# §10.18 Modulus and Phase Functions

## §10.18(i) Definitions

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

 10.18.1 $\displaystyle\mathop{M_{\nu}\/}\nolimits\!\left(x\right)e^{i\!\mathop{\theta_{% \nu}\/}\nolimits\!\left(x\right)}$ $\displaystyle=\mathop{{H^{(1)}_{\nu}}\/}\nolimits\!\left(x\right),$ Defines: $\mathop{M_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{x}\right)$: modulus of Bessel functions Symbols: $\mathop{{H^{(1)}_{\NVar{\nu}}}\/}\nolimits\!\left(\NVar{z}\right)$: Bessel function of the third kind (or Hankel function), $\mathrm{e}$: base of exponential function, $\mathop{\theta_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{x}\right)$: 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) 10.18.2 $\displaystyle\mathop{N_{\nu}\/}\nolimits\!\left(x\right)e^{i\!\mathop{\phi_{% \nu}\/}\nolimits\!\left(x\right)}$ $\displaystyle=\mathop{{H^{(1)}_{\nu}}\/}\nolimits'\!\left(x\right),$ Defines: $\mathop{N_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{x}\right)$: modulus of derivatives of Bessel functions Symbols: $\mathop{{H^{(1)}_{\NVar{\nu}}}\/}\nolimits\!\left(\NVar{z}\right)$: Bessel function of the third kind (or Hankel function), $\mathrm{e}$: base of exponential function, $\mathop{\phi_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{x}\right)$: 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)

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 $\nu$ and $x$, with the branches of $\mathop{\theta_{\nu}\/}\nolimits\!\left(x\right)$ and $\mathop{\phi_{\nu}\/}\nolimits\!\left(x\right)$ fixed by

 10.18.3 $\displaystyle\mathop{\theta_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle\to-\tfrac{1}{2}\pi,$ $\displaystyle\mathop{\phi_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle\to\tfrac{1}{2}\pi$, $x\to 0+$. Defines: $\mathop{\phi_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{x}\right)$: phase of derivatives of Bessel functions and $\mathop{\theta_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{x}\right)$: 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(ii) Basic Properties

