10 Bessel Functions Bessel and Hankel Functions 10.12 Generating Function and Associated Series 10.14 Inequalities; Monotonicity

§10.13 Other Differential Equations

ⓘ

Keywords:: Bessel functions, cylinder functions, differential equations
Notes:: These results are obtainable from (10.2.1) by straightforward substitutions. See also §1.13(v).
Permalink:: http://dlmf.nist.gov/10.13
See also:: Annotations for Ch.10

In the following equations $\nu,\lambda,p,q$ , and $r$ are real or complex constants with $\lambda\neq 0$ , $p\neq 0$ , and $q\neq 0$ .

10.13.1		$w^{\prime\prime}+\left(\lambda^{2}-\frac{\nu^{2}-\tfrac{1}{4}}{z^{2}}\right)w=0,$
$w=z^{\frac{1}{2}}\mathscr{C}_{\nu}\left(\lambda z\right)$ ,
ⓘ Symbols: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.1.49 Referenced by: §1.18(vi), §10.36 Permalink: http://dlmf.nist.gov/10.13.E1 Encodings: TeX, pMML, png See also: Annotations for §10.13 and Ch.10

10.13.2		$w^{\prime\prime}+\left(\frac{\lambda^{2}}{4z}-\frac{\nu^{2}-1}{4z^{2}}\right)w% =0,$
$w=z^{\frac{1}{2}}\mathscr{C}_{\nu}\left(\lambda z^{\frac{1}{2}}\right)$ ,
ⓘ Symbols: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.1.50 Permalink: http://dlmf.nist.gov/10.13.E2 Encodings: TeX, pMML, png See also: Annotations for §10.13 and Ch.10

10.13.3		$w^{\prime\prime}+\lambda^{2}z^{p-2}w=0,$
$w=z^{\frac{1}{2}}\mathscr{C}_{1/p}\left(2\lambda z^{\frac{1}{2}p}/p\right)$ ,
ⓘ Symbols: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function and $z$ : complex variable A&S Ref: 9.1.51 Permalink: http://dlmf.nist.gov/10.13.E3 Encodings: TeX, pMML, png See also: Annotations for §10.13 and Ch.10

10.13.4		$w^{\prime\prime}+\frac{1\mp 2\nu}{z}w^{\prime}+\lambda^{2}w=0,$
$w=z^{\pm\nu}\mathscr{C}_{\nu}\left(\lambda z\right)$ ,
ⓘ Symbols: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.1.52 Referenced by: Erratum (V1.0.8) for Equation (10.13.4) Permalink: http://dlmf.nist.gov/10.13.E4 Encodings: TeX, pMML, png Generalization (effective with 1.0.8): This equation has been generalized to allow an additional case. Originally only the case $w=z^{\nu}\mathscr{C}_{\nu}\left(\lambda z\right)$ was covered. Suggested 2014-01-27 by Tom Koornwinder See also: Annotations for §10.13 and Ch.10

10.13.5		$z^{2}w^{\prime\prime}+(1-2r)zw^{\prime}+(\lambda^{2}q^{2}z^{2q}+r^{2}-\nu^{2}q% ^{2})w=0,$
$w=z^{r}\mathscr{C}_{\nu}\left(\lambda z^{q}\right)$ ,
ⓘ Symbols: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.1.53 Permalink: http://dlmf.nist.gov/10.13.E5 Encodings: TeX, pMML, png See also: Annotations for §10.13 and Ch.10

10.13.6		$w^{\prime\prime}+(\lambda^{2}e^{2z}-\nu^{2})w=0,$
$w=\mathscr{C}_{\nu}\left(\lambda e^{z}\right)$ ,
ⓘ Symbols: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $\mathrm{e}$ : base of natural logarithm, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.1.54 Referenced by: §10.36 Permalink: http://dlmf.nist.gov/10.13.E6 Encodings: TeX, pMML, png See also: Annotations for §10.13 and Ch.10

