§12.2 Differential Equations

PCFs are solutions of the differential equation

with three distinct standard forms

Each of these equations is transformable into the others. Standard solutions are $U(a,\pm z)$, $V(a,\pm z)$, $\overline{U}(a,\pm x)$ (not complex conjugate), $U(-a,\pm \mathrm{i}z)$ for (12.2.2); $W(a,\pm x)$ for (12.2.3); $D_{\nu }\left(\pm z\right)$ for (12.2.4), where

All solutions are entire functions of $z$ and entire functions of $a$ or $\nu$.

For real values of $z$ $(=x)$, numerically satisfactory pairs of solutions (§2.7(iv)) of (12.2.2) are $U(a,x)$ and $V(a,x)$ when $x$ is positive, or $U(a,-x)$ and $V(a,-x)$ when $x$ is negative. For (12.2.3) $W(a,x)$ and $W(a,-x)$ comprise a numerically satisfactory pair, for all $x\in ℝ$. The solutions $W(a,\pm x)$ are treated in §12.14.

In $ℂ$, for $j=0,1,2,3$, $U({(-1)}^{j-1}a,{(-\mathrm{i})}^{j-1}z)$ and $U({(-1)}^{j}a,{(-\mathrm{i})}^{j}z)$ comprise a numerically satisfactory pair of solutions in the half-plane $\frac{1}{4}(2j-3)\pi \le \mathrm{ph}z\le \frac{1}{4}(2j+1)\pi$.

For $n=0,1,\mathrm{\dots }$,

When $z$ $(=x)$ is real the solution $\overline{U}(a,x)$ is defined by

unless $a=\frac{1}{2},\frac{3}{2},\mathrm{\dots }$, in which case $\overline{U}(a,x)$ is undefined. Its importance is that when $a$ is negative and $|a|$ is large, $U(a,x)$ and $\overline{U}(a,x)$ asymptotically have the same envelope (modulus) and are $\frac{1}{2}\pi$ out of phase in the oscillatory interval . Properties of $\overline{U}(a,x)$ follow immediately from those of $V(a,x)$ via (12.2.21).

In the oscillatory interval we define

where $F(a,x)$ ($>$0), $\theta (a,x)$, $G(a,x)$ ($>$0), and $\psi (a,x)$ are real. $F$ or $G$ is the modulus and $\theta$ or $\psi$ is the corresponding phase.

For properties of the modulus and phase functions, including differential equations, see Miller (1955, pp. 72–73). For graphs of the modulus functions see §12.3(i).