§10.61 Definitions and Basic Properties

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

10.61.1		$$\begin{array}{ll}{\mathrm{ber}}_{\nu }x+\mathrm{i}{\mathrm{bei}}_{\nu }x& =J_{\nu }\left(x{\mathrm{e}}^{3\pi \mathrm{i}/4}\right)={\mathrm{e}}^{\nu \pi \mathrm{i}}{J}_{\nu }\left(x{\mathrm{e}}^{-\pi \mathrm{i}/4}\right)\\ & ={\mathrm{e}}^{\nu \pi \mathrm{i}/2}{I}_{\nu }\left(x{\mathrm{e}}^{\pi \mathrm{i}/4}\right)={\mathrm{e}}^{3\nu \pi \mathrm{i}/2}{I}_{\nu }\left(x{\mathrm{e}}^{-3\pi \mathrm{i}/4}\right),\end{array}$$
ⓘ Defines: ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function and ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function Symbols: $J_{\nu }\left(z\right)$: 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, $I_{\nu }\left(z\right)$: 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 See also: Annotations for §10.61(i), §10.61 and Ch.10

10.61.2		$$\begin{array}{ll}{\ker }_{\nu }x+\mathrm{i}{\mathrm{kei}}_{\nu }x& ={\mathrm{e}}^{-\nu \pi \mathrm{i}/2}{K}_{\nu }\left(x{\mathrm{e}}^{\pi \mathrm{i}/4}\right)=\frac{1}{2}\pi \mathrm{i}{H}_{\nu }^{(1)}\left(x{\mathrm{e}}^{3\pi \mathrm{i}/4}\right)\\ & =-\frac{1}{2}\pi \mathrm{i}{\mathrm{e}}^{-\nu \pi \mathrm{i}}{H}_{\nu }^{(2)}\left(x{\mathrm{e}}^{-\pi \mathrm{i}/4}\right).\end{array}$$
ⓘ Defines: ${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function and ${\ker }_{\nu }\left(x\right)$: Kelvin function Symbols: $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: 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, $K_{\nu }\left(z\right)$: 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 See also: Annotations for §10.61(i), §10.61 and Ch.10

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

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

10.61.3		$$x^2\frac{d^{2}w}{{dx}^2}+x\frac{dw}{dx}-(\mathrm{i}{x}^2+{\nu }^2)w=0,$$
ⓘ Symbols: ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\mathrm{i}$: imaginary unit, $x$: real variable and $\nu$: complex parameter A&S Ref: 9.9.3 Referenced by: §10.61(ii), §10.68(ii) Permalink: http://dlmf.nist.gov/10.61.E3 Encodings: TeX, pMML, png See also: Annotations for §10.61(ii), §10.61 and Ch.10

10.61.4		$$x^4\frac{d^{4}w}{{dx}^4}+2x^3\frac{d^{3}w}{{dx}^3}-(1+2{\nu }^2)\left(x^2\frac{d^{2}w}{{dx}^2}-x\frac{dw}{dx}\right)+({\nu }^4-4{\nu }^2+x^4)w=0,$$
$w={\mathrm{ber}}_{\pm \nu }x,{\mathrm{bei}}_{\pm \nu }x,{\ker }_{\pm \nu }x,{\mathrm{kei}}_{\pm \nu }x$.
ⓘ Symbols: ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $x$: real variable and $\nu$: complex parameter A&S Ref: 9.9.4 Referenced by: §10.61(ii) Permalink: http://dlmf.nist.gov/10.61.E4 Encodings: TeX, pMML, png See also: Annotations for §10.61(ii), §10.61 and Ch.10

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

10.61.5		${\mathrm{ber}}_n\left(-x\right)$	$={(-1)}^n{\mathrm{ber}}_{n}x,$
		${\mathrm{bei}}_n\left(-x\right)$	$={(-1)}^n{\mathrm{bei}}_{n}x.$
ⓘ Symbols: ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: 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

10.61.6		${\mathrm{ber}}_{-\nu }x$	$=\cos \left(\nu \pi \right){\mathrm{ber}}_{\nu }x+\sin \left(\nu \pi \right){\mathrm{bei}}_{\nu }x+(2/\pi )\sin \left(\nu \pi \right){\ker }_{\nu }x,$
		${\mathrm{bei}}_{-\nu }x$	$=-\sin \left(\nu \pi \right){\mathrm{ber}}_{\nu }x+\cos \left(\nu \pi \right){\mathrm{bei}}_{\nu }x+(2/\pi )\sin \left(\nu \pi \right){\mathrm{kei}}_{\nu }x.$
ⓘ Symbols: ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $x$: real variable and $\nu$: complex parameter A&S Ref: 9.9.5 Referenced by: §10.65(ii) Permalink: http://dlmf.nist.gov/10.61.E6 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.61(iv), §10.61 and Ch.10

