10 Bessel Functions Kelvin Functions 10.60 Sums 10.62 Graphs

§10.61 Definitions and Basic Properties

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Referenced by:: §10.61(i)
Permalink:: http://dlmf.nist.gov/10.61
See also:: Annotations for Ch.10

§10.61(i) Definitions

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Keywords:: Kelvin functions, definitions
Permalink:: http://dlmf.nist.gov/10.61.i
See also:: Annotations for §10.61 and Ch.10

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

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Keywords:: Kelvin functions, differential equations
Notes:: For (10.61.3) set $z=xe^{3\pi i/4}$ in (10.2.1). (10.61.4) follows by taking real and imaginary parts, and straightforward substitutions.
Permalink:: http://dlmf.nist.gov/10.61.ii
See also:: Annotations for §10.61 and Ch.10

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

§10.61(iii) Reflection Formulas for Arguments

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Keywords:: Kelvin functions, reflection formulas for arguments and orders
Notes:: See Whitehead (1911).
Permalink:: http://dlmf.nist.gov/10.61.iii
See also:: Annotations for §10.61 and Ch.10

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	$\displaystyle\operatorname{ber}_{n}\left(-x\right)$	$\displaystyle=(-1)^{n}\operatorname{ber}_{n}x,$
10.61.5	$\displaystyle\operatorname{bei}_{n}\left(-x\right)$	$\displaystyle=(-1)^{n}\operatorname{bei}_{n}x.$
ⓘ 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

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Keywords:: Kelvin functions, reflection formulas for arguments and orders
Notes:: See Whitehead (1911).
Referenced by:: §10.68(ii)
Permalink:: http://dlmf.nist.gov/10.61.iv
See also:: Annotations for §10.61 and Ch.10

10.61.6	$\displaystyle\operatorname{ber}_{-\nu}x$	$\displaystyle=\cos\left(\nu\pi\right)\operatorname{ber}_{\nu}x+\sin\left(\nu% \pi\right)\operatorname{bei}_{\nu}x+(2/\pi)\sin\left(\nu\pi\right)% \operatorname{ker}_{\nu}x,$
10.61.6	$\displaystyle\operatorname{bei}_{-\nu}x$	$\displaystyle=-\sin\left(\nu\pi\right)\operatorname{ber}_{\nu}x+\cos\left(\nu% \pi\right)\operatorname{bei}_{\nu}x+(2/\pi)\sin\left(\nu\pi\right)% \operatorname{kei}_{\nu}x.$
ⓘ Symbols: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : 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	$\displaystyle\operatorname{ker}_{-\nu}x$	$\displaystyle=\cos\left(\nu\pi\right)\operatorname{ker}_{\nu}x-\sin\left(\nu% \pi\right)\operatorname{kei}_{\nu}x,$
10.61.7	$\displaystyle\operatorname{kei}_{-\nu}x$	$\displaystyle=\sin\left(\nu\pi\right)\operatorname{ker}_{\nu}x+\cos\left(\nu% \pi\right)\operatorname{kei}_{\nu}x.$
ⓘ Symbols: $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : 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	$\displaystyle\operatorname{ber}_{-n}x$	$\displaystyle=(-1)^{n}\operatorname{ber}_{n}x,~\operatorname{bei}_{-n}x=(-1)^{% n}\operatorname{bei}_{n}x,$
10.61.8	$\displaystyle\operatorname{ker}_{-n}x$	$\displaystyle=(-1)^{n}\operatorname{ker}_{n}x,~\operatorname{kei}_{-n}x=(-1)^{% n}\operatorname{kei}_{n}x.$
ⓘ Symbols: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : 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(v) Orders $\pm\frac{1}{2}$

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Keywords:: Kelvin functions, orders $\pm\frac{1}{2}$
Notes:: (10.61.11) and (10.61.12) follow from the terminating forms of (10.67.1) and (10.67.2). Then (10.61.9) and (10.61.10) follow from these results and the terminating forms of (10.67.3) and (10.67.4). (Compare the derivation of the results given in §10.49(i) from (10.17.3)–(10.17.6).) The version of (10.61.9)–(10.61.10) given in Apelblat (1991) contains two sign errors.
Permalink:: http://dlmf.nist.gov/10.61.v
See also:: Annotations for §10.61 and Ch.10

10.61.9	$\displaystyle\operatorname{ber}_{\frac{1}{2}}\left(x\sqrt{2}\right)$	$\displaystyle=\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\cos\left(x+% \frac{\pi}{8}\right)-e^{-x}\cos\left(x-\frac{\pi}{8}\right)\right),$
10.61.9	$\displaystyle\operatorname{bei}_{\frac{1}{2}}\left(x\sqrt{2}\right)$	$\displaystyle=\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\sin\left(x+% \frac{\pi}{8}\right)+\,e^{-x}\sin\left(x-\frac{\pi}{8}\right)\right).$
ⓘ Symbols: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\sin\NVar{z}$ : 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	$\displaystyle\operatorname{ber}_{-\frac{1}{2}}\left(x\sqrt{2}\right)$	$\displaystyle=\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\sin\left(x+% \frac{\pi}{8}\right)-e^{-x}\sin\left(x-\frac{\pi}{8}\right)\right),$
10.61.10	$\displaystyle\operatorname{bei}_{-\frac{1}{2}}\left(x\sqrt{2}\right)$	$\displaystyle=-\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\cos\left(x+% \frac{\pi}{8}\right)+e^{-x}\cos\left(x-\frac{\pi}{8}\right)\right).$
ⓘ Symbols: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\sin\NVar{z}$ : 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	$\displaystyle\operatorname{ker}_{\frac{1}{2}}\left(x\sqrt{2}\right)$	$\displaystyle=\operatorname{kei}_{-\frac{1}{2}}\left(x\sqrt{2}\right)=-2^{-% \frac{3}{4}}\sqrt{\frac{\pi}{x}}e^{-x}\sin\left(x-\frac{\pi}{8}\right),$
ⓘ Symbols: $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\sin\NVar{z}$ : 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	$\displaystyle\operatorname{kei}_{\frac{1}{2}}\left(x\sqrt{2}\right)$	$\displaystyle=-\operatorname{ker}_{-\frac{1}{2}}\left(x\sqrt{2}\right)=-2^{-% \frac{3}{4}}\sqrt{\frac{\pi}{x}}e^{-x}\cos\left(x-\frac{\pi}{8}\right).$
ⓘ Symbols: $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : 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

10.60 Sums 10.62 Graphs

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§10.61 Definitions and Basic Properties

Contents

§10.61(i) Definitions

§10.61(ii) Differential Equations

§10.61(iii) Reflection Formulas for Arguments

§10.61(iv) Reflection Formulas for Orders

§10.61(v) Orders $\pm\frac{1}{2}$

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§10.61 Definitions and Basic Properties

Contents

§10.61(i) Definitions

§10.61(ii) Differential Equations

§10.61(iii) Reflection Formulas for Arguments

§10.61(iv) Reflection Formulas for Orders

§10.61(v) Orders ±12

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§10.61(v) Orders $\pm\frac{1}{2}$