# §10.18 Modulus and Phase Functions

## §10.18(i) Definitions

For $\nu\geq 0$ and $x>0$

 10.18.1 $\displaystyle M_{\nu}\left(x\right)e^{i\theta_{\nu}\left(x\right)}$ $\displaystyle={H^{(1)}_{\nu}}\left(x\right),$ ⓘ Defines: $M_{\NVar{\nu}}\left(\NVar{x}\right)$: modulus of Bessel functions Symbols: ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$: Bessel function of the third kind (or Hankel function), $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\theta_{\NVar{\nu}}\left(\NVar{x}\right)$: phase of Bessel functions, $x$: real variable and $\nu$: complex parameter Permalink: http://dlmf.nist.gov/10.18.E1 Encodings: TeX, pMML, png See also: Annotations for §10.18(i), §10.18 and Ch.10 10.18.2 $\displaystyle N_{\nu}\left(x\right)e^{i\phi_{\nu}\left(x\right)}$ $\displaystyle={H^{(1)}_{\nu}}'\left(x\right),$ ⓘ Defines: $N_{\NVar{\nu}}\left(\NVar{x}\right)$: modulus of derivatives of Bessel functions Symbols: ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$: Bessel function of the third kind (or Hankel function), $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\phi_{\NVar{\nu}}\left(\NVar{x}\right)$: phase of derivatives of Bessel functions, $x$: real variable and $\nu$: complex parameter Permalink: http://dlmf.nist.gov/10.18.E2 Encodings: TeX, pMML, png See also: Annotations for §10.18(i), §10.18 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 $\nu$ and $x$, with the branches of $\theta_{\nu}\left(x\right)$ and $\phi_{\nu}\left(x\right)$ fixed by

 10.18.3 $\displaystyle\theta_{\nu}\left(x\right)$ $\displaystyle\to-\tfrac{1}{2}\pi,$ $\displaystyle\phi_{\nu}\left(x\right)$ $\displaystyle\to\tfrac{1}{2}\pi$, $x\to 0+$. ⓘ Defines: $\phi_{\NVar{\nu}}\left(\NVar{x}\right)$: phase of derivatives of Bessel functions and $\theta_{\NVar{\nu}}\left(\NVar{x}\right)$: phase of Bessel functions Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $x$: real variable and $\nu$: complex parameter Referenced by: §10.18(i), §10.21(ii) Permalink: http://dlmf.nist.gov/10.18.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.18(i), §10.18 and Ch.10