10.61.7		${\ker }_{-\nu }x$	$=\cos \left(\nu \pi \right){\ker }_{\nu }x-\sin \left(\nu \pi \right){\mathrm{kei}}_{\nu }x,$
		${\mathrm{kei}}_{-\nu }x$	$=\sin \left(\nu \pi \right){\ker }_{\nu }x+\cos \left(\nu \pi \right){\mathrm{kei}}_{\nu }x.$
ⓘ Symbols: ${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $x$: real variable and $\nu$: complex parameter A&S Ref: 9.9.6 Permalink: http://dlmf.nist.gov/10.61.E7 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.61(iv), §10.61 and Ch.10

10.61.8		${\mathrm{ber}}_{-n}x$	$={(-1)}^n{\mathrm{ber}}_{n}x,{\mathrm{bei}}_{-n}x={(-1)}^n{\mathrm{bei}}_{n}x,$
		${\ker }_{-n}x$	$={(-1)}^n{\ker }_{n}x,{\mathrm{kei}}_{-n}x={(-1)}^n{\mathrm{kei}}_{n}x.$
ⓘ Symbols: ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $n$: integer and $x$: real variable A&S Ref: 9.9.7, 9.9.8 Permalink: http://dlmf.nist.gov/10.61.E8 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.61(iv), §10.61 and Ch.10

10.61.9		${\mathrm{ber}}_{\frac{1}{2}}\left(x\sqrt{2}\right)$	$={\displaystyle \frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}}\left({\mathrm{e}}^x\cos \left(x+{\displaystyle \frac{\pi }{8}}\right)-{\mathrm{e}}^{-x}\cos \left(x-{\displaystyle \frac{\pi }{8}}\right)\right),$
		${\mathrm{bei}}_{\frac{1}{2}}\left(x\sqrt{2}\right)$	$={\displaystyle \frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}}\left({\mathrm{e}}^x\sin \left(x+{\displaystyle \frac{\pi }{8}}\right)+{\mathrm{e}}^{-x}\sin \left(x-{\displaystyle \frac{\pi }{8}}\right)\right).$
ⓘ Symbols: ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\sin z$: sine function and $x$: real variable Referenced by: §10.61(v), Ch.10 Permalink: http://dlmf.nist.gov/10.61.E9 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.61(v), §10.61 and Ch.10

10.61.10		${\mathrm{ber}}_{-\frac{1}{2}}\left(x\sqrt{2}\right)$	$={\displaystyle \frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}}\left({\mathrm{e}}^x\sin \left(x+{\displaystyle \frac{\pi }{8}}\right)-{\mathrm{e}}^{-x}\sin \left(x-{\displaystyle \frac{\pi }{8}}\right)\right),$
		${\mathrm{bei}}_{-\frac{1}{2}}\left(x\sqrt{2}\right)$	$=-{\displaystyle \frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}}\left({\mathrm{e}}^x\cos \left(x+{\displaystyle \frac{\pi }{8}}\right)+{\mathrm{e}}^{-x}\cos \left(x-{\displaystyle \frac{\pi }{8}}\right)\right).$
ⓘ Symbols: ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\sin z$: sine function and $x$: real variable Referenced by: §10.61(v) Permalink: http://dlmf.nist.gov/10.61.E10 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.61(v), §10.61 and Ch.10

10.61.11		${\ker }_{\frac{1}{2}}\left(x\sqrt{2}\right)$	$={\mathrm{kei}}_{-\frac{1}{2}}\left(x\sqrt{2}\right)=-2^{-\frac{3}{4}}\sqrt{{\displaystyle \frac{\pi }{x}}}{\mathrm{e}}^{-x}\sin \left(x-{\displaystyle \frac{\pi }{8}}\right),$
ⓘ Symbols: ${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\sin z$: sine function and $x$: real variable Referenced by: §10.61(v) Permalink: http://dlmf.nist.gov/10.61.E11 Encodings: TeX, pMML, png See also: Annotations for §10.61(v), §10.61 and Ch.10
10.61.12		${\mathrm{kei}}_{\frac{1}{2}}\left(x\sqrt{2}\right)$	$=-{\ker }_{-\frac{1}{2}}\left(x\sqrt{2}\right)=-2^{-\frac{3}{4}}\sqrt{{\displaystyle \frac{\pi }{x}}}{\mathrm{e}}^{-x}\cos \left(x-{\displaystyle \frac{\pi }{8}}\right).$
ⓘ Symbols: ${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm and $x$: real variable Referenced by: §10.61(v), Ch.10 Permalink: http://dlmf.nist.gov/10.61.E12 Encodings: TeX, pMML, png See also: Annotations for §10.61(v), §10.61 and Ch.10