§10.68 Modulus and Phase Functions

Permalink:: http://dlmf.nist.gov/10.68

§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

Keywords:: Kelvin functions
Referenced by:: §10.62, §10.68(ii), ¶ ‣ §10.71(i)
Permalink:: http://dlmf.nist.gov/10.68.i

10.68.1 $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)e^{{i\!\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)}}=\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits x+i\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits x,$

Symbols:: $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $e$ : base of exponential function, $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of Bessel functions, $\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of Bessel functions, $x$ : real variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.68.E1
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10.68.2 $\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right)e^{{i\!\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right)}}=\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits x+i\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits x,$

Symbols:: $\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $e$ : base of exponential function, $\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of derivatives of Bessel functions, $\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of derivatives of Bessel functions, $x$ : real variable and $\nu$ : complex parameter
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where $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)\,(>0)$ , $\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right)\,(>0)$ , $\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)$ , and $\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right)$ are continuous real functions of $x$ and $\nu$ , with the branches of $\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)$ and $\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right)$ chosen to satisfy (10.68.18) and (10.68.21) as $x\to\infty$ . (See also §10.68(iv).)

§10.68(ii) Basic Properties

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.
Keywords:: Kelvin functions
Permalink:: http://dlmf.nist.gov/10.68.ii

10.68.3

$\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits x=\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)\mathop{\cos\/}\nolimits\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right),$

$\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits x=\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)\mathop{\sin\/}\nolimits\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right),$

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, $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of Bessel functions, $\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of Bessel functions, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.9, 9.10.19
Permalink:: http://dlmf.nist.gov/10.68.E3
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10.68.4

$\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits x=\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right)\mathop{\cos\/}\nolimits\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right),$

$\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits x=\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right)\mathop{\sin\/}\nolimits\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right).$

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, $\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of derivatives of Bessel functions, $\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of derivatives of Bessel functions, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real variable and $\nu$ : complex parameter
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10.68.5

$\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)=({\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits^{{2}}}x+{\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits^{{2}}}x)^{{\ifrac{1}{2}}},$

$\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right)=({\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits^{{2}}}x+{\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits^{{2}}}x)^{{\ifrac{1}{2}}},$

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, $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of Bessel functions, $\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of derivatives of Bessel functions, $x$ : 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
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10.68.6

$\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)=\mathop{\mathrm{Arctan}\/}\nolimits\!\left(\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits x/\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits x\right),$

$\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right)=\mathop{\mathrm{Arctan}\/}\nolimits\!\left(\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits x/\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits x\right).$

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, $\mathop{\mathrm{Arctan}\/}\nolimits z$ : general arctangent function, $\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of derivatives of Bessel functions, $\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of Bessel functions, $x$ : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.8, 9.10.18
Referenced by:: §10.68(iii)
Permalink:: http://dlmf.nist.gov/10.68.E6
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10.68.7

$\mathop{M_{{-n}}\/}\nolimits\!\left(x\right)=\mathop{M_{{n}}\/}\nolimits\!\left(x\right),$

$\mathop{\theta _{{-n}}\/}\nolimits\!\left(x\right)=\mathop{\theta _{{n}}\/}\nolimits\!\left(x\right)-n\pi.$

Symbols:: $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of Bessel functions, $\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of Bessel functions, $n$ : integer and $x$ : real variable
A&S Ref:: 9.10.10
Referenced by:: §10.68(ii)
Permalink:: http://dlmf.nist.gov/10.68.E7
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With arguments $(x)$ suppressed,

