§10.18 Modulus and Phase Functions

Referenced by:: §10.21(x), §10.3(i)
Permalink:: http://dlmf.nist.gov/10.18

§10.18(i) Definitions
§10.18(ii) Basic Properties
§10.18(iii) Asymptotic Expansions for Large Argument

§10.18(i) Definitions

Notes:: For (10.18.3) see §10.7(i).
Keywords:: Bessel functions
Referenced by:: §10.21(ii)
Permalink:: http://dlmf.nist.gov/10.18.i

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

10.18.1 $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)e^{{i\!\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)}}=\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(x\right),$

Defines:: $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of Bessel functions
Symbols:: $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $e$ : base of exponential function, $\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of Bessel functions, $x$ : real variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.18.E1
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10.18.2 $\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right)e^{{i\!\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right)}}={\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits^{{\prime}}}\!\left(x\right),$

Defines:: $\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of derivatives of Bessel functions
Symbols:: $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $e$ : base of exponential function, $\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of derivatives of Bessel functions, $x$ : real variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.18.E2
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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

$\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)\to-\tfrac{1}{2}\pi,$

$\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right)\to\tfrac{1}{2}\pi$ , $x\to 0+$ .

Defines:: $\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of derivatives of Bessel functions and $\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of Bessel functions
Symbols:: $x$ : real variable and $\nu$ : complex parameter
Referenced by:: §10.18(i), §10.21(ii)
Permalink:: http://dlmf.nist.gov/10.18.E3
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§10.18(ii) Basic Properties

Notes:: These results are verifiable by straightforward substitutions.
Keywords:: Bessel functions
Referenced by:: §9.8(ii)
Permalink:: http://dlmf.nist.gov/10.18.ii

10.18.4

$\mathop{J_{{\nu}}\/}\nolimits\!\left(x\right)=\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)\mathop{\cos\/}\nolimits\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right),$

$\mathop{Y_{{\nu}}\/}\nolimits\!\left(x\right)=\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)\mathop{\sin\/}\nolimits\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of Bessel functions, $\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of Bessel functions, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real variable and $\nu$ : complex parameter
A&S Ref:: 9.2.19
Permalink:: http://dlmf.nist.gov/10.18.E4
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10.18.5

${\mathop{J_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(x\right)=\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right)\mathop{\cos\/}\nolimits\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right),$

${\mathop{Y_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(x\right)=\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right)\mathop{\sin\/}\nolimits\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of derivatives of Bessel functions, $\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of derivatives of Bessel functions, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real variable and $\nu$ : complex parameter
A&S Ref:: 9.2.20
Permalink:: http://dlmf.nist.gov/10.18.E5
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10.18.6

$\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)=\left({\mathop{J_{{\nu}}\/}\nolimits^{{2}}}\!\left(x\right)+{\mathop{Y_{{\nu}}\/}\nolimits^{{2}}}\!\left(x\right)\right)^{{\frac{1}{2}}},$

$\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right)=\left({{\mathop{J_{{\nu}}\/}\nolimits^{{\prime}}}^{{2}}}\!\left(x\right)+{{\mathop{Y_{{\nu}}\/}\nolimits^{{\prime}}}^{{2}}}\!\left(x\right)\right)^{{\frac{1}{2}}},$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of Bessel functions, $\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right)$ : 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
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10.18.7

$\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)=\mathop{\mathrm{Arctan}\/}\nolimits\!\left(\mathop{Y_{{\nu}}\/}\nolimits\!\left(x\right)/\mathop{J_{{\nu}}\/}\nolimits\!\left(x\right)\right),$

$\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right)=\mathop{\mathrm{Arctan}\/}\nolimits\!\left({\mathop{Y_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(x\right)/{\mathop{J_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(x\right)\right).$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\mathrm{Arctan}\/}\nolimits z$ : general arctangent function, $\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of derivatives of Bessel functions, $\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)$ : 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
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10.18.8

${\mathop{M_{{\nu}}\/}\nolimits^{{2}}}\!\left(x\right){\mathop{\theta _{{\nu}}\/}\nolimits^{{\prime}}}\!\left(x\right)=\frac{2}{\pi x},$

${\mathop{N_{{\nu}}\/}\nolimits^{{2}}}\!\left(x\right){\mathop{\phi _{{\nu}}\/}\nolimits^{{\prime}}}\!\left(x\right)=\frac{2(x^{2}-\nu^{2})}{\pi x^{3}},$

Symbols:: $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of Bessel functions, $\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of derivatives of Bessel functions, $\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of derivatives of Bessel functions, $\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)$ : 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
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10.18.9 ${\mathop{N_{{\nu}}\/}\nolimits^{{2}}}\!\left(x\right)={{\mathop{M_{{\nu}}\/}\nolimits^{{\prime}}}^{{2}}}\!\left(x\right)+{\mathop{M_{{\nu}}\/}\nolimits^{{2}}}\!\left(x\right){{\mathop{\theta _{{\nu}}\/}\nolimits^{{\prime}}}^{{2}}}\!\left(x\right)={{\mathop{M_{{\nu}}\/}\nolimits^{{\prime}}}^{{2}}}\!\left(x\right)+\frac{4}{(\pi x\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right))^{2}},$

Symbols:: $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of Bessel functions, $\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of derivatives of Bessel functions, $\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)$ : 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

