# §9.8 Modulus and Phase

## §9.8(i) Definitions

Throughout this section $x$ is real and nonpositive.

 9.8.1 $\displaystyle\operatorname{Ai}\left(x\right)$ $\displaystyle=M\left(x\right)\sin\theta\left(x\right),$ ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$: Airy function, $M\left(\NVar{z}\right)$: Airy modulus function, $\theta\left(\NVar{z}\right)$: Airy phase function, $\sin\NVar{z}$: sine function and $x$: real variable Source: Olver (1997b, (2.01), p. 394) Referenced by: (9.8.13), (9.9.3) Permalink: http://dlmf.nist.gov/9.8.E1 Encodings: TeX, pMML, png See also: Annotations for §9.8(i), §9.8 and Ch.9 9.8.2 $\displaystyle\operatorname{Bi}\left(x\right)$ $\displaystyle=M\left(x\right)\cos\theta\left(x\right),$ ⓘ Symbols: $\operatorname{Bi}\left(\NVar{z}\right)$: Airy function, $\cos\NVar{z}$: cosine function, $M\left(\NVar{z}\right)$: Airy modulus function, $\theta\left(\NVar{z}\right)$: Airy phase function and $x$: real variable Source: Olver (1997b, (2.01), p. 394) Referenced by: (9.8.13), (9.9.3) Permalink: http://dlmf.nist.gov/9.8.E2 Encodings: TeX, pMML, png See also: Annotations for §9.8(i), §9.8 and Ch.9
 9.8.3 $\displaystyle M\left(x\right)$ $\displaystyle=\sqrt{{\operatorname{Ai}}^{2}\left(x\right)+{\operatorname{Bi}}^% {2}\left(x\right)},$ ⓘ Defines: $M\left(\NVar{z}\right)$: Airy modulus function Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$: Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$: Airy function, $x$: real variable and $z$: complex variable Source: Olver (1997b, (2.02), p. 395) Referenced by: (9.8.14), (9.8.15), (9.8.16), (9.8.17), (9.8.18), (9.8.9) Permalink: http://dlmf.nist.gov/9.8.E3 Encodings: TeX, pMML, png See also: Annotations for §9.8(i), §9.8 and Ch.9 9.8.4 $\displaystyle\theta\left(x\right)$ $\displaystyle=\operatorname{arctan}\left(\operatorname{Ai}\left(x\right)/% \operatorname{Bi}\left(x\right)\right).$ ⓘ Defines: $\theta\left(\NVar{z}\right)$: Airy phase function Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$: Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$: Airy function, $\operatorname{arctan}\NVar{z}$: arctangent function, $x$: real variable and $z$: complex variable Source: Olver (1997b, (2.02), p. 395) Referenced by: (9.11.19), (9.8.11), (9.8.15), (9.8.17), (9.8.19) Permalink: http://dlmf.nist.gov/9.8.E4 Encodings: TeX, pMML, png See also: Annotations for §9.8(i), §9.8 and Ch.9
 9.8.5 $\displaystyle\operatorname{Ai}'\left(x\right)$ $\displaystyle=N\left(x\right)\sin\phi\left(x\right),$ ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$: Airy function, $N\left(\NVar{z}\right)$: Airy modulus function, $\phi\left(\NVar{z}\right)$: Airy phase function, $\sin\NVar{z}$: sine function and $x$: real variable Source: Olver (1997b, (2.08), p. 396); with different notation Referenced by: (9.8.13), (9.9.4) Permalink: http://dlmf.nist.gov/9.8.E5 Encodings: TeX, pMML, png See also: Annotations for §9.8(i), §9.8 and Ch.9 9.8.6 $\displaystyle\operatorname{Bi}'\left(x\right)$ $\displaystyle=N\left(x\right)\cos\phi\left(x\right),$ ⓘ Symbols: $\operatorname{Bi}\left(\NVar{z}\right)$: Airy function, $\cos\NVar{z}$: cosine function, $N\left(\NVar{z}\right)$: Airy modulus function, $\phi\left(\NVar{z}\right)$: Airy phase function and $x$: real variable Source: Olver (1997b, (2.08), p. 396); with different notation Referenced by: (9.8.13), (9.9.4) Permalink: http://dlmf.nist.gov/9.8.E6 Encodings: TeX, pMML, png See also: Annotations for §9.8(i), §9.8 and Ch.9
 9.8.7 $\displaystyle N\left(x\right)$ $\displaystyle=\sqrt{{\operatorname{Ai}'}^{2}\left(x\right)+{\operatorname{Bi}'% }^{2}\left(x\right)},$ ⓘ Defines: $N\left(\NVar{z}\right)$: Airy modulus function Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$: Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$: Airy function, $x$: real variable and $z$: complex variable Source: Olver (1997b, (2.09–2.10), p. 396); with different notation Referenced by: (9.8.10), (9.8.14), (9.8.15), (9.8.16) Permalink: http://dlmf.nist.gov/9.8.E7 Encodings: TeX, pMML, png See also: Annotations for §9.8(i), §9.8 and Ch.9 9.8.8 $\displaystyle\phi\left(x\right)$ $\displaystyle=\operatorname{arctan}\left(\operatorname{Ai}'\left(x\right)/% \operatorname{Bi}'\left(x\right)\right).$ ⓘ Defines: $\phi\left(\NVar{z}\right)$: Airy phase function Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$: Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$: Airy function, $\operatorname{arctan}\NVar{z}$: arctangent function, $x$: real variable and $z$: complex variable Source: Olver (1997b, (2.09–2.10), p. 396); with different notation Referenced by: (9.11.19), (9.8.12), (9.8.16), (9.8.17) Permalink: http://dlmf.nist.gov/9.8.E8 Encodings: TeX, pMML, png See also: Annotations for §9.8(i), §9.8 and Ch.9

