numerically satisfactory pairs

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11: 2.9 Difference Equations

… ►As in the case of differential equations (§§2.7(iii), 2.7(iv)) recessive solutions are unique and dominant solutions are not; furthermore, one member of a numerically satisfactory pair has to be recessive. …

12: 2.7 Differential Equations

… ►This kind of cancellation cannot take place with $w_1(z)$ and $w_2(z)$, and for this reason, and following Miller (1950), we call $w_1(z)$ and $w_2(z)$ a numerically satisfactory pair of solutions. … ►This is characteristic of numerically satisfactory pairs. … ►In oscillatory intervals, and again following Miller (1950), we call a pair of solutions numerically satisfactory if asymptotically they have the same amplitude and are $\frac{1}{2}\pi$ out of phase.

13: 12.14 The Function $W(a,x)$

… ► $W(a,x)$ and $W(a,-x)$ form a numerically satisfactory pair of solutions when . …

14: 9.13 Generalized Airy Functions

… ►The function on the right-hand side is recessive in the sector $-(2j-1)\pi /m\le \mathrm{ph}z\le (2j+1)\pi /m$, and is therefore an essential member of any numerically satisfactory pair of solutions in this region. …

15: 13.2 Definitions and Basic Properties

… ►Fundamental pairs of solutions of (13.2.1) that are numerically satisfactory (§2.7(iv)) in the neighborhood of infinity are … ►A fundamental pair of solutions that is numerically satisfactory near the origin is … ►When $b=n+1=1,2,3,\mathrm{\dots }$, a fundamental pair that is numerically satisfactory near the origin is $M(a,n+1,z)$ and … ►When $b=-n=0,-1,-2,\mathrm{\dots }$, a fundamental pair that is numerically satisfactory near the origin is $z^{n+1}M(a+n+1,n+2,z)$ and …

16: 14.20 Conical (or Mehler) Functions

… ►For and $\tau >0$, a numerically satisfactory pair of real conical functions is ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$ and ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(-x\right)$. …

17: 13.14 Definitions and Basic Properties

… ►Fundamental pairs of solutions of (13.14.1) that are numerically satisfactory (§2.7(iv)) in the neighborhood of infinity are … ►A fundamental pair of solutions that is numerically satisfactory in the sector $|\mathrm{ph}z|\le \pi$ near the origin is …

18: 15.10 Hypergeometric Differential Equation

… ►It has regular singularities at $z=0,1,\mathrm{\infty }$, with corresponding exponent pairs $\{0,1-c\}$, $\{0,c-a-b\}$, $\{a,b\}$, respectively. When none of the exponent pairs differ by an integer, that is, when none of $c$, $c-a-b$, $a-b$ is an integer, we have the following pairs $f_1(z)$, $f_2(z)$ of fundamental solutions. They are also numerically satisfactory (§2.7(iv)) in the neighborhood of the corresponding singularity. … ►The three pairs of fundamental solutions given by (15.10.2), (15.10.4), and (15.10.6) can be transformed into 18 other solutions by means of (15.8.1), leading to a total of 24 solutions known as Kummer’s solutions. …