10.68.8 ${\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits^{{\prime}}}x=\tfrac{1}{2}\mathop{M_{{\nu+1}}\/}\nolimits\mathop{\cos\/}\nolimits\!\left(\mathop{\theta _{{\nu+1}}\/}\nolimits-\tfrac{1}{4}\pi\right)-\tfrac{1}{2}\mathop{M_{{\nu-1}}\/}\nolimits\mathop{\cos\/}\nolimits\!\left(\mathop{\theta _{{\nu-1}}\/}\nolimits-\tfrac{1}{4}\pi\right)=(\nu/x)\mathop{M_{{\nu}}\/}\nolimits\mathop{\cos\/}\nolimits\mathop{\theta _{{\nu}}\/}\nolimits+\mathop{M_{{\nu+1}}\/}\nolimits\mathop{\cos\/}\nolimits\!\left(\mathop{\theta _{{\nu+1}}\/}\nolimits-\tfrac{1}{4}\pi\right)=-(\nu/x)\mathop{M_{{\nu}}\/}\nolimits\mathop{\cos\/}\nolimits\mathop{\theta _{{\nu}}\/}\nolimits-\mathop{M_{{\nu-1}}\/}\nolimits\mathop{\cos\/}\nolimits\!\left(\mathop{\theta _{{\nu-1}}\/}\nolimits-\tfrac{1}{4}\pi\right),$

Symbols:: $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of Bessel functions, $\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of Bessel functions, $x$ : 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
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10.68.9 ${\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits^{{\prime}}}x=\tfrac{1}{2}\mathop{M_{{\nu+1}}\/}\nolimits\mathop{\sin\/}\nolimits\!\left(\mathop{\theta _{{\nu+1}}\/}\nolimits-\tfrac{1}{4}\pi\right)-\tfrac{1}{2}\mathop{M_{{\nu-1}}\/}\nolimits\mathop{\sin\/}\nolimits\!\left(\mathop{\theta _{{\nu-1}}\/}\nolimits-\tfrac{1}{4}\pi\right)=(\nu/x)\mathop{M_{{\nu}}\/}\nolimits\mathop{\sin\/}\nolimits\mathop{\theta _{{\nu}}\/}\nolimits+\mathop{M_{{\nu+1}}\/}\nolimits\mathop{\sin\/}\nolimits\!\left(\mathop{\theta _{{\nu+1}}\/}\nolimits-\tfrac{1}{4}\pi\right)=-(\nu/x)\mathop{M_{{\nu}}\/}\nolimits\mathop{\sin\/}\nolimits\mathop{\theta _{{\nu}}\/}\nolimits-\mathop{M_{{\nu-1}}\/}\nolimits\mathop{\sin\/}\nolimits\!\left(\mathop{\theta _{{\nu-1}}\/}\nolimits-\tfrac{1}{4}\pi\right).$

Symbols:: $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of Bessel functions, $\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of Bessel functions, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.12
Permalink:: http://dlmf.nist.gov/10.68.E9
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10.68.10

${\mathop{\mathrm{ber}\/}\nolimits^{{\prime}}}x=\mathop{M_{{1}}\/}\nolimits\mathop{\cos\/}\nolimits\!\left(\mathop{\theta _{{1}}\/}\nolimits-\tfrac{1}{4}\pi\right),$

${\mathop{\mathrm{bei}\/}\nolimits^{{\prime}}}x=\mathop{M_{{1}}\/}\nolimits\mathop{\sin\/}\nolimits\!\left(\mathop{\theta _{{1}}\/}\nolimits-\tfrac{1}{4}\pi\right).$

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, $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of Bessel functions, $\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of Bessel functions, $\mathop{\sin\/}\nolimits z$ : sine function and $x$ : real variable
A&S Ref:: 9.10.13
Permalink:: http://dlmf.nist.gov/10.68.E10
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10.68.11 ${\mathop{M_{{\nu}}\/}\nolimits^{{\prime}}}=(\nu/x)\mathop{M_{{\nu}}\/}\nolimits+\mathop{M_{{\nu+1}}\/}\nolimits\mathop{\cos\/}\nolimits\!\left(\mathop{\theta _{{\nu+1}}\/}\nolimits-\mathop{\theta _{{\nu}}\/}\nolimits-\tfrac{1}{4}\pi\right)=-(\nu/x)\mathop{M_{{\nu}}\/}\nolimits-\mathop{M_{{\nu-1}}\/}\nolimits\mathop{\cos\/}\nolimits\!\left(\mathop{\theta _{{\nu-1}}\/}\nolimits-\mathop{\theta _{{\nu}}\/}\nolimits-\tfrac{1}{4}\pi\right),$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of Bessel functions, $\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of Bessel functions, $x$ : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.14
Permalink:: http://dlmf.nist.gov/10.68.E11
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10.68.12 ${\mathop{\theta _{{\nu}}\/}\nolimits^{{\prime}}}=(\mathop{M_{{\nu+1}}\/}\nolimits/\mathop{M_{{\nu}}\/}\nolimits)\mathop{\sin\/}\nolimits\!\left(\mathop{\theta _{{\nu+1}}\/}\nolimits-\mathop{\theta _{{\nu}}\/}\nolimits-\tfrac{1}{4}\pi\right)=-(\mathop{M_{{\nu-1}}\/}\nolimits/\mathop{M_{{\nu}}\/}\nolimits)\mathop{\sin\/}\nolimits\!\left(\mathop{\theta _{{\nu-1}}\/}\nolimits-\mathop{\theta _{{\nu}}\/}\nolimits-\tfrac{1}{4}\pi\right).$

