§10.68 Modulus and Phase Functions

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Permalink:: http://dlmf.nist.gov/10.68
See also:: Annotations for Ch.10

§10.68(i) Definitions
§10.68(ii) Basic Properties
§10.68(iii) Asymptotic Expansions for Large Argument
§10.68(iv) Further Properties

§10.68(i) Definitions

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Keywords:: Kelvin functions, definitions, modulus and phase functions
Referenced by:: §10.62, §10.68(ii), §10.71(i)
Permalink:: http://dlmf.nist.gov/10.68.i
See also:: Annotations for §10.68 and Ch.10

10.68.1 $M_{\nu}\left(x\right)e^{i\theta_{\nu}\left(x\right)}=\operatorname{ber}_{\nu}x% +i\operatorname{bei}_{\nu}x,$

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Symbols:: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $M_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of Bessel functions, $\theta_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of Bessel functions, : real variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.68.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §10.68(i), §10.68 and Ch.10

10.68.2 $N_{\nu}\left(x\right)e^{i\phi_{\nu}\left(x\right)}=\operatorname{ker}_{\nu}x+i% \operatorname{kei}_{\nu}x,$

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Symbols:: $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $N_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of derivatives of Bessel functions, $\phi_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of derivatives of Bessel functions, : real variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.68.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §10.68(i), §10.68 and Ch.10

where $M_{\nu}\left(x\right)\,(>0)$ , $N_{\nu}\left(x\right)\,(>0)$ , $\theta_{\nu}\left(x\right)$ , and $\phi_{\nu}\left(x\right)$ are continuous real functions of and $\nu$ , with the branches of $\theta_{\nu}\left(x\right)$ and $\phi_{\nu}\left(x\right)$ chosen to satisfy (10.68.18) and (10.68.21) as $x\to\infty$ . (See also §10.68(iv).)

§10.68(ii) Basic Properties

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Keywords:: Kelvin functions, modulus and phase functions, properties
Notes:: These formulas are derived from the definitions §10.68(i), the differential equation (10.61.3), the reflection formulas in §10.61(iv), and recurrence relations in §10.63(i) by straightforward manipulations.
Permalink:: http://dlmf.nist.gov/10.68.ii
See also:: Annotations for §10.68 and Ch.10

10.68.3

$\operatorname{ber}_{\nu}x=M_{\nu}\left(x\right)\cos\theta_{\nu}\left(x\right),$

$\operatorname{bei}_{\nu}x=M_{\nu}\left(x\right)\sin\theta_{\nu}\left(x\right),$

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Symbols:: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\cos\NVar{z}$ : cosine function, $M_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of Bessel functions, $\theta_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of Bessel functions, $\sin\NVar{z}$ : sine function, : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.9, 9.10.19
Permalink:: http://dlmf.nist.gov/10.68.E3
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §10.68(ii), §10.68 and Ch.10

10.68.4

$\operatorname{ker}_{\nu}x=N_{\nu}\left(x\right)\cos\phi_{\nu}\left(x\right),$

$\operatorname{kei}_{\nu}x=N_{\nu}\left(x\right)\sin\phi_{\nu}\left(x\right).$

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Symbols:: $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\cos\NVar{z}$ : cosine function, $N_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of derivatives of Bessel functions, $\phi_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of derivatives of Bessel functions, $\sin\NVar{z}$ : sine function, : real variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.68.E4
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §10.68(ii), §10.68 and Ch.10

10.68.5

$M_{\nu}\left(x\right)=({\operatorname{ber}_{\nu}}^{2}x+{\operatorname{bei}_{% \nu}}^{2}x)^{\ifrac{1}{2}},$

$N_{\nu}\left(x\right)=({\operatorname{ker}_{\nu}}^{2}x+{\operatorname{kei}_{% \nu}}^{2}x)^{\ifrac{1}{2}},$

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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, $M_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of Bessel functions, $N_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of derivatives of Bessel functions, : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.8, 9.10.18
Referenced by:: §10.68(iii), §10.71(i)
Permalink:: http://dlmf.nist.gov/10.68.E5
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §10.68(ii), §10.68 and Ch.10

10.68.6

$\theta_{\nu}\left(x\right)=\operatorname{Arctan}\left(\operatorname{bei}_{\nu}% x/\operatorname{ber}_{\nu}x\right),$

