# §10.61 Definitions and Basic Properties

## §10.61(i) Definitions

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

 10.61.1 $\mathop{\mathrm{ber}_{\nu}\/}\nolimits x+i\mathop{\mathrm{bei}_{\nu}\/}% \nolimits x=\mathop{J_{\nu}\/}\nolimits\!\left(xe^{3\pi i/4}\right)=e^{\nu\pi i% }\mathop{J_{\nu}\/}\nolimits\!\left(xe^{-\pi i/4}\right)=e^{\nu\pi i/2}\mathop% {I_{\nu}\/}\nolimits\!\left(xe^{\pi i/4}\right)=e^{3\nu\pi i/2}\mathop{I_{\nu}% \/}\nolimits\!\left(xe^{-3\pi i/4}\right),$ Defines: $\mathop{\mathrm{bei}_{\nu}\/}\nolimits\!\left(x\right)$: Kelvin function and $\mathop{\mathrm{ber}_{\nu}\/}\nolimits\!\left(x\right)$: Kelvin function Symbols: $\mathop{J_{\nu}\/}\nolimits\!\left(z\right)$: Bessel function of the first kind, $e$: base of exponential function, $\mathop{I_{\nu}\/}\nolimits\!\left(z\right)$: 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
 10.61.2 $\mathop{\mathrm{ker}_{\nu}\/}\nolimits x+i\mathop{\mathrm{kei}_{\nu}\/}% \nolimits x=e^{-\nu\pi i/2}\mathop{K_{\nu}\/}\nolimits\!\left(xe^{\pi i/4}% \right)=\tfrac{1}{2}\pi i\mathop{{H^{(1)}_{\nu}}\/}\nolimits\!\left(xe^{3\pi i% /4}\right)=-\tfrac{1}{2}\pi ie^{-\nu\pi i}\mathop{{H^{(2)}_{\nu}}\/}\nolimits% \!\left(xe^{-\pi i/4}\right).$ Defines: $\mathop{\mathrm{kei}_{\nu}\/}\nolimits\!\left(x\right)$: Kelvin function and $\mathop{\mathrm{ker}_{\nu}\/}\nolimits\!\left(x\right)$: Kelvin function Symbols: $\mathop{{H^{(1)}_{\nu}}\/}\nolimits\!\left(z\right)$: Bessel function of the third kind (or Hankel function), $\mathop{{H^{(2)}_{\nu}}\/}\nolimits\!\left(z\right)$: Bessel function of the third kind (or Hankel function), $e$: base of exponential function, $\mathop{K_{\nu}\/}\nolimits\!\left(z\right)$: 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

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

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

## §10.61(ii) Differential Equations

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

## §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 $\mathop{\mathrm{ber}_{\nu}\/}\nolimits$ and $\mathop{\mathrm{bei}_{\nu}\/}\nolimits$ when $\nu$ is an integer. In particular,

 10.61.5 $\displaystyle\mathop{\mathrm{ber}_{n}\/}\nolimits\!\left(-x\right)$ $\displaystyle=(-1)^{n}\mathop{\mathrm{ber}_{n}\/}\nolimits x,$ $\displaystyle\mathop{\mathrm{bei}_{n}\/}\nolimits\!\left(-x\right)$ $\displaystyle=(-1)^{n}\mathop{\mathrm{bei}_{n}\/}\nolimits x.$

## §10.61(iv) Reflection Formulas for Orders

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

## §10.61(v) Orders $\pm\frac{1}{2}$

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