Symbols:: $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of Bessel functions, $\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of Bessel functions, $\mathop{\sin\/}\nolimits z$ : sine function and $\nu$ : complex parameter
A&S Ref:: 9.10.15
Permalink:: http://dlmf.nist.gov/10.68.E12
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10.68.13

${\mathop{M_{{0}}\/}\nolimits^{{\prime}}}=\mathop{M_{{1}}\/}\nolimits\mathop{\cos\/}\nolimits\!\left(\mathop{\theta _{{1}}\/}\nolimits-\mathop{\theta _{{0}}\/}\nolimits-\tfrac{1}{4}\pi\right),$

${\mathop{\theta _{{0}}\/}\nolimits^{{\prime}}}=(\mathop{M_{{1}}\/}\nolimits/\mathop{M_{{0}}\/}\nolimits)\mathop{\sin\/}\nolimits\!\left(\mathop{\theta _{{1}}\/}\nolimits-\mathop{\theta _{{0}}\/}\nolimits-\tfrac{1}{4}\pi\right).$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of Bessel functions, $\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of Bessel functions and $\mathop{\sin\/}\nolimits z$ : sine function
A&S Ref:: 9.10.16
Permalink:: http://dlmf.nist.gov/10.68.E13
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10.68.14

$\ifrac{d(x{\mathop{M_{{\nu}}\/}\nolimits^{{2}}}{\mathop{\theta _{{\nu}}\/}\nolimits^{{\prime}}})}{dx}=x{\mathop{M_{{\nu}}\/}\nolimits^{{2}}},$

$x^{2}{\mathop{M_{{\nu}}\/}\nolimits^{{\prime\prime}}}+x{\mathop{M_{{\nu}}\/}\nolimits^{{\prime}}}-\nu^{2}\mathop{M_{{\nu}}\/}\nolimits=x^{2}\mathop{M_{{\nu}}\/}\nolimits{{\mathop{\theta _{{\nu}}\/}\nolimits^{{\prime}}}^{{2}}}.$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of Bessel functions, $\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of Bessel functions, $x$ : 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
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Equations (10.68.8)–(10.68.14) also hold with the symbols $\mathop{\mathrm{ber}\/}\nolimits$ , $\mathop{\mathrm{bei}\/}\nolimits$ , $\mathop{M\/}\nolimits$ , and $\mathop{\theta\/}\nolimits$ replaced throughout by $\mathop{\mathrm{ker}\/}\nolimits$ , $\mathop{\mathrm{kei}\/}\nolimits$ , $\mathop{N\/}\nolimits$ , and $\mathop{\phi\/}\nolimits$ , respectively. In place of (10.68.7),

10.68.15

$\mathop{N_{{-\nu}}\/}\nolimits\!\left(x\right)=\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right),$

$\mathop{\phi _{{-\nu}}\/}\nolimits\!\left(x\right)=\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right)+\nu\pi.$

Symbols:: $\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of derivatives of Bessel functions, $\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of derivatives of Bessel functions, $x$ : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.20
Permalink:: http://dlmf.nist.gov/10.68.E15
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§10.68(iii) Asymptotic Expansions for Large Argument

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).
Keywords:: Kelvin functions
Permalink:: http://dlmf.nist.gov/10.68.iii

