§10.18 Modulus and Phase Functions

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

10.18.1		$M_{\nu }\left(x\right){\mathrm{e}}^{\mathrm{i}{\theta }_{\nu }\left(x\right)}$	$=H_{\nu }^{(1)}\left(x\right),$
ⓘ Defines: $M_{\nu }\left(x\right)$: modulus of Bessel functions Symbols: $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${\theta }_{\nu }\left(x\right)$: 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		$N_{\nu }\left(x\right){\mathrm{e}}^{\mathrm{i}{\varphi }_{\nu }\left(x\right)}$	$=H_{\nu }^{(1)}{}^{\prime }\left(x\right),$
ⓘ Defines: $N_{\nu }\left(x\right)$: modulus of derivatives of Bessel functions Symbols: $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${\varphi }_{\nu }\left(x\right)$: 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 ${\varphi }_{\nu }\left(x\right)$ are continuous real functions of $\nu$ and $x$, with the branches of ${\theta }_{\nu }\left(x\right)$ and ${\varphi }_{\nu }\left(x\right)$ fixed by

10.18.3		${\theta }_{\nu }\left(x\right)$	$\to -\frac{1}{2}\pi ,$
		${\varphi }_{\nu }\left(x\right)$	$\to \frac{1}{2}\pi$,
	$x\to 0+$.
ⓘ Defines: ${\varphi }_{\nu }\left(x\right)$: phase of derivatives of Bessel functions and ${\theta }_{\nu }\left(x\right)$: 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

10.18.4		$J_{\nu }\left(x\right)$	$=M_{\nu }\left(x\right)\cos {\theta }_{\nu }\left(x\right),$
		$Y_{\nu }\left(x\right)$	$=M_{\nu }\left(x\right)\sin {\theta }_{\nu }\left(x\right),$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\cos z$: 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: 9.2.19 Permalink: http://dlmf.nist.gov/10.18.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.18(ii), §10.18 and Ch.10

10.18.5		$J_{\nu }^{\prime }\left(x\right)$	$=N_{\nu }\left(x\right)\cos {\varphi }_{\nu }\left(x\right),$
		$Y_{\nu }^{\prime }\left(x\right)$	$=N_{\nu }\left(x\right)\sin {\varphi }_{\nu }\left(x\right),$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\cos z$: 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 A&S Ref: 9.2.20 Permalink: http://dlmf.nist.gov/10.18.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.18(ii), §10.18 and Ch.10

10.18.6		$M_{\nu }\left(x\right)$	$={\left(J_{\nu }^2\left(x\right)+Y_{\nu }^2\left(x\right)\right)}^{\frac{1}{2}},$
		$N_{\nu }\left(x\right)$	$={\left(J_{\nu }^{\prime }{}^2\left(x\right)+Y_{\nu }^{\prime }{}^2\left(x\right)\right)}^{\frac{1}{2}},$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $M_{\nu }\left(x\right)$: 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: 9.2.17, 9.2.18 Referenced by: §10.49(iv) Permalink: http://dlmf.nist.gov/10.18.E6 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.18(ii), §10.18 and Ch.10

10.18.7		${\theta }_{\nu }\left(x\right)$	$=\mathrm{Arctan}\left(Y_{\nu }\left(x\right)/J_{\nu }\left(x\right)\right),$
		${\varphi }_{\nu }\left(x\right)$	$=\mathrm{Arctan}\left(Y_{\nu }^{\prime }\left(x\right)/J_{\nu }^{\prime }\left(x\right)\right).$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\mathrm{Arctan}z$: 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: 9.2.17, 9.2.18 Referenced by: §10.18(iii) Permalink: http://dlmf.nist.gov/10.18.E7 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.18(ii), §10.18 and Ch.10