Graphs of $M\left(x\right)$ and $N\left(x\right)$ are included in §9.3(i). The branches of $\theta\left(x\right)$ and $\phi\left(x\right)$ are continuous and fixed by $\theta\left(0\right)=-\phi\left(0\right)=\tfrac{1}{6}\pi$. (These definitions of $\theta\left(x\right)$ and $\phi\left(x\right)$ 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 $\displaystyle|x|^{1/2}{M}^{2}\left(x\right)$ $\displaystyle=\tfrac{1}{2}\xi\left({J_{1/3}}^{2}\left(\xi\right)+{Y_{1/3}}^{2}% \left(\xi\right)\right),$ ⓘ 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, $M\left(\NVar{z}\right)$: Airy modulus function, $x$: real variable and $\xi$: change of variable Proof sketch: Combine definition (9.8.3) with (9.6.17) and (9.6.19) using (10.4.3). Referenced by: (9.8.20), (9.8.21), (9.8.22), (9.8.23) Permalink: http://dlmf.nist.gov/9.8.E9 Encodings: TeX, pMML, png See also: Annotations for §9.8(i), §9.8 and Ch.9 9.8.10 $\displaystyle|x|^{-1/2}{N}^{2}\left(x\right)$ $\displaystyle=\tfrac{1}{2}\xi\left({J_{2/3}}^{2}\left(\xi\right)+{Y_{2/3}}^{2}% \left(\xi\right)\right),$ ⓘ 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, $N\left(\NVar{z}\right)$: Airy modulus function, $x$: real variable and $\xi$: change of variable Proof sketch: Combine definition (9.8.7) with (9.6.18) and (9.6.20) using (10.4.3). Permalink: http://dlmf.nist.gov/9.8.E10 Encodings: TeX, pMML, png See also: Annotations for §9.8(i), §9.8 and Ch.9
 9.8.11 $\displaystyle\theta\left(x\right)$ $\displaystyle=\tfrac{2}{3}\pi+\operatorname{arctan}\left(Y_{1/3}\left(\xi% \right)/J_{1/3}\left(\xi\right)\right),$ 9.8.12 $\displaystyle\phi\left(x\right)$ $\displaystyle=\tfrac{1}{3}\pi+\operatorname{arctan}\left(Y_{2/3}\left(\xi% \right)/J_{2/3}\left(\xi\right)\right).$