## §10.18(ii) Basic Properties

 10.18.4 $\displaystyle J_{\nu}\left(x\right)$ $\displaystyle=M_{\nu}\left(x\right)\cos\theta_{\nu}\left(x\right),$ $\displaystyle Y_{\nu}\left(x\right)$ $\displaystyle=M_{\nu}\left(x\right)\sin\theta_{\nu}\left(x\right),$
 10.18.5 $\displaystyle J_{\nu}'\left(x\right)$ $\displaystyle=N_{\nu}\left(x\right)\cos\phi_{\nu}\left(x\right),$ $\displaystyle Y_{\nu}'\left(x\right)$ $\displaystyle=N_{\nu}\left(x\right)\sin\phi_{\nu}\left(x\right),$
 10.18.6 $\displaystyle M_{\nu}\left(x\right)$ $\displaystyle=\left({J_{\nu}}^{2}\left(x\right)+{Y_{\nu}}^{2}\left(x\right)% \right)^{\frac{1}{2}},$ $\displaystyle N_{\nu}\left(x\right)$ $\displaystyle=\left({J_{\nu}'}^{2}\left(x\right)+{Y_{\nu}'}^{2}\left(x\right)% \right)^{\frac{1}{2}},$
 10.18.7 $\displaystyle\theta_{\nu}\left(x\right)$ $\displaystyle=\operatorname{Arctan}\left(Y_{\nu}\left(x\right)/J_{\nu}\left(x% \right)\right),$ $\displaystyle\phi_{\nu}\left(x\right)$ $\displaystyle=\operatorname{Arctan}\left(Y_{\nu}'\left(x\right)/J_{\nu}'\left(% x\right)\right).$
 10.18.8 $\displaystyle{M_{\nu}}^{2}\left(x\right)\theta_{\nu}'\left(x\right)$ $\displaystyle=\frac{2}{\pi x},$ $\displaystyle{N_{\nu}}^{2}\left(x\right)\phi_{\nu}'\left(x\right)$ $\displaystyle=\frac{2(x^{2}-\nu^{2})}{\pi x^{3}},$
 10.18.9 ${N_{\nu}}^{2}\left(x\right)={M_{\nu}'}^{2}\left(x\right)+{M_{\nu}}^{2}\left(x% \right){\theta_{\nu}'}^{2}\left(x\right)={M_{\nu}'}^{2}\left(x\right)+\frac{4}% {(\pi xM_{\nu}\left(x\right))^{2}},$
 10.18.10 $(x^{2}-\nu^{2})M_{\nu}\left(x\right)M_{\nu}'\left(x\right)+x^{2}N_{\nu}\left(x% \right)N_{\nu}'\left(x\right)+x{N_{\nu}}^{2}\left(x\right)=0.$ ⓘ Symbols: $M_{\NVar{\nu}}\left(\NVar{x}\right)$: modulus of Bessel functions, $N_{\NVar{\nu}}\left(\NVar{x}\right)$: modulus of derivatives of Bessel functions, $x$: real variable and $\nu$: complex parameter A&S Ref: 9.2.23 Referenced by: §10.18(iii) Permalink: http://dlmf.nist.gov/10.18.E10 Encodings: TeX, pMML, png See also: Annotations for §10.18(ii), §10.18 and Ch.10
 10.18.11 $\tan\left(\phi_{\nu}\left(x\right)-\theta_{\nu}\left(x\right)\right)=\frac{M_{% \nu}\left(x\right)\theta_{\nu}'\left(x\right)}{M_{\nu}'\left(x\right)}=\frac{2% }{\pi xM_{\nu}\left(x\right)M_{\nu}'\left(x\right)},$
 10.18.12 $M_{\nu}\left(x\right)N_{\nu}\left(x\right)\sin\left(\phi_{\nu}\left(x\right)-% \theta_{\nu}\left(x\right)\right)=\frac{2}{\pi x}.$
 10.18.13 $x^{2}M_{\nu}''\left(x\right)+xM_{\nu}'\left(x\right)+(x^{2}-\nu^{2})M_{\nu}% \left(x\right)=\frac{4}{\pi^{2}{{M_{\nu}}^{3}(x)}},$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $M_{\NVar{\nu}}\left(\NVar{x}\right)$: modulus of Bessel functions, $x$: real variable and $\nu$: complex parameter A&S Ref: 9.2.25 Permalink: http://dlmf.nist.gov/10.18.E13 Encodings: TeX, pMML, png See also: Annotations for §10.18(ii), §10.18 and Ch.10
 10.18.14 $w^{\prime\prime}+\left(1+\frac{\frac{1}{4}-\nu^{2}}{x^{2}}\right)w=\frac{4}{% \pi^{2}w^{3}},$ $w=x^{\frac{1}{2}}M_{\nu}\left(x\right)$,
 10.18.15 $x^{3}w^{\prime\prime\prime}+x(4x^{2}+1-4\nu^{2})w^{\prime}+(4\nu^{2}-1)w=0,$ $w=x{M_{\nu}}^{2}\left(x\right)$. ⓘ Symbols: $M_{\NVar{\nu}}\left(\NVar{x}\right)$: modulus of Bessel functions, $x$: real variable and $\nu$: complex parameter A&S Ref: 9.2.26 Permalink: http://dlmf.nist.gov/10.18.E15 Encodings: TeX, pMML, png See also: Annotations for §10.18(ii), §10.18 and Ch.10
 10.18.16 ${\theta_{\nu}'}^{2}\left(x\right)+\frac{1}{2}\frac{\theta_{\nu}'''\left(x% \right)}{\theta_{\nu}'\left(x\right)}-\frac{3}{4}\left(\frac{\theta_{\nu}''% \left(x\right)}{\theta_{\nu}'\left(x\right)}\right)^{2}=1-\frac{\nu^{2}-\tfrac% {1}{4}}{x^{2}}.$ ⓘ Symbols: $\theta_{\NVar{\nu}}\left(\NVar{x}\right)$: phase of Bessel functions, $x$: real variable and $\nu$: complex parameter A&S Ref: 9.2.27 Permalink: http://dlmf.nist.gov/10.18.E16 Encodings: TeX, pMML, png See also: Annotations for §10.18(ii), §10.18 and Ch.10

