§9.8 Modulus and Phase

Referenced by:: §10.21(vii)
Permalink:: http://dlmf.nist.gov/9.8

§9.8(i) Definitions
§9.8(ii) Identities
§9.8(iii) Monotonicity
§9.8(iv) Asymptotic Expansions

§9.8(i) Definitions

Notes:: For (9.8.9)–(9.8.12) combine (9.6.17) and (9.6.18) with (10.4.3).
Defines:: $\mathop{M\/}\nolimits\!\left(z\right)$ : Airy modulus function, $\mathop{\theta\/}\nolimits\!\left(z\right)$ : Airy phase function, $\mathop{N\/}\nolimits\!\left(z\right)$ : Airy modulus function and $\mathop{\phi\/}\nolimits\!\left(z\right)$ : Airy phase function
Keywords:: Airy functions
Referenced by:: §9.11(v), Figure 9.3.1, Figure 9.3.1, Figure 9.3.2, Figure 9.3.2
Permalink:: http://dlmf.nist.gov/9.8.i

Throughout this section $x$ is real and nonpositive.

9.8.1 $\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)=\mathop{M\/}\nolimits\!\left(x\right)\mathop{\sin\/}\nolimits\mathop{\theta\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{M\/}\nolimits\!\left(z\right)$ : Airy modulus function, $\mathop{\theta\/}\nolimits\!\left(z\right)$ : Airy phase function, $\mathop{\sin\/}\nolimits z$ : sine function and $x$ : real variable
Permalink:: http://dlmf.nist.gov/9.8.E1
Encodings:: TeX, pMML, png

9.8.2 $\mathop{\mathrm{Bi}\/}\nolimits\!\left(x\right)=\mathop{M\/}\nolimits\!\left(x\right)\mathop{\cos\/}\nolimits\mathop{\theta\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{M\/}\nolimits\!\left(z\right)$ : Airy modulus function, $\mathop{\theta\/}\nolimits\!\left(z\right)$ : Airy phase function and $x$ : real variable
Permalink:: http://dlmf.nist.gov/9.8.E2
Encodings:: TeX, pMML, png

9.8.3 $\mathop{M\/}\nolimits\!\left(x\right)=\sqrt{{\mathop{\mathrm{Ai}\/}\nolimits^{{2}}}\!\left(x\right)+{\mathop{\mathrm{Bi}\/}\nolimits^{{2}}}\!\left(x\right)},$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{M\/}\nolimits\!\left(z\right)$ : Airy modulus function and $x$ : real variable
Permalink:: http://dlmf.nist.gov/9.8.E3
Encodings:: TeX, pMML, png

9.8.4 $\mathop{\theta\/}\nolimits\!\left(x\right)=\mathop{\mathrm{arctan}\/}\nolimits\!\left(\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)/\mathop{\mathrm{Bi}\/}\nolimits\!\left(x\right)\right).$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, $\mathop{\theta\/}\nolimits\!\left(z\right)$ : Airy phase function and $x$ : real variable
Referenced by:: §9.11(v)
Permalink:: http://dlmf.nist.gov/9.8.E4
Encodings:: TeX, pMML, png

9.8.5 ${\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(x\right)=\mathop{N\/}\nolimits\!\left(x\right)\mathop{\sin\/}\nolimits\mathop{\phi\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{N\/}\nolimits\!\left(z\right)$ : Airy modulus function, $\mathop{\phi\/}\nolimits\!\left(z\right)$ : Airy phase function, $\mathop{\sin\/}\nolimits z$ : sine function and $x$ : real variable
Permalink:: http://dlmf.nist.gov/9.8.E5
Encodings:: TeX, pMML, png

9.8.6 ${\mathop{\mathrm{Bi}\/}\nolimits^{{\prime}}}\!\left(x\right)=\mathop{N\/}\nolimits\!\left(x\right)\mathop{\cos\/}\nolimits\mathop{\phi\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{N\/}\nolimits\!\left(z\right)$ : Airy modulus function, $\mathop{\phi\/}\nolimits\!\left(z\right)$ : Airy phase function and $x$ : real variable
Permalink:: http://dlmf.nist.gov/9.8.E6
Encodings:: TeX, pMML, png

