# §10.68 Modulus and Phase Functions

## §10.68(i) Definitions

 10.68.1 $M_{\nu}\left(x\right)e^{i\theta_{\nu}\left(x\right)}=\operatorname{ber}_{\nu}x% +i\operatorname{bei}_{\nu}x,$
 10.68.2 $N_{\nu}\left(x\right)e^{i\phi_{\nu}\left(x\right)}=\operatorname{ker}_{\nu}x+i% \operatorname{kei}_{\nu}x,$

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 $x$ 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

 10.68.3 $\displaystyle\operatorname{ber}_{\nu}x$ $\displaystyle=M_{\nu}\left(x\right)\cos\theta_{\nu}\left(x\right),$ $\displaystyle\operatorname{bei}_{\nu}x$ $\displaystyle=M_{\nu}\left(x\right)\sin\theta_{\nu}\left(x\right),$
 10.68.4 $\displaystyle\operatorname{ker}_{\nu}x$ $\displaystyle=N_{\nu}\left(x\right)\cos\phi_{\nu}\left(x\right),$ $\displaystyle\operatorname{kei}_{\nu}x$ $\displaystyle=N_{\nu}\left(x\right)\sin\phi_{\nu}\left(x\right).$
 10.68.5 $\displaystyle M_{\nu}\left(x\right)$ $\displaystyle=({\operatorname{ber}_{\nu}}^{2}x+{\operatorname{bei}_{\nu}}^{2}x% )^{\ifrac{1}{2}},$ $\displaystyle N_{\nu}\left(x\right)$ $\displaystyle=({\operatorname{ker}_{\nu}}^{2}x+{\operatorname{kei}_{\nu}}^{2}x% )^{\ifrac{1}{2}},$
 10.68.6 $\displaystyle\theta_{\nu}\left(x\right)$ $\displaystyle=\operatorname{Arctan}\left(\operatorname{bei}_{\nu}x/% \operatorname{ber}_{\nu}x\right),$ $\displaystyle\phi_{\nu}\left(x\right)$ $\displaystyle=\operatorname{Arctan}\left(\operatorname{kei}_{\nu}x/% \operatorname{ker}_{\nu}x\right).$
 10.68.7 $\displaystyle M_{-n}\left(x\right)$ $\displaystyle=M_{n}\left(x\right),$ $\displaystyle\theta_{-n}\left(x\right)$ $\displaystyle=\theta_{n}\left(x\right)-n\pi.$ ⓘ 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, $n$: 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 $\operatorname{ber}_{\nu}'x=\tfrac{1}{2}M_{\nu+1}\cos\left(\theta_{\nu+1}-% \tfrac{1}{4}\pi\right)-\tfrac{1}{2}M_{\nu-1}\cos\left(\theta_{\nu-1}-\tfrac{1}% {4}\pi\right)=(\nu/x)M_{\nu}\cos\theta_{\nu}+M_{\nu+1}\cos\left(\theta_{\nu+1}% -\tfrac{1}{4}\pi\right)=-(\nu/x)M_{\nu}\cos\theta_{\nu}-M_{\nu-1}\cos\left(% \theta_{\nu-1}-\tfrac{1}{4}\pi\right),$
 10.68.9 $\operatorname{bei}_{\nu}'x=\tfrac{1}{2}M_{\nu+1}\sin\left(\theta_{\nu+1}-% \tfrac{1}{4}\pi\right)-\tfrac{1}{2}M_{\nu-1}\sin\left(\theta_{\nu-1}-\tfrac{1}% {4}\pi\right)=(\nu/x)M_{\nu}\sin\theta_{\nu}+M_{\nu+1}\sin\left(\theta_{\nu+1}% -\tfrac{1}{4}\pi\right)=-(\nu/x)M_{\nu}\sin\theta_{\nu}-M_{\nu-1}\sin\left(% \theta_{\nu-1}-\tfrac{1}{4}\pi\right).$
 10.68.10 $\displaystyle\operatorname{ber}'x$ $\displaystyle=M_{1}\cos\left(\theta_{1}-\tfrac{1}{4}\pi\right),$ $\displaystyle\operatorname{bei}'x$ $\displaystyle=M_{1}\sin\left(\theta_{1}-\tfrac{1}{4}\pi\right).$
 10.68.11 $M_{\nu}'=(\nu/x)M_{\nu}+M_{\nu+1}\cos\left(\theta_{\nu+1}-\theta_{\nu}-\tfrac{% 1}{4}\pi\right)=-(\nu/x)M_{\nu}-M_{\nu-1}\cos\left(\theta_{\nu-1}-\theta_{\nu}% -\tfrac{1}{4}\pi\right),$
 10.68.12 $\theta_{\nu}'=(M_{\nu+1}/M_{\nu})\sin\left(\theta_{\nu+1}-\theta_{\nu}-\tfrac{% 1}{4}\pi\right)=-(M_{\nu-1}/M_{\nu})\sin\left(\theta_{\nu-1}-\theta_{\nu}-% \tfrac{1}{4}\pi\right).$
 10.68.13 $\displaystyle M_{0}'$ $\displaystyle=M_{1}\cos\left(\theta_{1}-\theta_{0}-\tfrac{1}{4}\pi\right),$ $\displaystyle\theta_{0}'$ $\displaystyle=(M_{1}/M_{0})\sin\left(\theta_{1}-\theta_{0}-\tfrac{1}{4}\pi% \right).$
 10.68.14 $\displaystyle\ifrac{\mathrm{d}(x{M_{\nu}}^{2}\theta_{\nu}')}{\mathrm{d}x}$ $\displaystyle=x{M_{\nu}}^{2},$ $\displaystyle x^{2}M_{\nu}''+xM_{\nu}'-\nu^{2}M_{\nu}$ $\displaystyle=x^{2}M_{\nu}{\theta_{\nu}'}^{2}.$

