10 Bessel Functions Bessel and Hankel Functions 10.17 Asymptotic Expansions for Large Argument 10.19 Asymptotic Expansions for Large Order

§10.18 Modulus and Phase Functions

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Referenced by:: §10.21(x), §10.3(i)
Permalink:: http://dlmf.nist.gov/10.18
See also:: Annotations for Ch.10

§10.18(i) Definitions

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Keywords:: Bessel functions, definitions, modulus and phase functions
Notes:: For (10.18.3) see §10.7(i).
Referenced by:: §10.21(ii)
Permalink:: http://dlmf.nist.gov/10.18.i
See also:: Annotations for §10.18 and Ch.10

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

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Keywords:: Bessel functions, basic properties, modulus and phase functions
Notes:: These results are verifiable by straightforward substitutions.
Permalink:: http://dlmf.nist.gov/10.18.ii
See also:: Annotations for §10.18 and Ch.10

10.18.4	$\displaystyle J_{\nu}\left(x\right)$	$\displaystyle=M_{\nu}\left(x\right)\cos\theta_{\nu}\left(x\right),$
10.18.4	$\displaystyle Y_{\nu}\left(x\right)$	$\displaystyle=M_{\nu}\left(x\right)\sin\theta_{\nu}\left(x\right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\cos\NVar{z}$ : cosine function, $M_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of Bessel functions, $\theta_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of Bessel functions, $\sin\NVar{z}$ : 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	$\displaystyle J_{\nu}'\left(x\right)$	$\displaystyle=N_{\nu}\left(x\right)\cos\phi_{\nu}\left(x\right),$
10.18.5	$\displaystyle Y_{\nu}'\left(x\right)$	$\displaystyle=N_{\nu}\left(x\right)\sin\phi_{\nu}\left(x\right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\cos\NVar{z}$ : cosine function, $N_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of derivatives of Bessel functions, $\phi_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of derivatives of Bessel functions, $\sin\NVar{z}$ : 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	$\displaystyle M_{\nu}\left(x\right)$	$\displaystyle=\left({J_{\nu}}^{2}\left(x\right)+{Y_{\nu}}^{2}\left(x\right)% \right)^{\frac{1}{2}},$
10.18.6	$\displaystyle N_{\nu}\left(x\right)$	$\displaystyle=\left({J_{\nu}'}^{2}\left(x\right)+{Y_{\nu}'}^{2}\left(x\right)% \right)^{\frac{1}{2}},$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $M_{\NVar{\nu}}\left(\NVar{x}\right)$ : 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: 9.2.17, 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	$\displaystyle\theta_{\nu}\left(x\right)$	$\displaystyle=\operatorname{Arctan}\left(Y_{\nu}\left(x\right)/J_{\nu}\left(x% \right)\right),$
10.18.7	$\displaystyle\phi_{\nu}\left(x\right)$	$\displaystyle=\operatorname{Arctan}\left(Y_{\nu}'\left(x\right)/J_{\nu}'\left(% x\right)\right).$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\operatorname{Arctan}\NVar{z}$ : general arctangent function, $\phi_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of derivatives of Bessel functions, $\theta_{\NVar{\nu}}\left(\NVar{x}\right)$ : 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	$\displaystyle{M_{\nu}}^{2}\left(x\right)\theta_{\nu}'\left(x\right)$	$\displaystyle=\frac{2}{\pi x},$
10.18.8	$\displaystyle{N_{\nu}}^{2}\left(x\right)\phi_{\nu}'\left(x\right)$	$\displaystyle=\frac{2(x^{2}-\nu^{2})}{\pi x^{3}},$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $M_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of Bessel functions, $N_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of derivatives of Bessel functions, $\phi_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of derivatives of Bessel functions, $\theta_{\NVar{\nu}}\left(\NVar{x}\right)$ : 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		${N_{\nu}}^{2}\left(x\right)={M_{\nu}'}^{2}\left(x\right)+{M_{\nu}}^{2}\left(x% \right){\theta_{\nu}'}^{2}\left(x\right)={M_{\nu}'}^{2}\left(x\right)+\frac{4}% {(\pi xM_{\nu}\left(x\right))^{2}},$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $M_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of Bessel functions, $N_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of derivatives of Bessel functions, $\theta_{\NVar{\nu}}\left(\NVar{x}\right)$ : 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		$(x^{2}-\nu^{2})M_{\nu}\left(x\right)M_{\nu}'\left(x\right)+x^{2}N_{\nu}\left(x% \right)N_{\nu}'\left(x\right)+x{N_{\nu}}^{2}\left(x\right)=0.$
ⓘ 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		$\tan\left(\phi_{\nu}\left(x\right)-\theta_{\nu}\left(x\right)\right)=\frac{M_{% \nu}\left(x\right)\theta_{\nu}'\left(x\right)}{M_{\nu}'\left(x\right)}=\frac{2% }{\pi xM_{\nu}\left(x\right)M_{\nu}'\left(x\right)},$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $M_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of Bessel functions, $\phi_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of derivatives of Bessel functions, $\theta_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of Bessel functions, $\tan\NVar{z}$ : 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		$M_{\nu}\left(x\right)N_{\nu}\left(x\right)\sin\left(\phi_{\nu}\left(x\right)-% \theta_{\nu}\left(x\right)\right)=\frac{2}{\pi x}.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $M_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of Bessel functions, $N_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of derivatives of Bessel functions, $\phi_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of derivatives of Bessel functions, $\theta_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of Bessel functions, $\sin\NVar{z}$ : 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		$x^{2}M_{\nu}''\left(x\right)+xM_{\nu}'\left(x\right)+(x^{2}-\nu^{2})M_{\nu}% \left(x\right)=\frac{4}{\pi^{2}{{M_{\nu}}^{3}(x)}},$
ⓘ 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		$w^{\prime\prime}+\left(1+\frac{\frac{1}{4}-\nu^{2}}{x^{2}}\right)w=\frac{4}{% \pi^{2}w^{3}},$
$w=x^{\frac{1}{2}}M_{\nu}\left(x\right)$ ,
ⓘ 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 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: $M_{\NVar{\nu}}\left(\NVar{x}\right)$ : 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