9.8.7 $\mathop{N\/}\nolimits\!\left(x\right)=\sqrt{{{\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}^{{2}}}\!\left(x\right)+{{\mathop{\mathrm{Bi}\/}\nolimits^{{\prime}}}^{{2}}}\!\left(x\right)},$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{N\/}\nolimits\!\left(z\right)$ : Airy modulus function and $x$ : real variable
Permalink:: http://dlmf.nist.gov/9.8.E7
Encodings:: TeX, pMML, png

9.8.8 $\mathop{\phi\/}\nolimits\!\left(x\right)=\mathop{\mathrm{arctan}\/}\nolimits\!\left({\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(x\right)/{\mathop{\mathrm{Bi}\/}\nolimits^{{\prime}}}\!\left(x\right)\right).$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, $\mathop{\phi\/}\nolimits\!\left(z\right)$ : Airy phase function and $x$ : real variable
Referenced by:: §9.11(v)
Permalink:: http://dlmf.nist.gov/9.8.E8
Encodings:: TeX, pMML, png

Graphs of $\mathop{M\/}\nolimits\!\left(x\right)$ and $\mathop{N\/}\nolimits\!\left(x\right)$ are included in §9.3(i). The branches of $\mathop{\theta\/}\nolimits\!\left(x\right)$ and $\mathop{\phi\/}\nolimits\!\left(x\right)$ are continuous and fixed by $\mathop{\theta\/}\nolimits\!\left(0\right)=-\mathop{\phi\/}\nolimits\!\left(0\right)=\tfrac{1}{6}\pi$ . (These definitions of $\mathop{\theta\/}\nolimits\!\left(x\right)$ and $\mathop{\phi\/}\nolimits\!\left(x\right)$ differ from Abramowitz and Stegun (1964, Chapter 10), and agree more closely with those used in Miller (1946) and Olver (1997b, Chapter 11).)

In terms of Bessel functions, and with $\xi=\tfrac{2}{3}|x|^{{3/2}}$ ,

9.8.9 $|x|^{{1/2}}{\mathop{M\/}\nolimits^{{2}}}\!\left(x\right)=\tfrac{1}{2}\xi\left({\mathop{J_{{1/3}}\/}\nolimits^{{2}}}\!\left(\xi\right)+{\mathop{Y_{{1/3}}\/}\nolimits^{{2}}}\!\left(\xi\right)\right),$

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, $\mathop{M\/}\nolimits\!\left(z\right)$ : Airy modulus function, $x$ : real variable and $xi$ : change of variable
Referenced by:: §9.8(i), §9.8(ii), §9.8(iv)
Permalink:: http://dlmf.nist.gov/9.8.E9
Encodings:: TeX, pMML, png

9.8.10 $|x|^{{-1/2}}{\mathop{N\/}\nolimits^{{2}}}\!\left(x\right)=\tfrac{1}{2}\xi\left({\mathop{J_{{2/3}}\/}\nolimits^{{2}}}\!\left(\xi\right)+{\mathop{Y_{{2/3}}\/}\nolimits^{{2}}}\!\left(\xi\right)\right),$

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, $\mathop{N\/}\nolimits\!\left(z\right)$ : Airy modulus function, $x$ : real variable and $xi$ : change of variable
Permalink:: http://dlmf.nist.gov/9.8.E10
Encodings:: TeX, pMML, png

9.8.11 $\mathop{\theta\/}\nolimits\!\left(x\right)=\tfrac{2}{3}\pi+\mathop{\mathrm{arctan}\/}\nolimits\!\left(\mathop{Y_{{1/3}}\/}\nolimits\!\left(\xi\right)/\mathop{J_{{1/3}}\/}\nolimits\!\left(\xi\right)\right),$

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, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, $\mathop{\theta\/}\nolimits\!\left(z\right)$ : Airy phase function, $x$ : real variable and $xi$ : change of variable
Permalink:: http://dlmf.nist.gov/9.8.E11
Encodings:: TeX, pMML, png

9.8.12 $\mathop{\phi\/}\nolimits\!\left(x\right)=\tfrac{1}{3}\pi+\mathop{\mathrm{arctan}\/}\nolimits\!\left(\mathop{Y_{{2/3}}\/}\nolimits\!\left(\xi\right)/\mathop{J_{{2/3}}\/}\nolimits\!\left(\xi\right)\right).$