10.18.8		$M_{\nu }^2\left(x\right){\theta }_{\nu }^{\prime }\left(x\right)$	$={\displaystyle \frac{2}{\pi x}},$
		$N_{\nu }^2\left(x\right){\varphi }_{\nu }^{\prime }\left(x\right)$	$={\displaystyle \frac{2(x^2-{\nu }^2)}{\pi x^3}},$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $M_{\nu }\left(x\right)$: modulus of Bessel functions, $N_{\nu }\left(x\right)$: modulus of derivatives of Bessel functions, ${\varphi }_{\nu }\left(x\right)$: 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: 9.2.21 Referenced by: §10.18(iii), §10.21(ii) Permalink: http://dlmf.nist.gov/10.18.E8 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.18(ii), §10.18 and Ch.10

10.18.9		$$N_{\nu }^2\left(x\right)=M_{\nu }^{\prime }{}^2\left(x\right)+M_{\nu }^2\left(x\right)\theta _{\nu }^{\prime }{}^2\left(x\right)=M_{\nu }^{\prime }{}^2\left(x\right)+\frac{4}{{(\pi x{M}_{\nu }\left(x\right))}^2},$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $M_{\nu }\left(x\right)$: modulus of Bessel functions, $N_{\nu }\left(x\right)$: modulus of derivatives of Bessel functions, ${\theta }_{\nu }\left(x\right)$: phase of Bessel functions, $x$: real variable and $\nu$: complex parameter A&S Ref: 9.2.22 Permalink: http://dlmf.nist.gov/10.18.E9 Encodings: TeX, pMML, png See also: Annotations for §10.18(ii), §10.18 and Ch.10

10.18.10		$$(x^2-{\nu }^2)M_{\nu }\left(x\right)M_{\nu }^{\prime }\left(x\right)+x^2{N}_{\nu }\left(x\right)N_{\nu }^{\prime }\left(x\right)+x{N}_{\nu }^2\left(x\right)=0.$$
ⓘ Symbols: $M_{\nu }\left(x\right)$: 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: 9.2.23 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		$$\tan \left({\varphi }_{\nu }\left(x\right)-{\theta }_{\nu }\left(x\right)\right)=\frac{M_{\nu }\left(x\right){\theta }_{\nu }^{\prime }\left(x\right)}{M_{\nu }^{\prime }\left(x\right)}=\frac{2}{\pi x{M}_{\nu }\left(x\right)M_{\nu }^{\prime }\left(x\right)},$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $M_{\nu }\left(x\right)$: modulus of Bessel functions, ${\varphi }_{\nu }\left(x\right)$: phase of derivatives of Bessel functions, ${\theta }_{\nu }\left(x\right)$: phase of Bessel functions, $\tan z$: tangent function, $x$: real variable and $\nu$: complex parameter A&S Ref: 9.2.24 Permalink: http://dlmf.nist.gov/10.18.E11 Encodings: TeX, pMML, png See also: Annotations for §10.18(ii), §10.18 and Ch.10

10.18.12		$$M_{\nu }\left(x\right)N_{\nu }\left(x\right)\sin \left({\varphi }_{\nu }\left(x\right)-{\theta }_{\nu }\left(x\right)\right)=\frac{2}{\pi x}.$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $M_{\nu }\left(x\right)$: modulus of Bessel functions, $N_{\nu }\left(x\right)$: modulus of derivatives of Bessel functions, ${\varphi }_{\nu }\left(x\right)$: phase of derivatives 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: 9.2.24 Permalink: http://dlmf.nist.gov/10.18.E12 Encodings: TeX, pMML, png See also: Annotations for §10.18(ii), §10.18 and Ch.10

10.18.13		$$x^2{M}_{\nu }^{\prime \prime }\left(x\right)+x{M}_{\nu }^{\prime }\left(x\right)+(x^2-{\nu }^2)M_{\nu }\left(x\right)=\frac{4}{{\pi }^2{M}_{\nu }^3(x)},$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $M_{\nu }\left(x\right)$: 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		$$w^{\prime \prime }+\left(1+\frac{\frac{1}{4}-{\nu }^2}{x^2}\right)w=\frac{4}{{\pi }^2{w}^3},$$
$w=x^{\frac{1}{2}}{M}_{\nu }\left(x\right)$,
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $M_{\nu }\left(x\right)$: modulus of Bessel functions, $x$: real variable and $\nu$: complex parameter Permalink: http://dlmf.nist.gov/10.18.E14 Encodings: TeX, pMML, png See also: Annotations for §10.18(ii), §10.18 and Ch.10