10.13.7		$z^{2}(z^{2}-\nu^{2})w^{\prime\prime}+z(z^{2}-3\nu^{2})w^{\prime}+((z^{2}-\nu^{% 2})^{2}-(z^{2}+\nu^{2}))w=0,$
$w=\mathscr{C}_{\nu}'\left(z\right)$ ,
ⓘ Symbols: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.1.55 Permalink: http://dlmf.nist.gov/10.13.E7 Encodings: TeX, pMML, png See also: Annotations for §10.13 and Ch.10

10.13.8		$w^{(2n)}=(-1)^{n}\lambda^{2n}z^{-n}w,$
$w=z^{\frac{1}{2}n}\mathscr{C}_{n}\left(2\lambda e^{k\pi\mathrm{i}/n}z^{\frac{1% }{2}}\right)$ , $k=0,1,\dotsc,2n-1$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $n$ : integer, $k$ : nonnegative integer and $z$ : complex variable A&S Ref: 9.1.56 Referenced by: §10.36 Permalink: http://dlmf.nist.gov/10.13.E8 Encodings: TeX, pMML, png See also: Annotations for §10.13 and Ch.10

In (10.13.9)–(10.13.11) $\mathscr{C}_{\nu}\left(z\right)$ , $\mathscr{D}_{\mu}(z)$ are any cylinder functions of orders $\nu,\mu$ , respectively, and $\vartheta=z(\!\ifrac{\mathrm{d}}{\mathrm{d}z})$ .

10.13.9		${z^{2}w^{\prime\prime\prime}+3zw^{\prime\prime}+(4z^{2}+1-4\nu^{2})w^{\prime}+% 4zw=0},$
$w=\mathscr{C}_{\nu}\left(z\right)\mathscr{D}_{\nu}(z)$ ,
ⓘ Symbols: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $z$ : complex variable, $\nu$ : complex parameter and $\mathscr{D}_{\nu}(z)$ : cylinder function A&S Ref: 9.1.58 (modified) Referenced by: §10.13, §10.36 Permalink: http://dlmf.nist.gov/10.13.E9 Encodings: TeX, pMML, png See also: Annotations for §10.13 and Ch.10

10.13.10		${z^{3}w^{\prime\prime\prime}+z(4z^{2}+1-4\nu^{2})w^{\prime}+(4\nu^{2}-1)w=0},$
$w=z\mathscr{C}_{\nu}\left(z\right)\mathscr{D}_{\nu}(z)$ ,
ⓘ Symbols: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $z$ : complex variable, $\nu$ : complex parameter and $\mathscr{D}_{\nu}(z)$ : cylinder function A&S Ref: 9.1.59 Permalink: http://dlmf.nist.gov/10.13.E10 Encodings: TeX, pMML, png See also: Annotations for §10.13 and Ch.10

10.13.11		$\left(\vartheta^{4}-2(\nu^{2}+\mu^{2})\vartheta^{2}+(\nu^{2}-\mu^{2})^{2}% \right)w+4z^{2}(\vartheta+1)(\vartheta+2)w=0,$
$w=\mathscr{C}_{\nu}\left(z\right)\mathscr{D}_{\mu}(z).$
ⓘ Symbols: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $z$ : complex variable, $\nu$ : complex parameter, $\vartheta=z(\ifrac{\mathrm{d}}{\mathrm{d}z})$ and $\mathscr{D}_{\nu}(z)$ : cylinder function A&S Ref: 9.1.57 Referenced by: §10.13, §10.36 Permalink: http://dlmf.nist.gov/10.13.E11 Encodings: TeX, pMML, png See also: Annotations for §10.13 and Ch.10

For further differential equations see Kamke (1977, pp. 440–451). See also Watson (1944, pp. 95–100).

10.12 Generating Function and Associated Series 10.14 Inequalities; Monotonicity

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