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10 Bessel FunctionsBessel and Hankel Functions

§10.13 Other Differential Equations

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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.

10.13.1w^{{\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),
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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
10.13.2w^{{\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),
10.13.3w^{{\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),
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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
10.13.4w^{{\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),
10.13.5z^{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),
10.13.6w^{{\prime\prime}}+(\lambda^{2}e^{{2z}}-\nu^{2})w=0,w=\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(\lambda e^{z}\right),
10.13.7z^{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),

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),
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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
10.13.11\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).

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

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