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

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $e$ : base of exponential function, $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of Bessel functions, $x$ : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.21
Permalink:: http://dlmf.nist.gov/10.68.E16
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10.68.17 $\mathop{\ln\/}\nolimits\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{x}{\sqrt{2}}-\frac{1}{2}\mathop{\ln\/}\nolimits\!\left(2\pi x\right)-\frac{\mu-1}{8\sqrt{2}}\frac{1}{x}-\frac{(\mu-1)(\mu-25)}{384\sqrt{2}}\frac{1}{x^{3}}-\frac{(\mu-1)(\mu-13)}{128}\frac{1}{x^{4}}+\mathop{O\/}\nolimits\!\left(\frac{1}{x^{5}}\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of Bessel functions, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $x$ : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.22
Permalink:: http://dlmf.nist.gov/10.68.E17
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10.68.18 $\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)=\frac{x}{\sqrt{2}}+\left(\frac{1}{2}\nu-\frac{1}{8}\right)\pi+\frac{\mu-1}{8\sqrt{2}}\frac{1}{x}+\frac{\mu-1}{16}\frac{1}{x^{2}}-\frac{(\mu-1)(\mu-25)}{384\sqrt{2}}\frac{1}{x^{3}}+\mathop{O\/}\nolimits\!\left(\frac{1}{x^{5}}\right).$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of Bessel functions, $x$ : 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
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10.68.19 $\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right)=e^{{-x/\sqrt{2}}}\left(\frac{\pi}{2x}\right)^{{\frac{1}{2}}}\left(1+\frac{\mu-1}{8\sqrt{2}}\frac{1}{x}+\frac{(\mu-1)^{2}}{256}\frac{1}{x^{2}}+\frac{(\mu-1)(\mu^{2}+14\mu-399)}{6144\sqrt{2}}\frac{1}{x^{3}}+\mathop{O\/}\nolimits\!\left(\frac{1}{x^{4}}\right)\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $e$ : base of exponential function, $\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of derivatives of Bessel functions, $x$ : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.24
Permalink:: http://dlmf.nist.gov/10.68.E19
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10.68.20 $\mathop{\ln\/}\nolimits\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right)=-\frac{x}{\sqrt{2}}+\frac{1}{2}\mathop{\ln\/}\nolimits\!\left(\frac{\pi}{2x}\right)+\frac{\mu-1}{8\sqrt{2}}\frac{1}{x}+\frac{(\mu-1)(\mu-25)}{384\sqrt{2}}\frac{1}{x^{3}}-\frac{(\mu-1)(\mu-13)}{128}\frac{1}{x^{4}}+\mathop{O\/}\nolimits\!\left(\frac{1}{x^{5}}\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of derivatives of Bessel functions, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $x$ : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.25
Permalink:: http://dlmf.nist.gov/10.68.E20
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10.68.21 $\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right)=-\frac{x}{\sqrt{2}}-\left(\frac{1}{2}\nu+\frac{1}{8}\right)\pi-\frac{\mu-1}{8\sqrt{2}}\frac{1}{x}+\frac{\mu-1}{16}\frac{1}{x^{2}}+\frac{(\mu-1)(\mu-25)}{384\sqrt{2}}\frac{1}{x^{3}}+\mathop{O\/}\nolimits\!\left(\frac{1}{x^{5}}\right).$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of derivatives of Bessel functions, $x$ : 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
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§10.68(iv) Further Properties

Referenced by:: §10.68(i), §10.75(xi)
Permalink:: http://dlmf.nist.gov/10.68.iv

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 $\mathop{\phi _{{1}}\/}\nolimits\!\left(0\right)=\tfrac{5}{4}\pi$ (Eq. (6.10)), and $\lim _{{x\to\infty}}(\mathop{\phi _{{1}}\/}\nolimits\!\left(x\right)+(x/\sqrt{2}))=-\tfrac{5}{8}\pi$ (Eqs. (10.20) and (Eqs. (10.26b)). However, numerical tabulations show that if the second of these equations applies and $\mathop{\phi _{{1}}\/}\nolimits\!\left(x\right)$ is continuous, then $\mathop{\phi _{{1}}\/}\nolimits\!\left(0\right)=-\tfrac{3}{4}\pi$ ; 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