§9.8 Modulus and Phase

Throughout this section $x$ is real and nonpositive.

9.8.1		$\mathrm{Ai}\left(x\right)$	$=M\left(x\right)\sin \theta \left(x\right),$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $M\left(z\right)$: Airy modulus function, $\theta \left(z\right)$: Airy phase function, $\sin z$: 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		$\mathrm{Bi}\left(x\right)$	$=M\left(x\right)\cos \theta \left(x\right),$
ⓘ Symbols: $\mathrm{Bi}\left(z\right)$: Airy function, $\cos z$: cosine function, $M\left(z\right)$: Airy modulus function, $\theta \left(z\right)$: 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		$M\left(x\right)$	$=\sqrt{{\mathrm{Ai}}^2\left(x\right)+{\mathrm{Bi}}^2\left(x\right)},$
ⓘ Defines: $M\left(z\right)$: Airy modulus function Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: 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		$\theta \left(x\right)$	$=\arctan \left(\mathrm{Ai}\left(x\right)/\mathrm{Bi}\left(x\right)\right).$
ⓘ Defines: $\theta \left(z\right)$: Airy phase function Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\arctan z$: 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		${\mathrm{Ai}}^{\prime }\left(x\right)$	$=N\left(x\right)\sin \varphi \left(x\right),$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $N\left(z\right)$: Airy modulus function, $\varphi \left(z\right)$: Airy phase function, $\sin z$: 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		${\mathrm{Bi}}^{\prime }\left(x\right)$	$=N\left(x\right)\cos \varphi \left(x\right),$
ⓘ Symbols: $\mathrm{Bi}\left(z\right)$: Airy function, $\cos z$: cosine function, $N\left(z\right)$: Airy modulus function, $\varphi \left(z\right)$: 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		$N\left(x\right)$	$=\sqrt{\mathrm{Ai}^{\prime }{}^2\left(x\right)+\mathrm{Bi}^{\prime }{}^2\left(x\right)},$
ⓘ Defines: $N\left(z\right)$: Airy modulus function Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: 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		$\varphi \left(x\right)$	$=\arctan \left({\mathrm{Ai}}^{\prime }\left(x\right)/{\mathrm{Bi}}^{\prime }\left(x\right)\right).$
ⓘ Defines: $\varphi \left(z\right)$: Airy phase function Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\arctan z$: 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 $\varphi \left(x\right)$ are continuous and fixed by $\theta \left(0\right)=-\varphi \left(0\right)=\frac{1}{6}\pi$. (These definitions of $\theta \left(x\right)$ and $\varphi \left(x\right)$ 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 $\xi =\frac{2}{3}{|x|}^{3/2}$,

9.8.9		${|x|}^{1/2}{M}^2\left(x\right)$	$=\frac{1}{2}\xi \left(J_{1/3}^2\left(\xi \right)+Y_{1/3}^2\left(\xi \right)\right),$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $M\left(z\right)$: Airy modulus function, $x$: real variable and $\xi$: change of variable Source: 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		${|x|}^{-1/2}{N}^2\left(x\right)$	$=\frac{1}{2}\xi \left(J_{2/3}^2\left(\xi \right)+Y_{2/3}^2\left(\xi \right)\right),$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $N\left(z\right)$: Airy modulus function, $x$: real variable and $\xi$: change of variable Source: 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		$\theta \left(x\right)$	$=\frac{2}{3}\pi +\arctan \left(Y_{1/3}\left(\xi \right)/J_{1/3}\left(\xi \right)\right),$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\arctan z$: arctangent function, $\theta \left(z\right)$: Airy phase function, $x$: real variable and $\xi$: change of variable Source: Combine definition (9.8.4) with (9.6.17) and (9.6.19) using (10.4.3). Permalink: http://dlmf.nist.gov/9.8.E11 Encodings: TeX, pMML, png See also: Annotations for §9.8(i), §9.8 and Ch.9
9.8.12		$\varphi \left(x\right)$	$=\frac{1}{3}\pi +\arctan \left(Y_{2/3}\left(\xi \right)/J_{2/3}\left(\xi \right)\right).$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\arctan z$: arctangent function, $\varphi \left(z\right)$: Airy phase function, $x$: real variable and $\xi$: change of variable Source: Combine definition (9.8.8) with (9.6.18) and (9.6.20) 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.E12 Encodings: TeX, pMML, png See also: Annotations for §9.8(i), §9.8 and Ch.9

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

9.8.13		$$M\left(x\right)N\left(x\right)\sin \left(\theta \left(x\right)-\varphi \left(x\right)\right)={\pi }^{-1},$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $M\left(z\right)$: Airy modulus function, $N\left(z\right)$: Airy modulus function, $\varphi \left(z\right)$: Airy phase function, $\theta \left(z\right)$: Airy phase function, $\sin z$: sine function and $x$: real variable Source: First use (4.21.2) then (9.8.1), (9.8.2), (9.8.5), (9.8.6), and finally (9.2.7). Permalink: http://dlmf.nist.gov/9.8.E13 Encodings: TeX, pMML, png See also: Annotations for §9.8(ii), §9.8 and Ch.9

