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

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

Note: This service is not intended for secure transactions such as banking, social media, email, or purchasing. Use at your own risk. We assume no liability whatsoever for broken pages.

§10.13 Other Differential Equations

Pfad - The Proxy pFad of © 2024 Garber Painting. All rights reserved.