# §10.61 Definitions and Basic Properties

## §10.61(i) Definitions

Throughout §§10.61–§10.71 it is assumed that $x\geq 0$, $\nu\in\mathbb{R}$, and $n$ is a nonnegative integer.

 10.61.1 $\operatorname{ber}_{\nu}x+i\operatorname{bei}_{\nu}x=J_{\nu}\left(xe^{3\pi i/4% }\right)=e^{\nu\pi i}J_{\nu}\left(xe^{-\pi i/4}\right)=e^{\nu\pi i/2}I_{\nu}% \left(xe^{\pi i/4}\right)=e^{3\nu\pi i/2}I_{\nu}\left(xe^{-3\pi i/4}\right),$ ⓘ Defines: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$: Kelvin function and $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$: Kelvin function Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\NVar{\nu}}\left(\NVar{z}\right)$: modified Bessel function of the first kind, $x$: real variable and $\nu$: complex parameter A&S Ref: 9.9.1 Referenced by: §10.63(ii), §10.65(i), §10.65(ii), §10.67(i), §10.69 Permalink: http://dlmf.nist.gov/10.61.E1 Encodings: TeX, pMML, png See also: Annotations for §10.61(i), §10.61 and Ch.10
 10.61.2 $\operatorname{ker}_{\nu}x+i\operatorname{kei}_{\nu}x=e^{-\nu\pi i/2}K_{\nu}% \left(xe^{\pi i/4}\right)=\tfrac{1}{2}\pi i{H^{(1)}_{\nu}}\left(xe^{3\pi i/4}% \right)=-\tfrac{1}{2}\pi ie^{-\nu\pi i}{H^{(2)}_{\nu}}\left(xe^{-\pi i/4}% \right).$ ⓘ Defines: $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$: Kelvin function and $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$: Kelvin function Symbols: ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$: Bessel function of the third kind (or Hankel function), ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $K_{\NVar{\nu}}\left(\NVar{z}\right)$: modified Bessel function of the second kind, $x$: real variable and $\nu$: complex parameter A&S Ref: 9.9.2 Referenced by: §10.63(ii), §10.65(ii), §10.67(i), §10.69 Permalink: http://dlmf.nist.gov/10.61.E2 Encodings: TeX, pMML, png See also: Annotations for §10.61(i), §10.61 and Ch.10

When $\nu=0$ suffices on $\operatorname{ber}$, $\operatorname{bei}$, $\operatorname{ker}$, and $\operatorname{kei}$ are usually suppressed.

Most properties of $\operatorname{ber}_{\nu}x$, $\operatorname{bei}_{\nu}x$, $\operatorname{ker}_{\nu}x$, and $\operatorname{kei}_{\nu}x$ follow straightforwardly from the above definitions and results given in preceding sections of this chapter.

## §10.61(ii) Differential Equations

 10.61.3 $x^{2}\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}+x\frac{\mathrm{d}w}{\mathrm{d% }x}-(ix^{2}+\nu^{2})w=0,$ $w=\begin{array}[t]{cc}\operatorname{ber}_{\nu}x+i\operatorname{bei}_{\nu}x,&% \operatorname{ber}_{-\nu}x+i\operatorname{bei}_{-\nu}x\\ \operatorname{ker}_{\nu}x+i\operatorname{kei}_{\nu}x,&\operatorname{ker}_{-\nu% }x+i\operatorname{kei}_{-\nu}x.\end{array}$
 10.61.4 $x^{4}\frac{{\mathrm{d}}^{4}w}{{\mathrm{d}x}^{4}}+2x^{3}\frac{{\mathrm{d}}^{3}w% }{{\mathrm{d}x}^{3}}-(1+2\nu^{2})\left(x^{2}\frac{{\mathrm{d}}^{2}w}{{\mathrm{% d}x}^{2}}-x\frac{\mathrm{d}w}{\mathrm{d}x}\right)+(\nu^{4}-4\nu^{2}+x^{4})w=0,$ $w=\operatorname{ber}_{\pm\nu}x,\operatorname{bei}_{\pm\nu}x,\operatorname{ker}% _{\pm\nu}x,\operatorname{kei}_{\pm\nu}x$.