10.18.10 $(x^{2}-\nu^{2})\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right){\mathop{M_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(x\right)+x^{2}\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right){\mathop{N_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(x\right)+x{\mathop{N_{{\nu}}\/}\nolimits^{{2}}}\!\left(x\right)=0.$

Symbols:: $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of Bessel functions, $\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right)$ : 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

10.18.11 $\mathop{\tan\/}\nolimits\!\left(\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right)-\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)\right)=\frac{\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right){\mathop{\theta _{{\nu}}\/}\nolimits^{{\prime}}}\!\left(x\right)}{{\mathop{M_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(x\right)}=\frac{2}{\pi x\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right){\mathop{M_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(x\right)},$

Symbols:: $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of Bessel functions, $\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of derivatives of Bessel functions, $\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of Bessel functions, $\mathop{\tan\/}\nolimits z$ : 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

10.18.12 $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right)\mathop{\sin\/}\nolimits\!\left(\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right)-\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)\right)=\frac{2}{\pi x}.$

Symbols:: $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of Bessel functions, $\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of derivatives of Bessel functions, $\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of derivatives of Bessel functions, $\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of Bessel functions, $\mathop{\sin\/}\nolimits z$ : 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

10.18.13 $x^{2}{\mathop{M_{{\nu}}\/}\nolimits^{{\prime\prime}}}\!\left(x\right)+x{\mathop{M_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(x\right)+(x^{2}-\nu^{2})\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{4}{\pi^{2}{{\mathop{M_{{\nu}}\/}\nolimits^{{3}}}(x)}},$

Symbols:: $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ : 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

10.18.14 $w^{{\prime\prime}}+\left(1+\frac{\frac{1}{4}-\nu^{2}}{x^{2}}\right)w=\frac{4}{\pi^{2}w^{3}},$ $w=x^{{\frac{1}{2}}}\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ ,

Symbols:: $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of Bessel functions, $x$ : real variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.18.E14
Encodings:: TeX, pMML, png

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:: $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ : 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

10.18.16 ${{\mathop{\theta _{{\nu}}\/}\nolimits^{{\prime}}}^{{2}}}\!\left(x\right)+\frac{1}{2}\frac{{\mathop{\theta _{{\nu}}\/}\nolimits^{{\prime\prime\prime}}}\!\left(x\right)}{{\mathop{\theta _{{\nu}}\/}\nolimits^{{\prime}}}\!\left(x\right)}-\frac{3}{4}\left(\frac{{\mathop{\theta _{{\nu}}\/}\nolimits^{{\prime\prime}}}\!\left(x\right)}{{\mathop{\theta _{{\nu}}\/}\nolimits^{{\prime}}}\!\left(x\right)}\right)^{2}=1-\frac{\nu^{2}-\tfrac{1}{4}}{x^{2}}.$

Symbols:: $\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)$ : 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

§10.18(iii) Asymptotic Expansions for Large Argument

Notes:: For (10.18.17), and also the concluding paragraph, see Watson (1944, pp. 448–449). For (10.18.19) substitute into ${\mathop{N_{{\nu}}\/}\nolimits^{{2}}}\!\left(x\right)={\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits^{{\prime}}}\!\left(x\right){\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits^{{\prime}}}\!\left(x\right)$ 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 $\mathop{J_{{1/2}}\/}\nolimits\!\left(z\right)$ , $\mathop{Y_{{1/2}}\/}\nolimits\!\left(z\right)$ , and their derivatives (§10.21(i)). See also Bickley et al. (1952, p. xxxiv).
Keywords:: Bessel functions
Referenced by:: §9.8(iv)
Permalink:: http://dlmf.nist.gov/10.18.iii

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

10.18.17 ${\mathop{M_{{\nu}}\/}\nolimits^{{2}}}\!\left(x\right)\sim\frac{2}{\pi x}\left(1+\frac{1}{2}\frac{\mu-1}{(2x)^{2}}+\frac{1\cdot 3}{2\cdot 4}\frac{(\mu-1)(\mu-9)}{(2x)^{4}}+\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\frac{(\mu-1)(\mu-9)(\mu-25)}{(2x)^{6}}+\cdots\right),$

Symbols:: $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of Bessel functions, $\sim$ : asymptotic equality, $x$ : 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
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10.18.18 $\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)\sim x-\left(\frac{1}{2}\nu+\frac{1}{4}\right)\pi+\frac{\mu-1}{2(4x)}+\frac{(\mu-1)(\mu-25)}{6(4x)^{3}}+\frac{(\mu-1)(\mu^{2}-114\mu+1073)}{5(4x)^{5}}+\frac{(\mu-1)(5\mu^{3}-1535\mu^{2}+54703\mu-3\; 75733)}{14(4x)^{7}}+\cdots.$

Symbols:: $\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of Bessel functions, $\sim$ : asymptotic equality, $x$ : 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

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:: $\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of derivatives of Bessel functions, $\sim$ : asymptotic equality, $x$ : 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
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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:: $!!$ : $n!!$ : 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
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and

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

Symbols:: $\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of derivatives of Bessel functions, $\sim$ : asymptotic equality, $x$ : 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

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

§10.18 Modulus and Phase Functions

Contents

§10.18(i) Definitions

§10.18(ii) Basic Properties

§10.18(iii) Asymptotic Expansions for Large Argument