§10.68 Modulus and Phase Functions

10.68.1		$$M_{\nu }\left(x\right){\mathrm{e}}^{\mathrm{i}{\theta }_{\nu }\left(x\right)}={\mathrm{ber}}_{\nu }x+\mathrm{i}{\mathrm{bei}}_{\nu }x,$$
ⓘ Symbols: ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $M_{\nu }\left(x\right)$: modulus of Bessel functions, ${\theta }_{\nu }\left(x\right)$: phase of Bessel functions, $x$: 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		$$N_{\nu }\left(x\right){\mathrm{e}}^{\mathrm{i}{\varphi }_{\nu }\left(x\right)}={\ker }_{\nu }x+\mathrm{i}{\mathrm{kei}}_{\nu }x,$$
ⓘ Symbols: ${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $N_{\nu }\left(x\right)$: modulus of derivatives of Bessel functions, ${\varphi }_{\nu }\left(x\right)$: phase of derivatives of Bessel functions, $x$: 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 ${\varphi }_{\nu }\left(x\right)$ are continuous real functions of $x$ and $\nu$, with the branches of ${\theta }_{\nu }\left(x\right)$ and ${\varphi }_{\nu }\left(x\right)$ chosen to satisfy (10.68.18) and (10.68.21) as $x\to \mathrm{\infty }$. (See also §10.68(iv).)

10.68.3		${\mathrm{ber}}_{\nu }x$	$=M_{\nu }\left(x\right)\cos {\theta }_{\nu }\left(x\right),$
		${\mathrm{bei}}_{\nu }x$	$=M_{\nu }\left(x\right)\sin {\theta }_{\nu }\left(x\right),$
ⓘ Symbols: ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\cos z$: cosine function, $M_{\nu }\left(x\right)$: modulus of Bessel functions, ${\theta }_{\nu }\left(x\right)$: phase of Bessel functions, $\sin z$: 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 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.68(ii), §10.68 and Ch.10

10.68.4		${\ker }_{\nu }x$	$=N_{\nu }\left(x\right)\cos {\varphi }_{\nu }\left(x\right),$
		${\mathrm{kei}}_{\nu }x$	$=N_{\nu }\left(x\right)\sin {\varphi }_{\nu }\left(x\right).$
ⓘ Symbols: ${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\cos z$: cosine function, $N_{\nu }\left(x\right)$: modulus of derivatives of Bessel functions, ${\varphi }_{\nu }\left(x\right)$: phase of derivatives of Bessel functions, $\sin z$: sine function, $x$: 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		$M_{\nu }\left(x\right)$	$={({\mathrm{ber}}_{\nu }^{2}x+{\mathrm{bei}}_{\nu }^{2}x)}^{1/2},$
		$N_{\nu }\left(x\right)$	$={({\ker }_{\nu }^{2}x+{\mathrm{kei}}_{\nu }^{2}x)}^{1/2},$
ⓘ Symbols: ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $M_{\nu }\left(x\right)$: modulus of Bessel functions, $N_{\nu }\left(x\right)$: 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 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.68(ii), §10.68 and Ch.10

10.68.6		${\theta }_{\nu }\left(x\right)$	$=\mathrm{Arctan}\left({\mathrm{bei}}_{\nu }x/{\mathrm{ber}}_{\nu }x\right),$
		${\varphi }_{\nu }\left(x\right)$	$=\mathrm{Arctan}\left({\mathrm{kei}}_{\nu }x/{\ker }_{\nu }x\right).$
ⓘ Symbols: ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\mathrm{Arctan}z$: general arctangent function, ${\varphi }_{\nu }\left(x\right)$: phase of derivatives of Bessel functions, ${\theta }_{\nu }\left(x\right)$: 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 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.68(ii), §10.68 and Ch.10

10.68.7		$M_{-n}\left(x\right)$	$=M_n\left(x\right),$
		${\theta }_{-n}\left(x\right)$	$={\theta }_n\left(x\right)-n\pi .$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $M_{\nu }\left(x\right)$: modulus of Bessel functions, ${\theta }_{\nu }\left(x\right)$: 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 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.68(ii), §10.68 and Ch.10

With arguments $(x)$ suppressed,

10.68.8		$$\begin{array}{ll}{\mathrm{ber}}_{\nu }^{\prime }x& =\frac{1}{2}{M}_{\nu +1}\cos \left({\theta }_{\nu +1}-\frac{1}{4}\pi \right)-\frac{1}{2}{M}_{\nu -1}\cos \left({\theta }_{\nu -1}-\frac{1}{4}\pi \right)\\ & =(\nu /x)M_{\nu }\cos {\theta }_{\nu }+M_{\nu +1}\cos \left({\theta }_{\nu +1}-\frac{1}{4}\pi \right)\\ & =-(\nu /x)M_{\nu }\cos {\theta }_{\nu }-M_{\nu -1}\cos \left({\theta }_{\nu -1}-\frac{1}{4}\pi \right),\end{array}$$
ⓘ Symbols: ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $M_{\nu }\left(x\right)$: modulus of Bessel functions, ${\theta }_{\nu }\left(x\right)$: 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 Encodings: TeX, pMML, png See also: Annotations for §10.68(ii), §10.68 and Ch.10

