# §10.13 Other Differential Equations

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 $w^{(2n)}=(-1)^{n}\lambda^{2n}z^{-n}w,$ $w=z^{\frac{1}{2}n}\mathscr{C}_{n}\left(2\lambda e^{k\pi\mathrm{i}/n}z^{\frac{1% }{2}}\right)$, $k=0,1,\dotsc,2n-1$.

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).