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

 10.68.15 $\displaystyle N_{-\nu}\left(x\right)$ $\displaystyle=N_{\nu}\left(x\right),$ $\displaystyle\phi_{-\nu}\left(x\right)$ $\displaystyle=\phi_{\nu}\left(x\right)+\nu\pi.$

## §10.68(iii) Asymptotic Expansions for Large Argument

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

 10.68.16 $\displaystyle M_{\nu}\left(x\right)$ $\displaystyle=\frac{e^{x/\sqrt{2}}}{(2\pi x)^{\frac{1}{2}}}\left(1-\frac{\mu-1% }{8\sqrt{2}}\frac{1}{x}+\frac{(\mu-1)^{2}}{256}\frac{1}{x^{2}}-\frac{(\mu-1)(% \mu^{2}+14\mu-399)}{6144\sqrt{2}}\frac{1}{x^{3}}+O\left(\frac{1}{x^{4}}\right)% \right),$ 10.68.17 $\displaystyle\ln M_{\nu}\left(x\right)$ $\displaystyle=\frac{x}{\sqrt{2}}-\frac{1}{2}\ln\left(2\pi x\right)-\frac{\mu-1% }{8\sqrt{2}}\frac{1}{x}-\frac{(\mu-1)(\mu-25)}{384\sqrt{2}}\frac{1}{x^{3}}-% \frac{(\mu-1)(\mu-13)}{128}\frac{1}{x^{4}}+O\left(\frac{1}{x^{5}}\right),$ 10.68.18 $\displaystyle\theta_{\nu}\left(x\right)$ $\displaystyle=\frac{x}{\sqrt{2}}+\left(\frac{1}{2}\nu-\frac{1}{8}\right)\pi+% \frac{\mu-1}{8\sqrt{2}}\frac{1}{x}+\frac{\mu-1}{16}\frac{1}{x^{2}}-\frac{(\mu-% 1)(\mu-25)}{384\sqrt{2}}\frac{1}{x^{3}}+O\left(\frac{1}{x^{5}}\right).$ ⓘ 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, $x$: 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 $\displaystyle N_{\nu}\left(x\right)$ $\displaystyle=e^{-x/\sqrt{2}}\left(\frac{\pi}{2x}\right)^{\frac{1}{2}}\left(1+% \frac{\mu-1}{8\sqrt{2}}\frac{1}{x}+\frac{(\mu-1)^{2}}{256}\frac{1}{x^{2}}+% \frac{(\mu-1)(\mu^{2}+14\mu-399)}{6144\sqrt{2}}\frac{1}{x^{3}}+O\left(\frac{1}% {x^{4}}\right)\right),$ 10.68.20 $\displaystyle\ln N_{\nu}\left(x\right)$ $\displaystyle=-\frac{x}{\sqrt{2}}+\frac{1}{2}\ln\left(\frac{\pi}{2x}\right)+% \frac{\mu-1}{8\sqrt{2}}\frac{1}{x}+\frac{(\mu-1)(\mu-25)}{384\sqrt{2}}\frac{1}% {x^{3}}-\frac{(\mu-1)(\mu-13)}{128}\frac{1}{x^{4}}+O\left(\frac{1}{x^{5}}% \right),$ 10.68.21 $\displaystyle\phi_{\nu}\left(x\right)$ $\displaystyle=-\frac{x}{\sqrt{2}}-\left(\frac{1}{2}\nu+\frac{1}{8}\right)\pi-% \frac{\mu-1}{8\sqrt{2}}\frac{1}{x}+\frac{\mu-1}{16}\frac{1}{x^{2}}+\frac{(\mu-% 1)(\mu-25)}{384\sqrt{2}}\frac{1}{x^{3}}+O\left(\frac{1}{x^{5}}\right).$ ⓘ 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, $x$: 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

## §10.68(iv) Further Properties

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).