$\phi_{\nu}\left(x\right)=\operatorname{Arctan}\left(\operatorname{kei}_{\nu}x/% \operatorname{ker}_{\nu}x\right).$

10.68.7

$M_{-n}\left(x\right)=M_{n}\left(x\right),$

$\theta_{-n}\left(x\right)=\theta_{n}\left(x\right)-n\pi.$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $M_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of Bessel functions, $\theta_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of Bessel functions, : integer and : real variable
A&S Ref:: 9.10.10
Referenced by:: §10.68(ii)
Permalink:: http://dlmf.nist.gov/10.68.E7
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §10.68(ii), §10.68 and Ch.10

With arguments suppressed,

10.68.8 $\operatorname{ber}_{\nu}'x=\tfrac{1}{2}M_{\nu+1}\cos\left(\theta_{\nu+1}-% \tfrac{1}{4}\pi\right)-\tfrac{1}{2}M_{\nu-1}\cos\left(\theta_{\nu-1}-\tfrac{1}% {4}\pi\right)=(\nu/x)M_{\nu}\cos\theta_{\nu}+M_{\nu+1}\cos\left(\theta_{\nu+1}% -\tfrac{1}{4}\pi\right)=-(\nu/x)M_{\nu}\cos\theta_{\nu}-M_{\nu-1}\cos\left(% \theta_{\nu-1}-\tfrac{1}{4}\pi\right),$

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Symbols:: $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $M_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of Bessel functions, $\theta_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of Bessel functions, : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.11
Referenced by:: §10.68(ii)
Permalink:: http://dlmf.nist.gov/10.68.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §10.68(ii), §10.68 and Ch.10

10.68.9 $\operatorname{bei}_{\nu}'x=\tfrac{1}{2}M_{\nu+1}\sin\left(\theta_{\nu+1}-% \tfrac{1}{4}\pi\right)-\tfrac{1}{2}M_{\nu-1}\sin\left(\theta_{\nu-1}-\tfrac{1}% {4}\pi\right)=(\nu/x)M_{\nu}\sin\theta_{\nu}+M_{\nu+1}\sin\left(\theta_{\nu+1}% -\tfrac{1}{4}\pi\right)=-(\nu/x)M_{\nu}\sin\theta_{\nu}-M_{\nu-1}\sin\left(% \theta_{\nu-1}-\tfrac{1}{4}\pi\right).$

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Symbols:: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\pi$ : the ratio of the circumference of a circle to its diameter, $M_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of Bessel functions, $\theta_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of Bessel functions, $\sin\NVar{z}$ : sine function, : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.12
Permalink:: http://dlmf.nist.gov/10.68.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §10.68(ii), §10.68 and Ch.10

10.68.10

$\operatorname{ber}'x=M_{1}\cos\left(\theta_{1}-\tfrac{1}{4}\pi\right),$

$\operatorname{bei}'x=M_{1}\sin\left(\theta_{1}-\tfrac{1}{4}\pi\right).$

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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, $M_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of Bessel functions, $\theta_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of Bessel functions, $\sin\NVar{z}$ : sine function and : real variable
A&S Ref:: 9.10.13
Permalink:: http://dlmf.nist.gov/10.68.E10
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §10.68(ii), §10.68 and Ch.10

10.68.11 $M_{\nu}'=(\nu/x)M_{\nu}+M_{\nu+1}\cos\left(\theta_{\nu+1}-\theta_{\nu}-\tfrac{% 1}{4}\pi\right)=-(\nu/x)M_{\nu}-M_{\nu-1}\cos\left(\theta_{\nu-1}-\theta_{\nu}% -\tfrac{1}{4}\pi\right),$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $M_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of Bessel functions, $\theta_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of Bessel functions, : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.14
Permalink:: http://dlmf.nist.gov/10.68.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §10.68(ii), §10.68 and Ch.10

10.68.12 $\theta_{\nu}'=(M_{\nu+1}/M_{\nu})\sin\left(\theta_{\nu+1}-\theta_{\nu}-\tfrac{% 1}{4}\pi\right)=-(M_{\nu-1}/M_{\nu})\sin\left(\theta_{\nu-1}-\theta_{\nu}-% \tfrac{1}{4}\pi\right).$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $M_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of Bessel functions, $\theta_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of Bessel functions, $\sin\NVar{z}$ : sine function and $\nu$ : complex parameter
A&S Ref:: 9.10.15
Permalink:: http://dlmf.nist.gov/10.68.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §10.68(ii), §10.68 and Ch.10

