§9.8 Modulus and Phase

Throughout this section $x$ is real and nonpositive.

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 $\xi =\frac{2}{3}{|x|}^{3/2}$,

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

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

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