§10.61 Definitions and Basic Properties

Referenced by:: §10.61(i)
Permalink:: http://dlmf.nist.gov/10.61

§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 $\pm\frac{1}{2}$

§10.61(i) Definitions

Keywords:: Kelvin functions
Permalink:: http://dlmf.nist.gov/10.61.i

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, $x$ : 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
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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, $x$ : 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
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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

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.
Keywords:: Kelvin functions, differential equations
Permalink:: http://dlmf.nist.gov/10.61.ii

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

Symbols:: $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $x$ : real variable and $\nu$ : complex parameter
A&S Ref:: 9.9.3
Referenced by:: §10.61(ii), §10.68(ii)
Permalink:: http://dlmf.nist.gov/10.61.E3
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10.61.4 $x^{4}\frac{{d}^{4}w}{{dx}^{4}}+2x^{3}\frac{{d}^{3}w}{{dx}^{3}}-(1+2\nu^{2})\left(x^{2}\frac{{d}^{2}w}{{dx}^{2}}-x\frac{dw}{dx}\right)+(\nu^{4}-4\nu^{2}+x^{4})w=0,$ $w=\mathop{\mathrm{ber}_{{\pm\nu}}\/}\nolimits x,\mathop{\mathrm{bei}_{{\pm\nu}}\/}\nolimits x,\mathop{\mathrm{ker}_{{\pm\nu}}\/}\nolimits x,\mathop{\mathrm{kei}_{{\pm\nu}}\/}\nolimits x$ .

Symbols:: $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits\!\left(x\right)$ : 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
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§10.61(iii) Reflection Formulas for Arguments

Notes:: See Whitehead (1911).
Keywords:: Kelvin functions
Permalink:: http://dlmf.nist.gov/10.61.iii

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

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

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

Symbols:: $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $n$ : integer and $x$ : real variable
A&S Ref:: 9.9.13
Permalink:: http://dlmf.nist.gov/10.61.E5
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§10.61(iv) Reflection Formulas for Orders

Notes:: See Whitehead (1911).
Keywords:: Kelvin functions
Referenced by:: §10.68(ii)
Permalink:: http://dlmf.nist.gov/10.61.iv

10.61.6

$\mathop{\mathrm{ber}_{{-\nu}}\/}\nolimits x=\mathop{\cos\/}\nolimits\!\left(\nu\pi\right)\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits x+\mathop{\sin\/}\nolimits\!\left(\nu\pi\right)\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits x+(2/\pi)\mathop{\sin\/}\nolimits\!\left(\nu\pi\right)\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits x,$

$\mathop{\mathrm{bei}_{{-\nu}}\/}\nolimits x=-\mathop{\sin\/}\nolimits\!\left(\nu\pi\right)\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits x+\mathop{\cos\/}\nolimits\!\left(\nu\pi\right)\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits x+(2/\pi)\mathop{\sin\/}\nolimits\!\left(\nu\pi\right)\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits x.$

Symbols:: $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : 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
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10.61.7

$\mathop{\mathrm{ker}_{{-\nu}}\/}\nolimits x=\mathop{\cos\/}\nolimits\!\left(\nu\pi\right)\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits x-\mathop{\sin\/}\nolimits\!\left(\nu\pi\right)\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits x,$

$\mathop{\mathrm{kei}_{{-\nu}}\/}\nolimits x=\mathop{\sin\/}\nolimits\!\left(\nu\pi\right)\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits x+\mathop{\cos\/}\nolimits\!\left(\nu\pi\right)\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits x.$

Symbols:: $\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real variable and $\nu$ : complex parameter
A&S Ref:: 9.9.6
Permalink:: http://dlmf.nist.gov/10.61.E7
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10.61.8

$\mathop{\mathrm{ber}_{{-n}}\/}\nolimits x=(-1)^{n}\mathop{\mathrm{ber}_{{n}}\/}\nolimits x,~\mathop{\mathrm{bei}_{{-n}}\/}\nolimits x=(-1)^{n}\mathop{\mathrm{bei}_{{n}}\/}\nolimits x,$

$\mathop{\mathrm{ker}_{{-n}}\/}\nolimits x=(-1)^{n}\mathop{\mathrm{ker}_{{n}}\/}\nolimits x,~\mathop{\mathrm{kei}_{{-n}}\/}\nolimits x=(-1)^{n}\mathop{\mathrm{kei}_{{n}}\/}\nolimits x.$