## §9.8(ii) Identities

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

 9.8.13 $M\left(x\right)N\left(x\right)\sin\left(\theta\left(x\right)-\phi\left(x\right% )\right)=\pi^{-1},$
 9.8.14 $\displaystyle{M}^{2}\left(x\right)\theta'\left(x\right)$ $\displaystyle=-\pi^{-1}$, $\displaystyle{N}^{2}\left(x\right)\phi'\left(x\right)$ $\displaystyle=\pi^{-1}x$, $\displaystyle N\left(x\right)N'\left(x\right)$ $\displaystyle=xM\left(x\right)M'\left(x\right)$, ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $M\left(\NVar{z}\right)$: Airy modulus function, $N\left(\NVar{z}\right)$: Airy modulus function, $\phi\left(\NVar{z}\right)$: Airy phase function, $\theta\left(\NVar{z}\right)$: Airy phase function and $x$: real variable Proof sketch: For the first two equations see Olver (1997b, p. 404). For the third one use (9.8.3), (9.8.7) and (9.2.1). Referenced by: (9.11.19), (9.8.15), (9.8.16) Permalink: http://dlmf.nist.gov/9.8.E14 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §9.8(ii), §9.8 and Ch.9
 9.8.15 $\displaystyle{N}^{2}\left(x\right)$ $\displaystyle={M'}^{2}\left(x\right)+{M}^{2}\left(x\right){\theta'}^{2}\left(x% \right)={M'}^{2}(x)+\pi^{-2}{M}^{-2}\left(x\right),$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $M\left(\NVar{z}\right)$: Airy modulus function, $N\left(\NVar{z}\right)$: Airy modulus function, $\theta\left(\NVar{z}\right)$: Airy phase function and $x$: real variable Proof sketch: Use (9.8.3), (9.8.4), (9.8.7) and (9.8.14). Permalink: http://dlmf.nist.gov/9.8.E15 Encodings: TeX, pMML, png See also: Annotations for §9.8(ii), §9.8 and Ch.9 9.8.16 $\displaystyle x^{2}{M}^{2}\left(x\right)$ $\displaystyle={N'}^{2}\left(x\right)+{N}^{2}\left(x\right){\phi'}^{2}\left(x% \right)={N'}^{2}\left(x\right)+\pi^{-2}x^{2}{N}^{-2}\left(x\right),$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $M\left(\NVar{z}\right)$: Airy modulus function, $N\left(\NVar{z}\right)$: Airy modulus function, $\phi\left(\NVar{z}\right)$: Airy phase function and $x$: real variable Proof sketch: Use (9.8.3), (9.8.7), (9.8.8), (9.8.14) and (9.2.1). Permalink: http://dlmf.nist.gov/9.8.E16 Encodings: TeX, pMML, png See also: Annotations for §9.8(ii), §9.8 and Ch.9
 9.8.17 $\tan\left(\theta\left(x\right)-\phi\left(x\right)\right)=1/(\pi M\left(x\right% )M'\left(x\right))=-M\left(x\right)\theta'\left(x\right)/M'\left(x\right),$ ⓘ
 9.8.18 $\displaystyle M''\left(x\right)$ $\displaystyle=xM\left(x\right)+\pi^{-2}{M}^{-3}\left(x\right)$, $\displaystyle{{M}^{2}}'''\left(x\right)-4x{{M}^{2}}'\left(x\right)-2{M}^{2}% \left(x\right)$ $\displaystyle=0,$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $M\left(\NVar{z}\right)$: Airy modulus function and $x$: real variable Proof sketch: For the first equation see Olver (1997b, p. 404). For the second one use (9.8.3) and (9.2.1). Permalink: http://dlmf.nist.gov/9.8.E18 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §9.8(ii), §9.8 and Ch.9
 9.8.19 ${\theta'}^{2}\left(x\right)+\tfrac{1}{2}(\theta'''\left(x\right)/\theta'\left(% x\right))-\tfrac{3}{4}(\theta''\left(x\right)/\theta'\left(x\right))^{2}=-x.$ ⓘ Symbols: $\theta\left(\NVar{z}\right)$: Airy phase function and $x$: real variable Proof sketch: Combine (9.8.4) with (9.2.1). Permalink: http://dlmf.nist.gov/9.8.E19 Encodings: TeX, pMML, png See also: Annotations for §9.8(ii), §9.8 and Ch.9