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Keywords:: Bessel functions, asymptotic expansions for large argument, modulus and phase functions
Notes:: For (10.18.17), and also the concluding paragraph, see Watson (1944, pp. 448–449). For (10.18.19) substitute into ${N_{\nu}}^{2}\left(x\right)={H^{(1)}_{\nu}}'\left(x\right){H^{(2)}_{\nu}}'% \left(x\right)$ by means of (10.17.11), (10.17.12). The general term in (10.18.20) can be verified via (10.18.10). For (10.18.18) the first two terms can be found from (10.18.7), (10.17.3), (10.17.4), except for an arbitrary integer multiple of $\pi$ . Higher terms can be calculated via (10.18.8), (10.18.17). By continuity, the multiple of $\pi$ is independent of $\nu$ , hence it may be determined, e.g. by setting $\nu=\tfrac{1}{2}$ and referring to (10.16.1). Similar methods can be used for (10.18.21), together with the interlacing properties of the zeros of $J_{1/2}\left(z\right)$ , $Y_{1/2}\left(z\right)$ , and their derivatives (§10.21(i)). See also Bickley et al. (1952, p. xxxiv).
Referenced by:: (9.8.20), (9.8.21), (9.8.22), (9.8.23), Erratum (V1.1.8) for Section 10.18(iii)
Permalink:: http://dlmf.nist.gov/10.18.iii
Modification (effective with 1.1.8):: The paragraph immediately following (10.18.21) was updated to include new information about (10.18.17)-(10.18.18) from Nemes (2021).
Suggested 2022-12-05 by Gergő Nemes
See also:: Annotations for §10.18 and Ch.10

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		$\phi_{\nu}\left(x\right)\sim x-\left(\frac{1}{2}\nu-\frac{1}{4}\right)\pi+% \frac{\mu+3}{2(4x)}+\frac{\mu^{2}+46\mu-63}{6(4x)^{3}}+\frac{\mu^{3}+185\mu^{2% }-2053\mu+1899}{5(4x)^{5}}+\dotsi.$
ⓘ Symbols: $\sim$ : Poincaré asymptotic expansion, $\pi$ : 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: 9.2.31 Referenced by: §10.18(iii), Erratum (V1.1.8) for Section 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

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).

10.17 Asymptotic Expansions for Large Argument 10.19 Asymptotic Expansions for Large Order

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§10.18 Modulus and Phase Functions

Contents

§10.18(i) Definitions

§10.18(ii) Basic Properties

§10.18(iii) Asymptotic Expansions for Large Argument

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