§10.13 Other Differential Equations

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

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 $w^{{\prime\prime}}+\left(\lambda^{2}-\frac{\nu^{2}-\tfrac{1}{4}}{z^{2}}\right)w=0,$ $w=z^{{\frac{1}{2}}}\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(\lambda z\right),$

Symbols:: $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)$ : cylinder function, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.49
Referenced by:: §10.36
Permalink:: http://dlmf.nist.gov/10.13.E1
Encodings:: TeX, pMML, png

10.13.2 $w^{{\prime\prime}}+\left(\frac{\lambda^{2}}{4z}-\frac{\nu^{2}-1}{4z^{2}}\right)w=0,$ $w=z^{{\frac{1}{2}}}\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(\lambda z^{{\frac{1}{2}}}\right),$

Symbols:: $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)$ : 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

10.13.3 $w^{{\prime\prime}}+\lambda^{2}z^{{p-2}}w=0,$ $w=z^{{\frac{1}{2}}}\mathop{\mathscr{C}_{{1/p}}\/}\nolimits\!\left(2\lambda z^{{\frac{1}{2}p}}/p\right),$

Symbols:: $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)$ : cylinder function and $z$ : complex variable
A&S Ref:: 9.1.51
Permalink:: http://dlmf.nist.gov/10.13.E3
Encodings:: TeX, pMML, png

10.13.4 $w^{{\prime\prime}}-\frac{2\nu-1}{z}w^{{\prime}}+\lambda^{2}w=0,$ $w=z^{\nu}\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(\lambda z\right),$

Symbols:: $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)$ : cylinder function, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.52
Permalink:: http://dlmf.nist.gov/10.13.E4
Encodings:: TeX, pMML, png

10.13.5 $z^{2}w^{{\prime\prime}}+(1-2r)zw^{{\prime}}+(\lambda^{2}q^{2}z^{{2q}}+r^{2}-\nu^{2}q^{2})w=0,$ $w=z^{r}\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(\lambda z^{q}\right),$

Symbols:: $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)$ : 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

10.13.6 $w^{{\prime\prime}}+(\lambda^{2}e^{{2z}}-\nu^{2})w=0,$ $w=\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(\lambda e^{z}\right),$

Symbols:: $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)$ : cylinder function, $e$ : base of exponential function, $z$ : 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

10.13.7 $z^{2}(z^{2}-\nu^{2})w^{{\prime\prime}}+z(z^{2}-3\nu^{2})w^{{\prime}}+((z^{2}-\nu^{2})^{2}-(z^{2}+\nu^{2}))w=0,$ $w={\mathop{\mathscr{C}_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(z\right),$

Symbols:: $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)$ : 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

10.13.8 $w^{{(2n)}}=(-1)^{n}\lambda^{{2n}}z^{{-n}}w,$ $w=z^{{\frac{1}{2}n}}\mathop{\mathscr{C}_{{n}}\/}\nolimits\!\left(2\lambda e^{{k\pi i/n}}z^{{\frac{1}{2}}}\right)$ , $k=0,1,\ldots,2n-1$ .

Symbols:: $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)$ : cylinder function, $e$ : base of exponential function, $n$ : 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

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

10.13.9 ${z^{2}w^{{\prime\prime\prime}}+3zw^{{\prime\prime}}+(4z^{2}+1-4\nu^{2})w^{{\prime}}+4zw=0},$ $w=\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)\mathscr{D}_{\nu}(z),$

Symbols:: $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)$ : 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

10.13.10 ${z^{3}w^{{\prime\prime\prime}}+z(4z^{2}+1-4\nu^{2})w^{{\prime}}+(4\nu^{2}-1)w=0},$ $w=z\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)\mathscr{D}_{\nu}(z),$

Symbols:: $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)$ : 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

10.13.11 $\left(\vartheta^{4}-2(\nu^{2}+\mu^{2})\vartheta^{2}+(\nu^{2}-\mu^{2})^{2}\right)w+4z^{2}(\vartheta+1)(\vartheta+2)w=0,$ $w=\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)\mathscr{D}_{\mu}(z).$

Symbols:: $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)$ : cylinder function, $z$ : complex variable, $\nu$ : complex parameter, $\vartheta=z(\ifrac{d}{dz})$ 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

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