## §9.8(iii) Monotonicity

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

## §9.8(iv) Asymptotic Expansions

As $x\rightarrow-\infty$

 9.8.20 $\displaystyle{M}^{2}\left(x\right)$ $\displaystyle\sim\frac{1}{\pi(-x)^{1/2}}\sum_{k=0}^{\infty}\frac{1\cdot 3\cdot 5% \cdots(6k-1)}{k!(96)^{k}}\frac{1}{x^{3k}},$ ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $M\left(\NVar{z}\right)$: Airy modulus function and $x$: real variable Proof sketch: Combine (9.8.9)–(9.8.12) with §10.18(iii) Referenced by: §9.8(iv) Permalink: http://dlmf.nist.gov/9.8.E20 Encodings: TeX, pMML, png See also: Annotations for §9.8(iv), §9.8 and Ch.9 9.8.21 $\displaystyle{N}^{2}\left(x\right)$ $\displaystyle\sim\frac{(-x)^{1/2}}{\pi}\sum_{k=0}^{\infty}\frac{1\cdot 3\cdot 5% \cdots(6k-1)}{k!(96)^{k}}\frac{1+6k}{1-6k}\frac{1}{x^{3k}},$ ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $N\left(\NVar{z}\right)$: Airy modulus function and $x$: real variable Proof sketch: Combine (9.8.9)–(9.8.12) with §10.18(iii) Referenced by: §9.8(iv) Permalink: http://dlmf.nist.gov/9.8.E21 Encodings: TeX, pMML, png See also: Annotations for §9.8(iv), §9.8 and Ch.9 9.8.22 $\displaystyle\theta\left(x\right)$ $\displaystyle\sim\frac{\pi}{4}+\frac{2}{3}(-x)^{3/2}\left(1+\frac{5}{32}\frac{% 1}{x^{3}}+\frac{1105}{6144}\frac{1}{x^{6}}+\frac{82825}{65536}\frac{1}{x^{9}}+% \frac{12820\;31525}{587\;20256}\frac{1}{x^{12}}+\cdots\right),$ ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\theta\left(\NVar{z}\right)$: Airy phase function and $x$: real variable Proof sketch: Combine (9.8.9)–(9.8.12) with §10.18(iii). See also Nemes (2021, Theorem 2.1 and p. 4333). Referenced by: §9.8(iv), §9.8(iv), Erratum (V1.1.8) for Section 9.8(iv) Permalink: http://dlmf.nist.gov/9.8.E22 Encodings: TeX, pMML, png See also: Annotations for §9.8(iv), §9.8 and Ch.9 9.8.23 $\displaystyle\phi\left(x\right)$ $\displaystyle\sim-\frac{\pi}{4}+\frac{2}{3}(-x)^{3/2}\left(1-\frac{7}{32}\frac% {1}{x^{3}}-\frac{1463}{6144}\frac{1}{x^{6}}-\frac{4\;95271}{3\;27680}\frac{1}{% x^{9}}-\frac{2065\;30429}{83\;88608}\frac{1}{x^{12}}-\cdots\right).$ ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\phi\left(\NVar{z}\right)$: Airy phase function and $x$: real variable Proof sketch: Combine (9.8.9)–(9.8.12) with §10.18(iii). See also Nemes (2021, Theorem 2.1 and p. 4333). Referenced by: §9.8(iv), §9.8(iv), §9.8(iv), Erratum (V1.1.8) for Section 9.8(iv) Permalink: http://dlmf.nist.gov/9.8.E23 Encodings: TeX, pMML, png See also: Annotations for §9.8(iv), §9.8 and Ch.9

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), (9.8.22) and (9.8.23), 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).