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, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, $\mathop{\phi\/}\nolimits\!\left(z\right)$ : Airy phase function, $x$ : real variable and $xi$ : change of variable
Referenced by:: §9.8(i), §9.8(ii), §9.8(iv)
Permalink:: http://dlmf.nist.gov/9.8.E12
Encodings:: TeX, pMML, png

§9.8(ii) Identities

Notes:: Combine (9.8.9)–(9.8.12) with §10.18(ii). See also Olver (1997b, p. 404), Miller (1946, p. B10). For (9.8.13) use (9.2.7).
Keywords:: Airy functions
Permalink:: http://dlmf.nist.gov/9.8.ii

Primes denote differentiations with respect to $x$ , which is continued to be assumed real and nonpositive.

9.8.13 $\mathop{M\/}\nolimits\!\left(x\right)\mathop{N\/}\nolimits\!\left(x\right)\mathop{\sin\/}\nolimits\!\left(\mathop{\theta\/}\nolimits\!\left(x\right)-\mathop{\phi\/}\nolimits\!\left(x\right)\right)=\pi^{{-1}},$

Symbols:: $\mathop{M\/}\nolimits\!\left(z\right)$ : Airy modulus function, $\mathop{N\/}\nolimits\!\left(z\right)$ : Airy modulus function, $\mathop{\phi\/}\nolimits\!\left(z\right)$ : Airy phase function, $\mathop{\theta\/}\nolimits\!\left(z\right)$ : Airy phase function, $\mathop{\sin\/}\nolimits z$ : sine function and $x$ : real variable
Referenced by:: §9.8(ii)
Permalink:: http://dlmf.nist.gov/9.8.E13
Encodings:: TeX, pMML, png

9.8.14

${\mathop{M\/}\nolimits^{{2}}}\!\left(x\right){\mathop{\theta\/}\nolimits^{{\prime}}}\!\left(x\right)=-\pi^{{-1}}$ ,

${\mathop{N\/}\nolimits^{{2}}}\!\left(x\right){\mathop{\phi\/}\nolimits^{{\prime}}}\!\left(x\right)=\pi^{{-1}}x$ ,

$\mathop{N\/}\nolimits\!\left(x\right){\mathop{N\/}\nolimits^{{\prime}}}\!\left(x\right)=x\mathop{M\/}\nolimits\!\left(x\right){\mathop{M\/}\nolimits^{{\prime}}}\!\left(x\right)$ ,

Symbols:: $\mathop{M\/}\nolimits\!\left(z\right)$ : Airy modulus function, $\mathop{N\/}\nolimits\!\left(z\right)$ : Airy modulus function, $\mathop{\phi\/}\nolimits\!\left(z\right)$ : Airy phase function, $\mathop{\theta\/}\nolimits\!\left(z\right)$ : Airy phase function and $x$ : real variable
Referenced by:: §9.11(v)
Permalink:: http://dlmf.nist.gov/9.8.E14
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

9.8.15 ${\mathop{N\/}\nolimits^{{2}}}\!\left(x\right)={{\mathop{M\/}\nolimits^{{\prime}}}^{{2}}}\!\left(x\right)+{\mathop{M\/}\nolimits^{{2}}}\!\left(x\right){{\mathop{\theta\/}\nolimits^{{\prime}}}^{{2}}}\!\left(x\right)={{\mathop{M\/}\nolimits^{{\prime}}}^{{2}}}(x)+\pi^{{-2}}{\mathop{M\/}\nolimits^{{-2}}}\!\left(x\right),$

Symbols:: $\mathop{M\/}\nolimits\!\left(z\right)$ : Airy modulus function, $\mathop{N\/}\nolimits\!\left(z\right)$ : Airy modulus function, $\mathop{\theta\/}\nolimits\!\left(z\right)$ : Airy phase function and $x$ : real variable
Permalink:: http://dlmf.nist.gov/9.8.E15
Encodings:: TeX, pMML, png

9.8.16 $x^{2}{\mathop{M\/}\nolimits^{{2}}}\!\left(x\right)={{\mathop{N\/}\nolimits^{{\prime}}}^{{2}}}\!\left(x\right)+{\mathop{N\/}\nolimits^{{2}}}\!\left(x\right){{\mathop{\phi\/}\nolimits^{{\prime}}}^{{2}}}\!\left(x\right)={{\mathop{N\/}\nolimits^{{\prime}}}^{{2}}}\!\left(x\right)+\pi^{{-2}}x^{2}{\mathop{N\/}\nolimits^{{-2}}}\!\left(x\right),$

