: modified Bessel functions
Referenced by:: §10.43(ii)
Permalink:: http://dlmf.nist.gov/10.29.i

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

10.29.1

$\mathop{\mathscr{Z}_{{\nu-1}}\/}\nolimits\!\left(z\right)-\mathop{\mathscr{Z}_% {{\nu+1}}\/}\nolimits\!\left(z\right)=(2\nu/z)\mathop{\mathscr{Z}_{{\nu}}\/}% \nolimits\!\left(z\right),$

$\mathop{\mathscr{Z}_{{\nu-1}}\/}\nolimits\!\left(z\right)+\mathop{\mathscr{Z}_% {{\nu+1}}\/}\nolimits\!\left(z\right)=2\!{\mathop{\mathscr{Z}_{{\nu}}\/}% \nolimits^{{\prime}}}\!\left(z\right).$

Symbols:: $\mathop{\mathscr{Z}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified cylinder function, : 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
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10.29.2

${\mathop{\mathscr{Z}_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(z\right)=\mathop{% \mathscr{Z}_{{\nu-1}}\/}\nolimits\!\left(z\right)-(\nu/z)\mathop{\mathscr{Z}_{% {\nu}}\/}\nolimits\!\left(z\right),$

${\mathop{\mathscr{Z}_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(z\right)=\mathop{% \mathscr{Z}_{{\nu+1}}\/}\nolimits\!\left(z\right)+(\nu/z)\mathop{\mathscr{Z}_{% {\nu}}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{\mathscr{Z}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified cylinder function, : 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
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10.29.3

${\mathop{I_{{0}}\/}\nolimits^{{\prime}}}\!\left(z\right)=\mathop{I_{{1}}\/}% \nolimits\!\left(z\right),$

${\mathop{K_{{0}}\/}\nolimits^{{\prime}}}\!\left(z\right)=-\mathop{K_{{1}}\/}% \nolimits\!\left(z\right).$

Symbols:: $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function and : complex variable
A&S Ref:: 9.6.27
Permalink:: http://dlmf.nist.gov/10.29.E3
Encodings:: TeX, TeX, pMML, pMML, png, png

§10.29(ii) Derivatives

Notes:: See Watson (1944, p. 79). For (10.29.5) use induction combined with the second of (10.29.1).
Keywords:: derivatives, modified Bessel functions
Permalink:: http://dlmf.nist.gov/10.29.ii

For $k=0,1,2,\ldots$ ,

10.29.4

$\left(\frac{1}{z}\frac{d}{dz}\right)^{k}(z^{\nu}\mathop{\mathscr{Z}_{{\nu}}\/}% \nolimits\!\left(z\right))=z^{{\nu-k}}\mathop{\mathscr{Z}_{{\nu-k}}\/}% \nolimits\!\left(z\right),$

$\left(\frac{1}{z}\frac{d}{dz}\right)^{k}(z^{{-\nu}}\mathop{\mathscr{Z}_{{\nu}}% \/}\nolimits\!\left(z\right))=z^{{-\nu-k}}\mathop{\mathscr{Z}_{{\nu+k}}\/}% \nolimits\!\left(z\right).$

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , $\mathop{\mathscr{Z}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified cylinder function, : nonnegative integer, : 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

10.29.5 ${\mathop{\mathscr{Z}_{{\nu}}\/}\nolimits^{{(k)}}}\!\left(z\right)=\frac{1}{2^{% k}}\left(\mathop{\mathscr{Z}_{{\nu-k}}\/}\nolimits\!\left(z\right)+\binom{k}{1% }\mathop{\mathscr{Z}_{{\nu-k+2}}\/}\nolimits\!\left(z\right)+\binom{k}{2}% \mathop{\mathscr{Z}_{{\nu-k+4}}\/}\nolimits\!\left(z\right)+\cdots+\mathop{% \mathscr{Z}_{{\nu+k}}\/}\nolimits\!\left(z\right)\right).$

Symbols:: $\binom{m}{n}$ : binomial coefficient, $\mathop{\mathscr{Z}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified cylinder function, : nonnegative integer, : 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

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 10.28 Wronskians and Cross-Products 10.30 Limiting Forms

§10.29 Recurrence Relations and Derivatives

Contents

§10.29(i) Recurrence Relations

§10.29(ii) Derivatives