10 Bessel Functions Modified Bessel Functions 10.35 Generating Function and Associated Series 10.37 Inequalities; Monotonicity

§10.36 Other Differential Equations

Keywords:: differential equations, modified Bessel functions
Permalink:: http://dlmf.nist.gov/10.36

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

Symbols:: $\mathop{\mathscr{Z}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified cylinder function, : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.36.E1
Encodings:: TeX, pMML, png

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:: : base of exponential function, $\mathop{\mathscr{Z}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified cylinder function, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.6.41
Permalink:: http://dlmf.nist.gov/10.36.E2
Encodings:: TeX, pMML, png

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

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