Symbols:: $\mathop{M\/}\nolimits\!\left(z\right)$ : Airy modulus function, $\mathop{N\/}\nolimits\!\left(z\right)$ : Airy modulus function, $\mathop{\phi\/}\nolimits\!\left(z\right)$ : Airy phase function and $x$ : real variable
Permalink:: http://dlmf.nist.gov/9.8.E16
Encodings:: TeX, pMML, png

9.8.17 $\mathop{\tan\/}\nolimits\!\left(\mathop{\theta\/}\nolimits\!\left(x\right)-\mathop{\phi\/}\nolimits\!\left(x\right)\right)=1/(\pi\mathop{M\/}\nolimits\!\left(x\right){\mathop{M\/}\nolimits^{{\prime}}}\!\left(x\right))=-\mathop{M\/}\nolimits\!\left(x\right){\mathop{\theta\/}\nolimits^{{\prime}}}\!\left(x\right)/{\mathop{M\/}\nolimits^{{\prime}}}\!\left(x\right),$

Symbols:: $\mathop{M\/}\nolimits\!\left(z\right)$ : Airy modulus function, $\mathop{\phi\/}\nolimits\!\left(z\right)$ : Airy phase function, $\mathop{\theta\/}\nolimits\!\left(z\right)$ : Airy phase function, $\mathop{\tan\/}\nolimits z$ : tangent function and $x$ : real variable
Permalink:: http://dlmf.nist.gov/9.8.E17
Encodings:: TeX, pMML, png

9.8.18

${\mathop{M\/}\nolimits^{{\prime\prime}}}\!\left(x\right)=x\mathop{M\/}\nolimits\!\left(x\right)+\pi^{{-2}}{\mathop{M\/}\nolimits^{{-3}}}\!\left(x\right)$ ,

${{\mathop{M\/}\nolimits^{{2}}}^{{\prime\prime\prime}}}\!\left(x\right)-4x{{\mathop{M\/}\nolimits^{{2}}}^{{\prime}}}\!\left(x\right)-2{\mathop{M\/}\nolimits^{{2}}}\!\left(x\right)=0,$

Symbols:: $\mathop{M\/}\nolimits\!\left(z\right)$ : Airy modulus function and $x$ : real variable
Permalink:: http://dlmf.nist.gov/9.8.E18
Encodings:: TeX, TeX, pMML, pMML, png, png

9.8.19 ${{\mathop{\theta\/}\nolimits^{{\prime}}}^{{2}}}\!\left(x\right)+\tfrac{1}{2}({\mathop{\theta\/}\nolimits^{{\prime\prime\prime}}}\!\left(x\right)/{\mathop{\theta\/}\nolimits^{{\prime}}}\!\left(x\right))-\tfrac{3}{4}({\mathop{\theta\/}\nolimits^{{\prime\prime}}}\!\left(x\right)/{\mathop{\theta\/}\nolimits^{{\prime}}}\!\left(x\right))^{2}=-x.$

Symbols:: $\mathop{\theta\/}\nolimits\!\left(z\right)$ : Airy phase function and $x$ : real variable
Permalink:: http://dlmf.nist.gov/9.8.E19
Encodings:: TeX, pMML, png

§9.8(iii) Monotonicity

Notes:: See Olver (1997b, p. 404).
Keywords:: Airy functions, monotonicity
Permalink:: http://dlmf.nist.gov/9.8.iii

As $x$ increases from $-\infty$ to 0 each of the functions $\mathop{M\/}\nolimits\!\left(x\right)$ , ${\mathop{M\/}\nolimits^{{\prime}}}\!\left(x\right)$ , $|x|^{{-1/4}}\mathop{N\/}\nolimits\!\left(x\right)$ , $\mathop{M\/}\nolimits\!\left(x\right)\mathop{N\/}\nolimits\!\left(x\right)$ , ${\mathop{\theta\/}\nolimits^{{\prime}}}\!\left(x\right)$ , ${\mathop{\phi\/}\nolimits^{{\prime}}}\!\left(x\right)$ is increasing, and each of the functions $|x|^{{1/4}}\mathop{M\/}\nolimits\!\left(x\right)$ , $\mathop{\theta\/}\nolimits\!\left(x\right)$ , $\mathop{\phi\/}\nolimits\!\left(x\right)$ is decreasing.