## §10.61(iii) Reflection Formulas for Arguments

In general, Kelvin functions have a branch point at $x=0$ and functions with arguments $xe^{\pm\pi i}$ are complex. The branch point is absent, however, in the case of $\operatorname{ber}_{\nu}$ and $\operatorname{bei}_{\nu}$ when $\nu$ is an integer. In particular,

 10.61.5 $\displaystyle\operatorname{ber}_{n}\left(-x\right)$ $\displaystyle=(-1)^{n}\operatorname{ber}_{n}x,$ $\displaystyle\operatorname{bei}_{n}\left(-x\right)$ $\displaystyle=(-1)^{n}\operatorname{bei}_{n}x.$ ⓘ Symbols: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$: Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$: Kelvin function, $n$: integer and $x$: real variable A&S Ref: 9.9.13 Permalink: http://dlmf.nist.gov/10.61.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.61(iii), §10.61 and Ch.10

## §10.61(iv) Reflection Formulas for Orders

 10.61.6 $\displaystyle\operatorname{ber}_{-\nu}x$ $\displaystyle=\cos\left(\nu\pi\right)\operatorname{ber}_{\nu}x+\sin\left(\nu% \pi\right)\operatorname{bei}_{\nu}x+(2/\pi)\sin\left(\nu\pi\right)% \operatorname{ker}_{\nu}x,$ $\displaystyle\operatorname{bei}_{-\nu}x$ $\displaystyle=-\sin\left(\nu\pi\right)\operatorname{ber}_{\nu}x+\cos\left(\nu% \pi\right)\operatorname{bei}_{\nu}x+(2/\pi)\sin\left(\nu\pi\right)% \operatorname{kei}_{\nu}x.$
 10.61.7 $\displaystyle\operatorname{ker}_{-\nu}x$ $\displaystyle=\cos\left(\nu\pi\right)\operatorname{ker}_{\nu}x-\sin\left(\nu% \pi\right)\operatorname{kei}_{\nu}x,$ $\displaystyle\operatorname{kei}_{-\nu}x$ $\displaystyle=\sin\left(\nu\pi\right)\operatorname{ker}_{\nu}x+\cos\left(\nu% \pi\right)\operatorname{kei}_{\nu}x.$
 10.61.8 $\displaystyle\operatorname{ber}_{-n}x$ $\displaystyle=(-1)^{n}\operatorname{ber}_{n}x,~\operatorname{bei}_{-n}x=(-1)^{% n}\operatorname{bei}_{n}x,$ $\displaystyle\operatorname{ker}_{-n}x$ $\displaystyle=(-1)^{n}\operatorname{ker}_{n}x,~\operatorname{kei}_{-n}x=(-1)^{% n}\operatorname{kei}_{n}x.$

## §10.61(v) Orders $\pm\frac{1}{2}$

 10.61.9 $\displaystyle\operatorname{ber}_{\frac{1}{2}}\left(x\sqrt{2}\right)$ $\displaystyle=\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\cos\left(x+% \frac{\pi}{8}\right)-e^{-x}\cos\left(x-\frac{\pi}{8}\right)\right),$ $\displaystyle\operatorname{bei}_{\frac{1}{2}}\left(x\sqrt{2}\right)$ $\displaystyle=\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\sin\left(x+% \frac{\pi}{8}\right)+\,e^{-x}\sin\left(x-\frac{\pi}{8}\right)\right).$
 10.61.10 $\displaystyle\operatorname{ber}_{-\frac{1}{2}}\left(x\sqrt{2}\right)$ $\displaystyle=\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\sin\left(x+% \frac{\pi}{8}\right)-e^{-x}\sin\left(x-\frac{\pi}{8}\right)\right),$ $\displaystyle\operatorname{bei}_{-\frac{1}{2}}\left(x\sqrt{2}\right)$ $\displaystyle=-\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\cos\left(x+% \frac{\pi}{8}\right)+e^{-x}\cos\left(x-\frac{\pi}{8}\right)\right).$
 10.61.11 $\displaystyle\operatorname{ker}_{\frac{1}{2}}\left(x\sqrt{2}\right)$ $\displaystyle=\operatorname{kei}_{-\frac{1}{2}}\left(x\sqrt{2}\right)=-2^{-% \frac{3}{4}}\sqrt{\frac{\pi}{x}}e^{-x}\sin\left(x-\frac{\pi}{8}\right),$ 10.61.12 $\displaystyle\operatorname{kei}_{\frac{1}{2}}\left(x\sqrt{2}\right)$ $\displaystyle=-\operatorname{ker}_{-\frac{1}{2}}\left(x\sqrt{2}\right)=-2^{-% \frac{3}{4}}\sqrt{\frac{\pi}{x}}e^{-x}\cos\left(x-\frac{\pi}{8}\right).$