# §9.8 Modulus and Phase

## §9.8(i) Definitions

Throughout this section $x$ is real and nonpositive.

 9.8.1 $\displaystyle\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)$ $\displaystyle=\mathop{M\/}\nolimits\!\left(x\right)\mathop{\sin\/}\nolimits% \mathop{\theta\/}\nolimits\!\left(x\right),$ 9.8.2 $\displaystyle\mathop{\mathrm{Bi}\/}\nolimits\!\left(x\right)$ $\displaystyle=\mathop{M\/}\nolimits\!\left(x\right)\mathop{\cos\/}\nolimits% \mathop{\theta\/}\nolimits\!\left(x\right),$
 9.8.3 $\displaystyle\mathop{M\/}\nolimits\!\left(x\right)$ $\displaystyle=\sqrt{{\mathop{\mathrm{Ai}\/}\nolimits^{2}}\!\left(x\right)+{% \mathop{\mathrm{Bi}\/}\nolimits^{2}}\!\left(x\right)},$ 9.8.4 $\displaystyle\mathop{\theta\/}\nolimits\!\left(x\right)$ $\displaystyle=\mathop{\mathrm{arctan}\/}\nolimits\!\left(\mathop{\mathrm{Ai}\/% }\nolimits\!\left(x\right)/\mathop{\mathrm{Bi}\/}\nolimits\!\left(x\right)% \right).$
 9.8.5 $\displaystyle\mathop{\mathrm{Ai}\/}\nolimits'\!\left(x\right)$ $\displaystyle=\mathop{N\/}\nolimits\!\left(x\right)\mathop{\sin\/}\nolimits% \mathop{\phi\/}\nolimits\!\left(x\right),$ 9.8.6 $\displaystyle\mathop{\mathrm{Bi}\/}\nolimits'\!\left(x\right)$ $\displaystyle=\mathop{N\/}\nolimits\!\left(x\right)\mathop{\cos\/}\nolimits% \mathop{\phi\/}\nolimits\!\left(x\right),$
 9.8.7 $\displaystyle\mathop{N\/}\nolimits\!\left(x\right)$ $\displaystyle=\sqrt{{\mathop{\mathrm{Ai}\/}\nolimits'^{2}}\!\left(x\right)+{% \mathop{\mathrm{Bi}\/}\nolimits'^{2}}\!\left(x\right)},$ 9.8.8 $\displaystyle\mathop{\phi\/}\nolimits\!\left(x\right)$ $\displaystyle=\mathop{\mathrm{arctan}\/}\nolimits\!\left(\mathop{\mathrm{Ai}\/% }\nolimits'\!\left(x\right)/\mathop{\mathrm{Bi}\/}\nolimits'\!\left(x\right)% \right).$

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 $\displaystyle|x|^{1/2}{\mathop{M\/}\nolimits^{2}}\!\left(x\right)$ $\displaystyle=\tfrac{1}{2}\xi\left({\mathop{J_{1/3}\/}\nolimits^{2}}\!\left(% \xi\right)+{\mathop{Y_{1/3}\/}\nolimits^{2}}\!\left(\xi\right)\right),$ 9.8.10 $\displaystyle|x|^{-1/2}{\mathop{N\/}\nolimits^{2}}\!\left(x\right)$ $\displaystyle=\tfrac{1}{2}\xi\left({\mathop{J_{2/3}\/}\nolimits^{2}}\!\left(% \xi\right)+{\mathop{Y_{2/3}\/}\nolimits^{2}}\!\left(\xi\right)\right),$
 9.8.11 $\displaystyle\mathop{\theta\/}\nolimits\!\left(x\right)$ $\displaystyle=\tfrac{2}{3}\pi+\mathop{\mathrm{arctan}\/}\nolimits\!\left(% \mathop{Y_{1/3}\/}\nolimits\!\left(\xi\right)/\mathop{J_{1/3}\/}\nolimits\!% \left(\xi\right)\right),$ 9.8.12 $\displaystyle\mathop{\phi\/}\nolimits\!\left(x\right)$ $\displaystyle=\tfrac{1}{3}\pi+\mathop{\mathrm{arctan}\/}\nolimits\!\left(% \mathop{Y_{2/3}\/}\nolimits\!\left(\xi\right)/\mathop{J_{2/3}\/}\nolimits\!% \left(\xi\right)\right).$