10.68.9		$$\begin{array}{ll}{\mathrm{bei}}_{\nu }^{\prime }x& =\frac{1}{2}{M}_{\nu +1}\sin \left({\theta }_{\nu +1}-\frac{1}{4}\pi \right)-\frac{1}{2}{M}_{\nu -1}\sin \left({\theta }_{\nu -1}-\frac{1}{4}\pi \right)\\ & =(\nu /x)M_{\nu }\sin {\theta }_{\nu }+M_{\nu +1}\sin \left({\theta }_{\nu +1}-\frac{1}{4}\pi \right)\\ & =-(\nu /x)M_{\nu }\sin {\theta }_{\nu }-M_{\nu -1}\sin \left({\theta }_{\nu -1}-\frac{1}{4}\pi \right).\end{array}$$
ⓘ Symbols: ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $M_{\nu }\left(x\right)$: modulus of Bessel functions, ${\theta }_{\nu }\left(x\right)$: phase of Bessel functions, $\sin z$: sine function, $x$: 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		${\mathrm{ber}}^{\prime }x$	$=M_1\cos \left({\theta }_1-\frac{1}{4}\pi \right),$
		${\mathrm{bei}}^{\prime }x$	$=M_1\sin \left({\theta }_1-\frac{1}{4}\pi \right).$
ⓘ Symbols: ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $M_{\nu }\left(x\right)$: modulus of Bessel functions, ${\theta }_{\nu }\left(x\right)$: phase of Bessel functions, $\sin z$: sine function and $x$: 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		$$M_{\nu }^{\prime }=(\nu /x)M_{\nu }+M_{\nu +1}\cos \left({\theta }_{\nu +1}-{\theta }_{\nu }-\frac{1}{4}\pi \right)=-(\nu /x)M_{\nu }-M_{\nu -1}\cos \left({\theta }_{\nu -1}-{\theta }_{\nu }-\frac{1}{4}\pi \right),$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $M_{\nu }\left(x\right)$: modulus of Bessel functions, ${\theta }_{\nu }\left(x\right)$: 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 Encodings: TeX, pMML, png See also: Annotations for §10.68(ii), §10.68 and Ch.10

10.68.12		$${\theta }_{\nu }^{\prime }=(M_{\nu +1}/M_{\nu })\sin \left({\theta }_{\nu +1}-{\theta }_{\nu }-\frac{1}{4}\pi \right)=-(M_{\nu -1}/M_{\nu })\sin \left({\theta }_{\nu -1}-{\theta }_{\nu }-\frac{1}{4}\pi \right).$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $M_{\nu }\left(x\right)$: modulus of Bessel functions, ${\theta }_{\nu }\left(x\right)$: phase of Bessel functions, $\sin z$: 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		$M_0^{\prime }$	$=M_1\cos \left({\theta }_1-{\theta }_0-\frac{1}{4}\pi \right),$
		${\theta }_0^{\prime }$	$=(M_1/M_0)\sin \left({\theta }_1-{\theta }_0-\frac{1}{4}\pi \right).$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $M_{\nu }\left(x\right)$: modulus of Bessel functions, ${\theta }_{\nu }\left(x\right)$: phase of Bessel functions and $\sin z$: 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		$d(x{M}_{\nu }^2{\theta }_{\nu }^{\prime })/dx$	$=x{M}_{\nu }^2,$
		$x^2{M}_{\nu }^{\prime \prime }+x{M}_{\nu }^{\prime }-{\nu }^2{M}_{\nu }$	$=x^2{M}_{\nu }\theta _{\nu }^{\prime }{}^2.$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $M_{\nu }\left(x\right)$: modulus of Bessel functions, ${\theta }_{\nu }\left(x\right)$: 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 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 $\mathrm{ber}$, $\mathrm{bei}$, $M$, and $\theta$ replaced throughout by $\ker$, $\mathrm{kei}$, $N$, and $\varphi$, respectively. In place of (10.68.7),

10.68.15		$N_{-\nu }\left(x\right)$	$=N_{\nu }\left(x\right),$
		${\varphi }_{-\nu }\left(x\right)$	$={\varphi }_{\nu }\left(x\right)+\nu \pi .$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $N_{\nu }\left(x\right)$: modulus of derivatives of Bessel functions, ${\varphi }_{\nu }\left(x\right)$: 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 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.68(ii), §10.68 and Ch.10