9.8.14		$M^2\left(x\right){\theta }^{\prime }\left(x\right)$	$=-{\pi }^{-1}$,
		$N^2\left(x\right){\varphi }^{\prime }\left(x\right)$	$={\pi }^{-1}x$,
		$N\left(x\right)N^{\prime }\left(x\right)$	$=xM\left(x\right)M^{\prime }\left(x\right)$,
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $M\left(z\right)$: Airy modulus function, $N\left(z\right)$: Airy modulus function, $\varphi \left(z\right)$: Airy phase function, $\theta \left(z\right)$: Airy phase function and $x$: real variable Source: 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		$N^2\left(x\right)$	$=M^{\prime }{}^2\left(x\right)+M^2\left(x\right)\theta ^{\prime }{}^2\left(x\right)=M^{\prime }{}^2(x)+{\pi }^{-2}{M}^{-2}\left(x\right),$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $M\left(z\right)$: Airy modulus function, $N\left(z\right)$: Airy modulus function, $\theta \left(z\right)$: Airy phase function and $x$: real variable Source: 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		$x^2{M}^2\left(x\right)$	$=N^{\prime }{}^2\left(x\right)+N^2\left(x\right)\varphi ^{\prime }{}^2\left(x\right)=N^{\prime }{}^2\left(x\right)+{\pi }^{-2}{x}^2{N}^{-2}\left(x\right),$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $M\left(z\right)$: Airy modulus function, $N\left(z\right)$: Airy modulus function, $\varphi \left(z\right)$: Airy phase function and $x$: real variable Source: 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		$$\tan \left(\theta \left(x\right)-\varphi \left(x\right)\right)=1/(\pi M\left(x\right)M^{\prime }\left(x\right))=-M\left(x\right){\theta }^{\prime }\left(x\right)/M^{\prime }\left(x\right),$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $M\left(z\right)$: Airy modulus function, $\varphi \left(z\right)$: Airy phase function, $\theta \left(z\right)$: Airy phase function, $\tan z$: tangent function and $x$: real variable Source: Combine (4.21.4) with (9.8.3), (9.8.4), (9.8.8) and (9.2.7). Permalink: http://dlmf.nist.gov/9.8.E17 Encodings: TeX, pMML, png See also: Annotations for §9.8(ii), §9.8 and Ch.9

9.8.18		$M^{\prime \prime }\left(x\right)$	$=xM\left(x\right)+{\pi }^{-2}{M}^{-3}\left(x\right)$,
		$M^2{}^{\prime \prime \prime }\left(x\right)-4xM^2{}^{\prime }\left(x\right)-2M^2\left(x\right)$	$=0,$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $M\left(z\right)$: Airy modulus function and $x$: real variable Source: 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		$$\theta ^{\prime }{}^2\left(x\right)+\frac{1}{2}({\theta }^{\prime \prime \prime }\left(x\right)/{\theta }^{\prime }\left(x\right))-\frac{3}{4}{({\theta }^{\prime \prime }\left(x\right)/{\theta }^{\prime }\left(x\right))}^2=-x.$$
ⓘ Symbols: $\theta \left(z\right)$: Airy phase function and $x$: real variable Source: 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

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

As $x\to -\mathrm{\infty }$

9.8.20		$M^2\left(x\right)$	$\sim {\displaystyle \frac{1}{\pi {(-x)}^{1/2}}}{\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{\displaystyle \frac{1\cdot 3\cdot 5\mathrm{\cdots }(6k-1)}{k!{(96)}^k}}{\displaystyle \frac{1}{x^{3k}}},$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $M\left(z\right)$: Airy modulus function and $x$: real variable Source: 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		$N^2\left(x\right)$	$\sim {\displaystyle \frac{{(-x)}^{1/2}}{\pi }}{\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{\displaystyle \frac{1\cdot 3\cdot 5\mathrm{\cdots }(6k-1)}{k!{(96)}^k}}{\displaystyle \frac{1+6k}{1-6k}}{\displaystyle \frac{1}{x^{3k}}},$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $N\left(z\right)$: Airy modulus function and $x$: real variable Source: 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		$\theta \left(x\right)$	$\sim {\displaystyle \frac{\pi }{4}}+{\displaystyle \frac{2}{3}}{(-x)}^{3/2}\left(1+{\displaystyle \frac{5}{32}}{\displaystyle \frac{1}{x^3}}+{\displaystyle \frac{1105}{6144}}{\displaystyle \frac{1}{x^6}}+{\displaystyle \frac{82825}{65536}}{\displaystyle \frac{1}{x^9}}+{\displaystyle \frac{\mathrm{12820\hspace{0.33em}31525}}{\mathrm{587\hspace{0.33em}20256}}}{\displaystyle \frac{1}{x^{12}}}+\mathrm{\cdots }\right),$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\theta \left(z\right)$: Airy phase function and $x$: real variable Source: Combine (9.8.9)–(9.8.12) with §10.18(iii) Referenced by: §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		$\varphi \left(x\right)$	$\sim -{\displaystyle \frac{\pi }{4}}+{\displaystyle \frac{2}{3}}{(-x)}^{3/2}\left(1-{\displaystyle \frac{7}{32}}{\displaystyle \frac{1}{x^3}}-{\displaystyle \frac{1463}{6144}}{\displaystyle \frac{1}{x^6}}-{\displaystyle \frac{\mathrm{4\hspace{0.33em}95271}}{\mathrm{3\hspace{0.33em}27680}}}{\displaystyle \frac{1}{x^9}}-{\displaystyle \frac{\mathrm{2065\hspace{0.33em}30429}}{\mathrm{83\hspace{0.33em}88608}}}{\displaystyle \frac{1}{x^{12}}}-\mathrm{\cdots }\right).$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\varphi \left(z\right)$: Airy phase function and $x$: real variable Source: Combine (9.8.9)–(9.8.12) with §10.18(iii) Referenced by: §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

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 $n\ge 0$ for (9.8.20) and $n\ge 1$ 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 $x^{-15}$, $x^{-18}$, $\mathrm{\dots }$, $x^{-56}$ are available in Sherry (1959).