## §9.8(ii) Identities

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

 9.8.13 $\mathop{M\/}\nolimits\!\left(x\right)\mathop{N\/}\nolimits\!\left(x\right)% \mathop{\sin\/}\nolimits\!\left(\mathop{\theta\/}\nolimits\!\left(x\right)-% \mathop{\phi\/}\nolimits\!\left(x\right)\right)=\pi^{-1},$
 9.8.14 $\displaystyle{\mathop{M\/}\nolimits^{2}}\!\left(x\right)\mathop{\theta\/}% \nolimits'\!\left(x\right)$ $\displaystyle=-\pi^{-1}$, $\displaystyle{\mathop{N\/}\nolimits^{2}}\!\left(x\right)\mathop{\phi\/}% \nolimits'\!\left(x\right)$ $\displaystyle=\pi^{-1}x$, $\displaystyle\mathop{N\/}\nolimits\!\left(x\right)\mathop{N\/}\nolimits'\!% \left(x\right)$ $\displaystyle=x\mathop{M\/}\nolimits\!\left(x\right)\mathop{M\/}\nolimits'\!% \left(x\right)$,
 9.8.15 $\displaystyle{\mathop{N\/}\nolimits^{2}}\!\left(x\right)$ $\displaystyle={\mathop{M\/}\nolimits'^{2}}\!\left(x\right)+{\mathop{M\/}% \nolimits^{2}}\!\left(x\right){\mathop{\theta\/}\nolimits'^{2}}\!\left(x\right% )={\mathop{M\/}\nolimits'^{2}}(x)+\pi^{-2}{\mathop{M\/}\nolimits^{-2}}\!\left(% x\right),$ 9.8.16 $\displaystyle x^{2}{\mathop{M\/}\nolimits^{2}}\!\left(x\right)$ $\displaystyle={\mathop{N\/}\nolimits'^{2}}\!\left(x\right)+{\mathop{N\/}% \nolimits^{2}}\!\left(x\right){\mathop{\phi\/}\nolimits'^{2}}\!\left(x\right)=% {\mathop{N\/}\nolimits'^{2}}\!\left(x\right)+\pi^{-2}x^{2}{\mathop{N\/}% \nolimits^{-2}}\!\left(x\right),$
 9.8.17 $\mathop{\tan\/}\nolimits\!\left(\mathop{\theta\/}\nolimits\!\left(x\right)-% \mathop{\phi\/}\nolimits\!\left(x\right)\right)=1/(\pi\mathop{M\/}\nolimits\!% \left(x\right)\mathop{M\/}\nolimits'\!\left(x\right))=-\mathop{M\/}\nolimits\!% \left(x\right)\mathop{\theta\/}\nolimits'\!\left(x\right)/\mathop{M\/}% \nolimits'\!\left(x\right),$
 9.8.18 $\displaystyle\mathop{M\/}\nolimits''\!\left(x\right)$ $\displaystyle=x\mathop{M\/}\nolimits\!\left(x\right)+\pi^{-2}{\mathop{M\/}% \nolimits^{-3}}\!\left(x\right)$, $\displaystyle{\mathop{M\/}\nolimits^{2}}'''\!\left(x\right)-4x{\mathop{M\/}% \nolimits^{2}}'\!\left(x\right)-2{\mathop{M\/}\nolimits^{2}}\!\left(x\right)$ $\displaystyle=0,$ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{M\/}\nolimits\!\left(\NVar{z}\right)$: Airy modulus function and $x$: real variable Permalink: http://dlmf.nist.gov/9.8.E18 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 9.8(ii)
 9.8.19 ${\mathop{\theta\/}\nolimits'^{2}}\!\left(x\right)+\tfrac{1}{2}(\mathop{\theta% \/}\nolimits'''\!\left(x\right)/\mathop{\theta\/}\nolimits'\!\left(x\right))-% \tfrac{3}{4}(\mathop{\theta\/}\nolimits''\!\left(x\right)/\mathop{\theta\/}% \nolimits'\!\left(x\right))^{2}=-x.$ Symbols: $\mathop{\theta\/}\nolimits\!\left(\NVar{z}\right)$: Airy phase function and $x$: real variable Permalink: http://dlmf.nist.gov/9.8.E19 Encodings: TeX, pMML, png See also: Annotations for 9.8(ii)

## §9.8(iii) Monotonicity

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

As $x\rightarrow-\infty$

 9.8.20 $\displaystyle{\mathop{M\/}\nolimits^{2}}\!\left(x\right)$ $\displaystyle\sim\frac{1}{\pi(-x)^{1/2}}\sum_{k=0}^{\infty}\frac{1\cdot 3\cdot 5% \cdots(6k-1)}{k!(96)^{k}}\frac{1}{x^{3k}},$ 9.8.21 $\displaystyle{\mathop{N\/}\nolimits^{2}}\!\left(x\right)$ $\displaystyle\sim\frac{(-x)^{1/2}}{\pi}\sum_{k=0}^{\infty}\frac{1\cdot 3\cdot 5% \cdots(6k-1)}{k!(96)^{k}}\frac{1+6k}{1-6k}\frac{1}{x^{3k}},$ 9.8.22 $\displaystyle\mathop{\theta\/}\nolimits\!\left(x\right)$ $\displaystyle\sim\frac{\pi}{4}+\frac{2}{3}(-x)^{3/2}\left(1+\frac{5}{32}\frac{% 1}{x^{3}}+\frac{1105}{6144}\frac{1}{x^{6}}+\frac{82825}{65536}\frac{1}{x^{9}}+% \frac{12820\;31525}{587\;20256}\frac{1}{x^{12}}+\cdots\right),$ 9.8.23 $\displaystyle\mathop{\phi\/}\nolimits\!\left(x\right)$ $\displaystyle\sim-\frac{\pi}{4}+\frac{2}{3}(-x)^{3/2}\left(1-\frac{7}{32}\frac% {1}{x^{3}}-\frac{1463}{6144}\frac{1}{x^{6}}-\frac{4\;95271}{3\;27680}\frac{1}{% x^{9}}-\frac{2065\;30429}{83\;88608}\frac{1}{x^{12}}-\cdots\right).$

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).