# §10.36 Other Differential Equations

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 $\mathop{\mathscr{C}\/}\nolimits$ in the given solutions is replaced by $\mathop{\mathscr{Z}\/}\nolimits$. Also,

 10.36.1 $z^{2}(z^{2}+\nu^{2})w^{\prime\prime}+z(z^{2}+3\nu^{2})w^{\prime}-\left((z^{2}+% \nu^{2})^{2}+z^{2}-\nu^{2}\right)w=0,$ $w=\mathop{\mathscr{Z}_{\nu}\/}\nolimits'\!\left(z\right)$, Symbols: $\mathop{\mathscr{Z}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: 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
 10.36.2 ${z^{2}w^{\prime\prime}+z(1\pm 2z)w^{\prime}+(\pm z-\nu^{2})w=0},$ $w=e^{\mp z}\mathop{\mathscr{Z}_{\nu}\/}\nolimits\!\left(z\right)$. Symbols: $\mathrm{e}$: base of exponential function, $\mathop{\mathscr{Z}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: 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

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