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

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

 10.18.17 $\displaystyle{M_{\nu}}^{2}\left(x\right)$ $\displaystyle\sim\frac{2}{\pi x}\left(1+\frac{1}{2}\frac{\mu-1}{(2x)^{2}}+% \frac{1\cdot 3}{2\cdot 4}\frac{(\mu-1)(\mu-9)}{(2x)^{4}}+\frac{1\cdot 3\cdot 5% }{2\cdot 4\cdot 6}\frac{(\mu-1)(\mu-9)(\mu-25)}{(2x)^{6}}+\dotsb\right),$ ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $M_{\NVar{\nu}}\left(\NVar{x}\right)$: modulus of Bessel functions, $x$: real variable and $\nu$: complex parameter Proof sketch: See Nemes (2021, Theorem 2.1) A&S Ref: 9.2.28 Referenced by: §10.18(iii), §10.18(iii), §10.40(i), §10.40(i), §10.49(iv), Erratum (V1.1.8) for Section 10.18(iii) Permalink: http://dlmf.nist.gov/10.18.E17 Encodings: TeX, pMML, png See also: Annotations for §10.18(iii), §10.18 and Ch.10 10.18.18 $\displaystyle\theta_{\nu}\left(x\right)$ $\displaystyle\sim x-\left(\frac{1}{2}\nu+\frac{1}{4}\right)\pi+\frac{\mu-1}{2(% 4x)}+\frac{(\mu-1)(\mu-25)}{6(4x)^{3}}+\frac{(\mu-1)(\mu^{2}-114\mu+1073)}{5(4% x)^{5}}+\frac{(\mu-1)(5\mu^{3}-1535\mu^{2}+54703\mu-3\;75733)}{14(4x)^{7}}+\dotsb.$ ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: 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 Proof sketch: See Nemes (2021, Theorem 2.1) A&S Ref: 9.2.29 Referenced by: §10.18(iii), §10.18(iii), Erratum (V1.1.8) for Section 10.18(iii) Permalink: http://dlmf.nist.gov/10.18.E18 Encodings: TeX, pMML, png See also: Annotations for §10.18(iii), §10.18 and Ch.10

Also,

 10.18.19 ${N_{\nu}}^{2}\left(x\right)\sim\frac{2}{\pi x}\left(1-\frac{1}{2}\frac{\mu-3}{% (2x)^{2}}-\frac{1}{2\cdot 4}\frac{(\mu-1)(\mu-45)}{(2x)^{4}}-\dotsb\right),$ ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $N_{\NVar{\nu}}\left(\NVar{x}\right)$: modulus of derivatives of Bessel functions, $x$: real variable and $\nu$: complex parameter A&S Ref: 9.2.30 (corrected) Referenced by: §10.18(iii), §10.40(i), §10.40(i) Permalink: http://dlmf.nist.gov/10.18.E19 Encodings: TeX, pMML, png See also: Annotations for §10.18(iii), §10.18 and Ch.10

the general term in this expansion being

 10.18.20 $-\frac{(2k-3)!!}{(2k)!!}\frac{(\mu-1)(\mu-9)\cdots(\mu-(2k-3)^{2})(\mu-(2k+1)(% 2k-1)^{2})}{(2x)^{2k}},$ $k\geq 2$, ⓘ Symbols: $!!$: double factorial, $k$: nonnegative integer and $x$: real variable Referenced by: §10.18(iii), §10.40(i) Permalink: http://dlmf.nist.gov/10.18.E20 Encodings: TeX, pMML, png See also: Annotations for §10.18(iii), §10.18 and Ch.10

and

 10.18.21 $\phi_{\nu}\left(x\right)\sim x-\left(\frac{1}{2}\nu-\frac{1}{4}\right)\pi+% \frac{\mu+3}{2(4x)}+\frac{\mu^{2}+46\mu-63}{6(4x)^{3}}+\frac{\mu^{3}+185\mu^{2% }-2053\mu+1899}{5(4x)^{5}}+\dotsi.$

In (10.18.17) and (10.18.18) the remainder after $n$ terms does not exceed the $(n+1)$th term in absolute value and is of the same sign, provided that $n>\nu-\frac{1}{2}$ for (10.18.17) and $-\frac{3}{2}\leq\nu\leq\frac{3}{2}$ for (10.18.18).