 10.18.4 $\displaystyle\mathop{J_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle=\mathop{M_{\nu}\/}\nolimits\!\left(x\right)\mathop{\cos\/}% \nolimits\mathop{\theta_{\nu}\/}\nolimits\!\left(x\right),$ $\displaystyle\mathop{Y_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle=\mathop{M_{\nu}\/}\nolimits\!\left(x\right)\mathop{\sin\/}% \nolimits\mathop{\theta_{\nu}\/}\nolimits\!\left(x\right),$
 10.18.5 $\displaystyle\mathop{J_{\nu}\/}\nolimits'\!\left(x\right)$ $\displaystyle=\mathop{N_{\nu}\/}\nolimits\!\left(x\right)\mathop{\cos\/}% \nolimits\mathop{\phi_{\nu}\/}\nolimits\!\left(x\right),$ $\displaystyle\mathop{Y_{\nu}\/}\nolimits'\!\left(x\right)$ $\displaystyle=\mathop{N_{\nu}\/}\nolimits\!\left(x\right)\mathop{\sin\/}% \nolimits\mathop{\phi_{\nu}\/}\nolimits\!\left(x\right),$
 10.18.6 $\displaystyle\mathop{M_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle=\left({\mathop{J_{\nu}\/}\nolimits^{2}}\!\left(x\right)+{\mathop% {Y_{\nu}\/}\nolimits^{2}}\!\left(x\right)\right)^{\frac{1}{2}},$ $\displaystyle\mathop{N_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle=\left({\mathop{J_{\nu}\/}\nolimits'^{2}}\!\left(x\right)+{% \mathop{Y_{\nu}\/}\nolimits'^{2}}\!\left(x\right)\right)^{\frac{1}{2}},$
 10.18.7 $\displaystyle\mathop{\theta_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle=\mathop{\mathrm{Arctan}\/}\nolimits\!\left(\mathop{Y_{\nu}\/}% \nolimits\!\left(x\right)/\mathop{J_{\nu}\/}\nolimits\!\left(x\right)\right),$ $\displaystyle\mathop{\phi_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle=\mathop{\mathrm{Arctan}\/}\nolimits\!\left(\mathop{Y_{\nu}\/}% \nolimits'\!\left(x\right)/\mathop{J_{\nu}\/}\nolimits'\!\left(x\right)\right).$
 10.18.8 $\displaystyle{\mathop{M_{\nu}\/}\nolimits^{2}}\!\left(x\right)\mathop{\theta_{% \nu}\/}\nolimits'\!\left(x\right)$ $\displaystyle=\frac{2}{\pi x},$ $\displaystyle{\mathop{N_{\nu}\/}\nolimits^{2}}\!\left(x\right)\mathop{\phi_{% \nu}\/}\nolimits'\!\left(x\right)$ $\displaystyle=\frac{2(x^{2}-\nu^{2})}{\pi x^{3}},$
 10.18.9 ${\mathop{N_{\nu}\/}\nolimits^{2}}\!\left(x\right)={\mathop{M_{\nu}\/}\nolimits% '^{2}}\!\left(x\right)+{\mathop{M_{\nu}\/}\nolimits^{2}}\!\left(x\right){% \mathop{\theta_{\nu}\/}\nolimits'^{2}}\!\left(x\right)={\mathop{M_{\nu}\/}% \nolimits'^{2}}\!\left(x\right)+\frac{4}{(\pi x\mathop{M_{\nu}\/}\nolimits\!% \left(x\right))^{2}},$
 10.18.10 $(x^{2}-\nu^{2})\mathop{M_{\nu}\/}\nolimits\!\left(x\right)\mathop{M_{\nu}\/}% \nolimits'\!\left(x\right)+x^{2}\mathop{N_{\nu}\/}\nolimits\!\left(x\right)% \mathop{N_{\nu}\/}\nolimits'\!\left(x\right)+x{\mathop{N_{\nu}\/}\nolimits^{2}% }\!\left(x\right)=0.$ Symbols: $\mathop{M_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{x}\right)$: modulus of Bessel functions, $\mathop{N_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{x}\right)$: 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)
 10.18.11 $\mathop{\tan\/}\nolimits\!\left(\mathop{\phi_{\nu}\/}\nolimits\!\left(x\right)% -\mathop{\theta_{\nu}\/}\nolimits\!\left(x\right)\right)=\frac{\mathop{M_{\nu}% \/}\nolimits\!\left(x\right)\mathop{\theta_{\nu}\/}\nolimits'\!\left(x\right)}% {\mathop{M_{\nu}\/}\nolimits'\!\left(x\right)}=\frac{2}{\pi x\mathop{M_{\nu}\/% }\nolimits\!\left(x\right)\mathop{M_{\nu}\/}\nolimits'\!\left(x\right)},$
 10.18.12 $\mathop{M_{\nu}\/}\nolimits\!\left(x\right)\mathop{N_{\nu}\/}\nolimits\!\left(% x\right)\mathop{\sin\/}\nolimits\!\left(\mathop{\phi_{\nu}\/}\nolimits\!\left(% x\right)-\mathop{\theta_{\nu}\/}\nolimits\!\left(x\right)\right)=\frac{2}{\pi x}.$
 10.18.13 $x^{2}\mathop{M_{\nu}\/}\nolimits''\!\left(x\right)+x\mathop{M_{\nu}\/}% \nolimits'\!\left(x\right)+(x^{2}-\nu^{2})\mathop{M_{\nu}\/}\nolimits\!\left(x% \right)=\frac{4}{\pi^{2}{{\mathop{M_{\nu}\/}\nolimits^{3}}(x)}},$
 10.18.14 $w^{\prime\prime}+\left(1+\frac{\frac{1}{4}-\nu^{2}}{x^{2}}\right)w=\frac{4}{% \pi^{2}w^{3}},$ $w=x^{\frac{1}{2}}\mathop{M_{\nu}\/}\nolimits\!\left(x\right)$,
 10.18.15 $x^{3}w^{\prime\prime\prime}+x(4x^{2}+1-4\nu^{2})w^{\prime}+(4\nu^{2}-1)w=0,$ $w=x{\mathop{M_{\nu}\/}\nolimits^{2}}\!\left(x\right)$. Symbols: $\mathop{M_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{x}\right)$: 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)
 10.18.16 ${\mathop{\theta_{\nu}\/}\nolimits'^{2}}\!\left(x\right)+\frac{1}{2}\frac{% \mathop{\theta_{\nu}\/}\nolimits'''\!\left(x\right)}{\mathop{\theta_{\nu}\/}% \nolimits'\!\left(x\right)}-\frac{3}{4}\left(\frac{\mathop{\theta_{\nu}\/}% \nolimits''\!\left(x\right)}{\mathop{\theta_{\nu}\/}\nolimits'\!\left(x\right)% }\right)^{2}=1-\frac{\nu^{2}-\tfrac{1}{4}}{x^{2}}.$ Symbols: $\mathop{\theta_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{x}\right)$: 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(iii) Asymptotic Expansions for Large Argument

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

 10.18.17 $\displaystyle{\mathop{M_{\nu}\/}\nolimits^{2}}\!\left(x\right)$ $\displaystyle\sim\frac{2}{\pi x}\left(1+\frac{1}{2}\frac{\mu-1}{(2x)^{2}}+% \frac{1\cdot 3}{2\cdot 4}\frac{(\mu-1)(\mu-9)}{(2x)^{4}}+\frac{1\cdot 3\cdot 5% }{2\cdot 4\cdot 6}\frac{(\mu-1)(\mu-9)(\mu-25)}{(2x)^{6}}+\cdots\right),$ 10.18.18 $\displaystyle\mathop{\theta_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle\sim x-\left(\frac{1}{2}\nu+\frac{1}{4}\right)\pi+\frac{\mu-1}{2(% 4x)}+\frac{(\mu-1)(\mu-25)}{6(4x)^{3}}+\frac{(\mu-1)(\mu^{2}-114\mu+1073)}{5(4% x)^{5}}+\frac{(\mu-1)(5\mu^{3}-1535\mu^{2}+54703\mu-3\;75733)}{14(4x)^{7}}+\cdots.$

Also,

 10.18.19 ${\mathop{N_{\nu}\/}\nolimits^{2}}\!\left(x\right)\sim\frac{2}{\pi x}\left(1-% \frac{1}{2}\frac{\mu-3}{(2x)^{2}}-\frac{1}{2\cdot 4}\frac{(\mu-1)(\mu-45)}{(2x% )^{4}}-\cdots\right),$ Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{N_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{x}\right)$: 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)

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 (as in $n!!$), $k$: 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)

and

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

The remainder after $k$ terms in (10.18.17) does not exceed the $(k+1)$th term in absolute value and is of the same sign, provided that $k>\nu-\tfrac{1}{2}$.