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

10.68.16		$M_{\nu }\left(x\right)$	$={\displaystyle \frac{{\mathrm{e}}^{x/\sqrt{2}}}{{(2\pi x)}^{\frac{1}{2}}}}\left(1-{\displaystyle \frac{\mu -1}{8\sqrt{2}}}{\displaystyle \frac{1}{x}}+{\displaystyle \frac{{(\mu -1)}^2}{256}}{\displaystyle \frac{1}{x^2}}-{\displaystyle \frac{(\mu -1)({\mu }^2+14\mu -399)}{6144\sqrt{2}}}{\displaystyle \frac{1}{x^3}}+O\left({\displaystyle \frac{1}{x^4}}\right)\right),$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $M_{\nu }\left(x\right)$: 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 Encodings: TeX, pMML, png See also: Annotations for §10.68(iii), §10.68 and Ch.10
10.68.17		$\ln M_{\nu }\left(x\right)$	$={\displaystyle \frac{x}{\sqrt{2}}}-{\displaystyle \frac{1}{2}}\ln \left(2\pi x\right)-{\displaystyle \frac{\mu -1}{8\sqrt{2}}}{\displaystyle \frac{1}{x}}-{\displaystyle \frac{(\mu -1)(\mu -25)}{384\sqrt{2}}}{\displaystyle \frac{1}{x^3}}-{\displaystyle \frac{(\mu -1)(\mu -13)}{128}}{\displaystyle \frac{1}{x^4}}+O\left({\displaystyle \frac{1}{x^5}}\right),$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $M_{\nu }\left(x\right)$: modulus of Bessel functions, $\ln z$: 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 Encodings: TeX, pMML, png See also: Annotations for §10.68(iii), §10.68 and Ch.10
10.68.18		${\theta }_{\nu }\left(x\right)$	$={\displaystyle \frac{x}{\sqrt{2}}}+\left({\displaystyle \frac{1}{2}}\nu -{\displaystyle \frac{1}{8}}\right)\pi +{\displaystyle \frac{\mu -1}{8\sqrt{2}}}{\displaystyle \frac{1}{x}}+{\displaystyle \frac{\mu -1}{16}}{\displaystyle \frac{1}{x^2}}-{\displaystyle \frac{(\mu -1)(\mu -25)}{384\sqrt{2}}}{\displaystyle \frac{1}{x^3}}+O\left({\displaystyle \frac{1}{x^5}}\right).$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, ${\theta }_{\nu }\left(x\right)$: 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 Encodings: TeX, pMML, png See also: Annotations for §10.68(iii), §10.68 and Ch.10
10.68.19		$N_{\nu }\left(x\right)$	$={\mathrm{e}}^{-x/\sqrt{2}}{\left({\displaystyle \frac{\pi }{2x}}\right)}^{\frac{1}{2}}\left(1+{\displaystyle \frac{\mu -1}{8\sqrt{2}}}{\displaystyle \frac{1}{x}}+{\displaystyle \frac{{(\mu -1)}^2}{256}}{\displaystyle \frac{1}{x^2}}+{\displaystyle \frac{(\mu -1)({\mu }^2+14\mu -399)}{6144\sqrt{2}}}{\displaystyle \frac{1}{x^3}}+O\left({\displaystyle \frac{1}{x^4}}\right)\right),$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $N_{\nu }\left(x\right)$: 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 Encodings: TeX, pMML, png See also: Annotations for §10.68(iii), §10.68 and Ch.10
10.68.20		$\ln N_{\nu }\left(x\right)$	$=-{\displaystyle \frac{x}{\sqrt{2}}}+{\displaystyle \frac{1}{2}}\ln \left({\displaystyle \frac{\pi }{2x}}\right)+{\displaystyle \frac{\mu -1}{8\sqrt{2}}}{\displaystyle \frac{1}{x}}+{\displaystyle \frac{(\mu -1)(\mu -25)}{384\sqrt{2}}}{\displaystyle \frac{1}{x^3}}-{\displaystyle \frac{(\mu -1)(\mu -13)}{128}}{\displaystyle \frac{1}{x^4}}+O\left({\displaystyle \frac{1}{x^5}}\right),$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $N_{\nu }\left(x\right)$: modulus of derivatives of Bessel functions, $\ln z$: 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 Encodings: TeX, pMML, png See also: Annotations for §10.68(iii), §10.68 and Ch.10
10.68.21		${\varphi }_{\nu }\left(x\right)$	$=-{\displaystyle \frac{x}{\sqrt{2}}}-\left({\displaystyle \frac{1}{2}}\nu +{\displaystyle \frac{1}{8}}\right)\pi -{\displaystyle \frac{\mu -1}{8\sqrt{2}}}{\displaystyle \frac{1}{x}}+{\displaystyle \frac{\mu -1}{16}}{\displaystyle \frac{1}{x^2}}+{\displaystyle \frac{(\mu -1)(\mu -25)}{384\sqrt{2}}}{\displaystyle \frac{1}{x^3}}+O\left({\displaystyle \frac{1}{x^5}}\right).$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, ${\varphi }_{\nu }\left(x\right)$: 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 Encodings: TeX, pMML, png See also: Annotations for §10.68(iii), §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 ${\varphi }_1\left(0\right)=\frac{5}{4}\pi$ (Eq. (6.10)), and ${lim}_{x\to \mathrm{\infty }}({\varphi }_1\left(x\right)+(x/\sqrt{2}))=-\frac{5}{8}\pi$ (Eqs. (10.20) and (Eqs. (10.26b)). However, numerical tabulations show that if the second of these equations applies and ${\varphi }_1\left(x\right)$ is continuous, then ${\varphi }_1\left(0\right)=-\frac{3}{4}\pi$; compare Abramowitz and Stegun (1964, p. 433).