Symbols:: $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits\!\left(x\right)$ : 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
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§10.61(v) 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.
Keywords:: Kelvin functions
Permalink:: http://dlmf.nist.gov/10.61.v

10.61.9

$\mathop{\mathrm{ber}_{{\frac{1}{2}}}\/}\nolimits\!\left(x\sqrt{2}\right)=\frac{2^{{-\frac{3}{4}}}}{\sqrt{\pi x}}\left(e^{x}\mathop{\cos\/}\nolimits\!\left(x+\frac{\pi}{8}\right)-e^{{-x}}\mathop{\cos\/}\nolimits\!\left(x-\frac{\pi}{8}\right)\right),$

$\mathop{\mathrm{bei}_{{\frac{1}{2}}}\/}\nolimits\!\left(x\sqrt{2}\right)=\frac{2^{{-\frac{3}{4}}}}{\sqrt{\pi x}}\left(e^{x}\mathop{\sin\/}\nolimits\!\left(x+\frac{\pi}{8}\right)+\, e^{{-x}}\mathop{\sin\/}\nolimits\!\left(x-\frac{\pi}{8}\right)\right).$

Symbols:: $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\cos\/}\nolimits z$ : cosine function, $e$ : base of exponential function, $\mathop{\sin\/}\nolimits z$ : sine function and $x$ : real variable
Referenced by:: §10.61(v), Ch.10
Permalink:: http://dlmf.nist.gov/10.61.E9
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10.61.10

$\mathop{\mathrm{ber}_{{-\frac{1}{2}}}\/}\nolimits\!\left(x\sqrt{2}\right)=\frac{2^{{-\frac{3}{4}}}}{\sqrt{\pi x}}\left(e^{x}\mathop{\sin\/}\nolimits\!\left(x+\frac{\pi}{8}\right)-e^{{-x}}\mathop{\sin\/}\nolimits\!\left(x-\frac{\pi}{8}\right)\right),$

$\mathop{\mathrm{bei}_{{-\frac{1}{2}}}\/}\nolimits\!\left(x\sqrt{2}\right)=-\frac{2^{{-\frac{3}{4}}}}{\sqrt{\pi x}}\left(e^{x}\mathop{\cos\/}\nolimits\!\left(x+\frac{\pi}{8}\right)+e^{{-x}}\mathop{\cos\/}\nolimits\!\left(x-\frac{\pi}{8}\right)\right).$

Symbols:: $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\cos\/}\nolimits z$ : cosine function, $e$ : base of exponential function, $\mathop{\sin\/}\nolimits z$ : sine function and $x$ : real variable
Referenced by:: §10.61(v)
Permalink:: http://dlmf.nist.gov/10.61.E10
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10.61.11 $\mathop{\mathrm{ker}_{{\frac{1}{2}}}\/}\nolimits\!\left(x\sqrt{2}\right)=\mathop{\mathrm{kei}_{{-\frac{1}{2}}}\/}\nolimits\!\left(x\sqrt{2}\right)=-2^{{-\frac{3}{4}}}\sqrt{\frac{\pi}{x}}e^{{-x}}\mathop{\sin\/}\nolimits\!\left(x-\frac{\pi}{8}\right),$

Symbols:: $\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $e$ : base of exponential function, $\mathop{\sin\/}\nolimits z$ : sine function and $x$ : real variable
Referenced by:: §10.61(v)
Permalink:: http://dlmf.nist.gov/10.61.E11
Encodings:: TeX, pMML, png

10.61.12 $\mathop{\mathrm{kei}_{{\frac{1}{2}}}\/}\nolimits\!\left(x\sqrt{2}\right)=-\mathop{\mathrm{ker}_{{-\frac{1}{2}}}\/}\nolimits\!\left(x\sqrt{2}\right)=-2^{{-\frac{3}{4}}}\sqrt{\frac{\pi}{x}}e^{{-x}}\mathop{\cos\/}\nolimits\!\left(x-\frac{\pi}{8}\right).$

Symbols:: $\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\cos\/}\nolimits z$ : cosine function, $e$ : base of exponential function and $x$ : real variable
Referenced by:: §10.61(v), Ch.10
Permalink:: http://dlmf.nist.gov/10.61.E12
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