10 Bessel Functions Kelvin Functions 10.67 Asymptotic Expansions for Large Argument 10.69 Uniform Asymptotic Expansions for Large Order

§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, : 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, : real variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.68.E1
Encodings:: TeX, pMML, png

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, : 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, : real variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.68.E2
Encodings:: TeX, pMML, png

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 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, : 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

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, : real variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.68.E4
Encodings:: TeX, TeX, pMML, pMML, png, png

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, : 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

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, : 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
Encodings:: TeX, TeX, pMML, pMML, png, png

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, : 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

With arguments 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, : 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

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, : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.12
Permalink:: http://dlmf.nist.gov/10.68.E9
Encodings:: TeX, pMML, png

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 : real variable
A&S Ref:: 9.10.13
Permalink:: http://dlmf.nist.gov/10.68.E10
Encodings:: TeX, TeX, pMML, pMML, png, png

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, : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.14
Permalink:: http://dlmf.nist.gov/10.68.E11
Encodings:: TeX, pMML, png

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
Encodings:: TeX, pMML, png

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
Encodings:: TeX, TeX, pMML, pMML, png, png

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 with respect to , $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of Bessel functions, $\mathop{\theta_{{\nu}}\/}\nolimits\!\left(x\right)$ : 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

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, : 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

§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, : base of exponential function, $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ : 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

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, : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.22
Permalink:: http://dlmf.nist.gov/10.68.E17
Encodings:: TeX, pMML, png

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, : 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

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, : base of exponential function, $\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right)$ : 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

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, : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.25
Permalink:: http://dlmf.nist.gov/10.68.E20
Encodings:: TeX, pMML, png

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, : 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

§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).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 10.67 Asymptotic Expansions for Large Argument 10.69 Uniform Asymptotic Expansions for Large Order

§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