10 Bessel Functions Modified Bessel Functions 10.35 Generating Function and Associated Series 10.37 Inequalities; Monotonicity

§10.36 Other Differential Equations

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Keywords:: differential equations, modified Bessel functions
Permalink:: http://dlmf.nist.gov/10.36
See also:: Annotations for Ch.10

The quantity $\lambda^{2}$ in (10.13.1)–(10.13.6) and (10.13.8) can be replaced by $-\lambda^{2}$ if at the same time the symbol $\mathscr{C}$ in the given solutions is replaced by $\mathscr{Z}$ . Also,

10.36.1		$z^{2}(z^{2}+\nu^{2})w^{\prime\prime}+z(z^{2}+3\nu^{2})w^{\prime}-\left((z^{2}+% \nu^{2})^{2}+z^{2}-\nu^{2}\right)w=0,$
$w=\mathscr{Z}_{\nu}'\left(z\right)$ ,
ⓘ Symbols: $\mathscr{Z}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified cylinder function, $z$ : complex variable and $\nu$ : complex parameter Permalink: http://dlmf.nist.gov/10.36.E1 Encodings: TeX, pMML, png See also: Annotations for §10.36 and Ch.10

10.36.2		${z^{2}w^{\prime\prime}+z(1\pm 2z)w^{\prime}+(\pm z-\nu^{2})w=0},$
$w=e^{\mp z}\mathscr{Z}_{\nu}\left(z\right)$ .
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $\mathscr{Z}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified cylinder function, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.6.41 Permalink: http://dlmf.nist.gov/10.36.E2 Encodings: TeX, pMML, png See also: Annotations for §10.36 and Ch.10

Differential equations for products can be obtained from (10.13.9)–(10.13.11) by replacing $z$ by $i z$ .

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