10.68.13

$M_{0}'=M_{1}\cos\left(\theta_{1}-\theta_{0}-\tfrac{1}{4}\pi\right),$

$\theta_{0}'=(M_{1}/M_{0})\sin\left(\theta_{1}-\theta_{0}-\tfrac{1}{4}\pi\right).$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $M_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of Bessel functions, $\theta_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of Bessel functions and $\sin\NVar{z}$ : sine function
A&S Ref:: 9.10.16
Permalink:: http://dlmf.nist.gov/10.68.E13
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §10.68(ii), §10.68 and Ch.10

10.68.14

$\ifrac{\mathrm{d}(x{M_{\nu}}^{2}\theta_{\nu}')}{\mathrm{d}x}=x{M_{\nu}}^{2},$

$x^{2}M_{\nu}''+xM_{\nu}'-\nu^{2}M_{\nu}=x^{2}M_{\nu}{\theta_{\nu}'}^{2}.$

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , $M_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of Bessel functions, $\theta_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of Bessel functions, : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.17
Referenced by:: §10.68(ii)
Permalink:: http://dlmf.nist.gov/10.68.E14
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §10.68(ii), §10.68 and Ch.10

Equations (10.68.8)–(10.68.14) also hold with the symbols $\operatorname{ber}$ , $\operatorname{bei}$ , , and $\theta$ replaced throughout by $\operatorname{ker}$ , $\operatorname{kei}$ , , and $\phi$ , respectively. In place of (10.68.7),

10.68.15

$N_{-\nu}\left(x\right)=N_{\nu}\left(x\right),$

$\phi_{-\nu}\left(x\right)=\phi_{\nu}\left(x\right)+\nu\pi.$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $N_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of derivatives of Bessel functions, $\phi_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of derivatives of Bessel functions, : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.20
Permalink:: http://dlmf.nist.gov/10.68.E15
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §10.68(ii), §10.68 and Ch.10

§10.68(iii) Asymptotic Expansions for Large Argument

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Keywords:: Kelvin functions, asymptotic expansions for large argument, modulus and phase functions
Notes:: Combine (10.68.5) and (10.68.6) with (10.67.1)–(10.67.4), ignoring the exponentially-small terms in (10.67.3) and (10.67.4). See also Whitehead (1911) and Young and Kirk (1964, pp. xiv–xv).
Permalink:: http://dlmf.nist.gov/10.68.iii
See also:: Annotations for §10.68 and Ch.10

When $\nu$ is fixed, $\mu=4\nu^{2}$ , and $x\to\infty$

10.68.16 $M_{\nu}\left(x\right)=\frac{e^{x/\sqrt{2}}}{(2\pi x)^{\frac{1}{2}}}\left(1-% \frac{\mu-1}{8\sqrt{2}}\frac{1}{x}+\frac{(\mu-1)^{2}}{256}\frac{1}{x^{2}}-% \frac{(\mu-1)(\mu^{2}+14\mu-399)}{6144\sqrt{2}}\frac{1}{x^{3}}+O\left(\frac{1}% {x^{4}}\right)\right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $M_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of Bessel functions, : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.21
Permalink:: http://dlmf.nist.gov/10.68.E16
Encodings:: TeX, pMML, png
See also:: Annotations for §10.68(iii), §10.68 and Ch.10

10.68.17 $\ln M_{\nu}\left(x\right)=\frac{x}{\sqrt{2}}-\frac{1}{2}\ln\left(2\pi x\right)% -\frac{\mu-1}{8\sqrt{2}}\frac{1}{x}-\frac{(\mu-1)(\mu-25)}{384\sqrt{2}}\frac{1% }{x^{3}}-\frac{(\mu-1)(\mu-13)}{128}\frac{1}{x^{4}}+O\left(\frac{1}{x^{5}}% \right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $M_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of Bessel functions, $\ln\NVar{z}$ : principal branch of logarithm function, : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.22
Permalink:: http://dlmf.nist.gov/10.68.E17
Encodings:: TeX, pMML, png
See also:: Annotations for §10.68(iii), §10.68 and Ch.10