§9.8(iv) Asymptotic Expansions

Notes:: Combine (9.8.9)–(9.8.12) with §10.18(iii). See also Miller (1946, p. B48).
Keywords:: Airy functions
Permalink:: http://dlmf.nist.gov/9.8.iv

As $x\rightarrow-\infty$

9.8.20 ${\mathop{M\/}\nolimits^{{2}}}\!\left(x\right)\sim\frac{1}{\pi(-x)^{{1/2}}}\sum _{{k=0}}^{{\infty}}\frac{1\cdot 3\cdot 5\cdots(6k-1)}{k!(96)^{k}}\frac{1}{x^{{3k}}},$

Symbols:: $!$ : $n!$ : factorial, $\mathop{M\/}\nolimits\!\left(z\right)$ : Airy modulus function, $\sim$ : asymptotic equality and $x$ : real variable
Referenced by:: §9.8(iv)
Permalink:: http://dlmf.nist.gov/9.8.E20
Encodings:: TeX, pMML, png

9.8.21 ${\mathop{N\/}\nolimits^{{2}}}\!\left(x\right)\sim\frac{(-x)^{{1/2}}}{\pi}\sum _{{k=0}}^{{\infty}}\frac{1\cdot 3\cdot 5\cdots(6k-1)}{k!(96)^{k}}\frac{1+6k}{1-6k}\frac{1}{x^{{3k}}},$

Symbols:: $!$ : $n!$ : factorial, $\mathop{N\/}\nolimits\!\left(z\right)$ : Airy modulus function, $\sim$ : asymptotic equality and $x$ : real variable
Referenced by:: §9.8(iv)
Permalink:: http://dlmf.nist.gov/9.8.E21
Encodings:: TeX, pMML, png

9.8.22 $\mathop{\theta\/}\nolimits\!\left(x\right)\sim\frac{\pi}{4}+\frac{2}{3}(-x)^{{3/2}}\left(1+\frac{5}{32}\frac{1}{x^{3}}+\frac{1105}{6144}\frac{1}{x^{6}}+\frac{82825}{65536}\frac{1}{x^{9}}+\frac{12820\; 31525}{587\; 20256}\frac{1}{x^{{12}}}+\cdots\right),$

Symbols:: $\mathop{\theta\/}\nolimits\!\left(z\right)$ : Airy phase function, $\sim$ : asymptotic equality and $x$ : real variable
Referenced by:: §9.8(iv)
Permalink:: http://dlmf.nist.gov/9.8.E22
Encodings:: TeX, pMML, png

9.8.23 $\mathop{\phi\/}\nolimits\!\left(x\right)\sim-\frac{\pi}{4}+\frac{2}{3}(-x)^{{3/2}}\left(1-\frac{7}{32}\frac{1}{x^{3}}-\frac{1463}{6144}\frac{1}{x^{6}}-\frac{4\; 95271}{3\; 27680}\frac{1}{x^{9}}-\frac{2065\; 30429}{83\; 88608}\frac{1}{x^{{12}}}-\cdots\right).$

Symbols:: $\mathop{\phi\/}\nolimits\!\left(z\right)$ : Airy phase function, $\sim$ : asymptotic equality and $x$ : real variable
Referenced by:: §9.8(iv)
Permalink:: http://dlmf.nist.gov/9.8.E23
Encodings:: TeX, pMML, png

In (9.8.20) and (9.8.21) the remainder after $n$ terms does not exceed the $(n+1)$ th term in absolute value and is of the same sign, provided that $n\geq 0$ for (9.8.20) and $n\geq 1$ for (9.8.21).

For higher terms in (9.8.22) and (9.8.23) see Fabijonas et al. (2004). Also, approximate values (25S) of the coefficients of the powers ${x^{{-15}}}$ , ${x^{{-18}}}$ , $\ldots,$ ${x^{{-56}}}$ are available in Sherry (1959).

§9.8 Modulus and Phase

Contents

§9.8(i) Definitions

§9.8(ii) Identities

§9.8(iii) Monotonicity

§9.8(iv) Asymptotic Expansions