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: $M_{\nu }\left(x\right)$: 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		$$\theta _{\nu }^{\prime }{}^2\left(x\right)+\frac{1}{2}\frac{{\theta }_{\nu }^{\prime \prime \prime }\left(x\right)}{{\theta }_{\nu }^{\prime }\left(x\right)}-\frac{3}{4}{\left(\frac{{\theta }_{\nu }^{\prime \prime }\left(x\right)}{{\theta }_{\nu }^{\prime }\left(x\right)}\right)}^2=1-\frac{{\nu }^2-\frac{1}{4}}{x^2}.$$
ⓘ Symbols: ${\theta }_{\nu }\left(x\right)$: 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

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

10.18.17		$M_{\nu }^2\left(x\right)$	$\sim {\displaystyle \frac{2}{\pi x}}\left(1+{\displaystyle \frac{1}{2}}{\displaystyle \frac{\mu -1}{{(2x)}^2}}+{\displaystyle \frac{1\cdot 3}{2\cdot 4}}{\displaystyle \frac{(\mu -1)(\mu -9)}{{(2x)}^4}}+{\displaystyle \frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}{\displaystyle \frac{(\mu -1)(\mu -9)(\mu -25)}{{(2x)}^6}}+\mathrm{\cdots }\right),$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $M_{\nu }\left(x\right)$: modulus of Bessel functions, $x$: real variable and $\nu$: complex parameter A&S Ref: 9.2.28 Referenced by: §10.18(iii), §10.18(iii), §10.40(i), §10.40(i), §10.49(iv) 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		${\theta }_{\nu }\left(x\right)$	$\sim \begin{array}{l}x-\left({\displaystyle \frac{1}{2}}\nu +{\displaystyle \frac{1}{4}}\right)\pi +{\displaystyle \frac{\mu -1}{2(4x)}}+{\displaystyle \frac{(\mu -1)(\mu -25)}{6{(4x)}^3}}+{\displaystyle \frac{(\mu -1)({\mu }^2-114\mu +1073)}{5{(4x)}^5}}\\ +{\displaystyle \frac{(\mu -1)(5{\mu }^3-1535{\mu }^2+54703\mu -\mathrm{3\hspace{0.33em}75733})}{14{(4x)}^7}}+\mathrm{\cdots }.\end{array}$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: 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: 9.2.29 Referenced by: §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		$$N_{\nu }^2\left(x\right)\sim \frac{2}{\pi x}\left(1-\frac{1}{2}\frac{\mu -3}{{(2x)}^2}-\frac{1}{2\cdot 4}\frac{(\mu -1)(\mu -45)}{{(2x)}^4}-\mathrm{\cdots }\right),$$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $N_{\nu }\left(x\right)$: 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		$$-\frac{(2k-3)!!}{(2k)!!}\frac{(\mu -1)(\mu -9)\mathrm{\cdots }(\mu -{(2k-3)}^2)(\mu -(2k+1){(2k-1)}^2)}{{(2x)}^{2k}},$$
$k\ge 2$,
ⓘ 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), §10.18 and Ch.10

and

10.18.21		$${\varphi }_{\nu }\left(x\right)\sim x-\left(\frac{1}{2}\nu -\frac{1}{4}\right)\pi +\frac{\mu +3}{2(4x)}+\frac{{\mu }^2+46\mu -63}{6{(4x)}^3}+\frac{{\mu }^3+185{\mu }^2-2053\mu +1899}{5{(4x)}^5}+\mathrm{\cdots }.$$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: 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: 9.2.31 Referenced by: §10.18(iii) Permalink: http://dlmf.nist.gov/10.18.E21 Encodings: TeX, pMML, png See also: Annotations for §10.18(iii), §10.18 and Ch.10

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 $k>\nu -\frac{1}{2}$.