10.68.18 $\theta_{\nu}\left(x\right)=\frac{x}{\sqrt{2}}+\left(\frac{1}{2}\nu-\frac{1}{8}% \right)\pi+\frac{\mu-1}{8\sqrt{2}}\frac{1}{x}+\frac{\mu-1}{16}\frac{1}{x^{2}}-% \frac{(\mu-1)(\mu-25)}{384\sqrt{2}}\frac{1}{x^{3}}+O\left(\frac{1}{x^{5}}% \right).$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\theta_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of Bessel functions, : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.23
Referenced by:: §10.68(i), §10.70
Permalink:: http://dlmf.nist.gov/10.68.E18
Encodings:: TeX, pMML, png
See also:: Annotations for §10.68(iii), §10.68 and Ch.10

10.68.19 $N_{\nu}\left(x\right)=e^{-x/\sqrt{2}}\left(\frac{\pi}{2x}\right)^{\frac{1}{2}}% \left(1+\frac{\mu-1}{8\sqrt{2}}\frac{1}{x}+\frac{(\mu-1)^{2}}{256}\frac{1}{x^{% 2}}+\frac{(\mu-1)(\mu^{2}+14\mu-399)}{6144\sqrt{2}}\frac{1}{x^{3}}+O\left(% \frac{1}{x^{4}}\right)\right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $N_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of derivatives of Bessel functions, : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.24
Permalink:: http://dlmf.nist.gov/10.68.E19
Encodings:: TeX, pMML, png
See also:: Annotations for §10.68(iii), §10.68 and Ch.10

10.68.20 $\ln N_{\nu}\left(x\right)=-\frac{x}{\sqrt{2}}+\frac{1}{2}\ln\left(\frac{\pi}{2% x}\right)+\frac{\mu-1}{8\sqrt{2}}\frac{1}{x}+\frac{(\mu-1)(\mu-25)}{384\sqrt{2% }}\frac{1}{x^{3}}-\frac{(\mu-1)(\mu-13)}{128}\frac{1}{x^{4}}+O\left(\frac{1}{x% ^{5}}\right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $N_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of derivatives of Bessel functions, $\ln\NVar{z}$ : principal branch of logarithm function, : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.25
Permalink:: http://dlmf.nist.gov/10.68.E20
Encodings:: TeX, pMML, png
See also:: Annotations for §10.68(iii), §10.68 and Ch.10

10.68.21 $\phi_{\nu}\left(x\right)=-\frac{x}{\sqrt{2}}-\left(\frac{1}{2}\nu+\frac{1}{8}% \right)\pi-\frac{\mu-1}{8\sqrt{2}}\frac{1}{x}+\frac{\mu-1}{16}\frac{1}{x^{2}}+% \frac{(\mu-1)(\mu-25)}{384\sqrt{2}}\frac{1}{x^{3}}+O\left(\frac{1}{x^{5}}% \right).$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\phi_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of derivatives of Bessel functions, : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.26
Referenced by:: §10.68(i), §10.70
Permalink:: http://dlmf.nist.gov/10.68.E21
Encodings:: TeX, pMML, png
See also:: Annotations for §10.68(iii), §10.68 and Ch.10

§10.68(iv) Further Properties

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Referenced by:: §10.68(i), 1st item
Permalink:: http://dlmf.nist.gov/10.68.iv
See also:: Annotations for §10.68 and Ch.10

Additional properties of the modulus and phase functions are given in Young and Kirk (1964, pp. xi–xv). However, care needs to be exercised with the branches of the phases. Thus this reference gives $\phi_{1}\left(0\right)=\tfrac{5}{4}\pi$ (Eq. (6.10)), and $\lim_{x\to\infty}(\phi_{1}\left(x\right)+(x/\sqrt{2}))=-\tfrac{5}{8}\pi$ (Eqs. (10.20) and (Eqs. (10.26b)). However, numerical tabulations show that if the second of these equations applies and $\phi_{1}\left(x\right)$ is continuous, then $\phi_{1}\left(0\right)=-\tfrac{3}{4}\pi$ ; compare Abramowitz and Stegun (1964, p. 433).

§10.68 Modulus and Phase Functions

Contents

§10.68(i) Definitions

§10.68(ii) Basic Properties

§10.68(iii) Asymptotic Expansions for Large Argument

§10.68(iv) Further Properties