numerically satisfactory pairs

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1: 10.25 Definitions
§10.25(iii) NumericallySatisfactoryPairs of Solutions
Table 10.25.1 lists numerically satisfactory pairs of solutions (§2.7(iv)) of (10.25.1). …
2: 9.2 Differential Equation
§9.2(iii) NumericallySatisfactoryPairs of Solutions
Table 9.2.1 lists numerically satisfactory pairs of solutions of (9.2.1) for the stated intervals or regions; compare §2.7(iv).
3: 10.2 Definitions
§10.2(iii) NumericallySatisfactoryPairs of Solutions
Table 10.2.1 lists numerically satisfactory pairs of solutions (§2.7(iv)) of (10.2.1) for the stated intervals or regions in the case $\Re\nu\geq 0$. …
4: 10.45 Functions of Imaginary Order
In consequence of (10.45.5)–(10.45.7), $\widetilde{I}_{\nu}\left(x\right)$ and $\widetilde{K}_{\nu}\left(x\right)$ comprise a numerically satisfactory pair of solutions of (10.45.1) when $x$ is large, and either $\widetilde{I}_{\nu}\left(x\right)$ and $(1/\pi)\sinh\left(\pi\nu\right)\widetilde{K}_{\nu}\left(x\right)$, or $\widetilde{I}_{\nu}\left(x\right)$ and $\widetilde{K}_{\nu}\left(x\right)$, comprise a numerically satisfactory pair when $x$ is small, depending whether $\nu\neq 0$ or $\nu=0$. …
5: 10.24 Functions of Imaginary Order
In consequence of (10.24.6), when $x$ is large $\widetilde{J}_{\nu}\left(x\right)$ and $\widetilde{Y}_{\nu}\left(x\right)$ comprise a numerically satisfactory pair of solutions of (10.24.1); compare §2.7(iv). Also, in consequence of (10.24.7)–(10.24.9), when $x$ is small either $\widetilde{J}_{\nu}\left(x\right)$ and $\tanh\left(\tfrac{1}{2}\pi\nu\right)\widetilde{Y}_{\nu}\left(x\right)$ or $\widetilde{J}_{\nu}\left(x\right)$ and $\widetilde{Y}_{\nu}\left(x\right)$ comprise a numerically satisfactory pair depending whether $\nu\neq 0$ or $\nu=0$. …
6: 10.47 Definitions and Basic Properties
§10.47(iii) NumericallySatisfactoryPairs of Solutions
For (10.47.1) numerically satisfactory pairs of solutions are given by Table 10.2.1 with the symbols $J$, $Y$, $H$, and $\nu$ replaced by $\mathsf{j}$, $\mathsf{y}$, $\mathsf{h}$, and $n$, respectively. For (10.47.2) numerically satisfactory pairs of solutions are ${\mathsf{i}^{(1)}_{n}}\left(z\right)$ and $\mathsf{k}_{n}\left(z\right)$ in the right half of the $z$-plane, and ${\mathsf{i}^{(1)}_{n}}\left(z\right)$ and $\mathsf{k}_{n}\left(-z\right)$ in the left half of the $z$-plane. …
7: 12.2 Differential Equations
For real values of $z$ $(=x)$, numerically satisfactory pairs of solutions (§2.7(iv)) of (12.2.2) are $U\left(a,x\right)$ and $V\left(a,x\right)$ when $x$ is positive, or $U\left(a,-x\right)$ and $V\left(a,-x\right)$ when $x$ is negative. For (12.2.3) $W\left(a,x\right)$ and $W\left(a,-x\right)$ comprise a numerically satisfactory pair, for all $x\in\mathbb{R}$. … In $\mathbb{C}$, for $j=0,1,2,3$, $U\left((-1)^{j-1}a,(-i)^{j-1}z\right)$ and $U\left((-1)^{j}a,(-i)^{j}z\right)$ comprise a numerically satisfactory pair of solutions in the half-plane $\tfrac{1}{4}(2j-3)\pi\leq\operatorname{ph}z\leq\tfrac{1}{4}(2j+1)\pi$. …
8: 14.2 Differential Equations
§14.2(iii) NumericallySatisfactory Solutions
Hence they comprise a numerically satisfactory pair of solutions (§2.7(iv)) of (14.2.2) in the interval $-1. When $\mu-\nu=0,-1,-2,\dots$, or $\mu+\nu=-1,-2,-3,\dots$, $\mathsf{P}^{-\mu}_{\nu}\left(x\right)$ and $\mathsf{P}^{-\mu}_{\nu}\left(-x\right)$ are linearly dependent, and in these cases either may be paired with almost any linearly independent solution to form a numerically satisfactory pair. … Hence they comprise a numerically satisfactory pair of solutions of (14.2.2) in the interval $1. With the same conditions, $P^{-\mu}_{\nu}\left(-x\right)$ and $\boldsymbol{Q}^{\mu}_{\nu}\left(-x\right)$ comprise a numerically satisfactory pair of solutions in the interval $-\infty. …
9: 14.21 Definitions and Basic Properties
When $\Re\nu\geq-\frac{1}{2}$ and $\Re\mu\geq 0$, a numerically satisfactory pair of solutions of (14.21.1) in the half-plane $|\operatorname{ph}z|\leq\frac{1}{2}\pi$ is given by $P^{-\mu}_{\nu}\left(z\right)$ and $\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)$. …
10: 28.8 Asymptotic Expansions for Large $q$
Barrett (1981) supplies asymptotic approximations for numerically satisfactory pairs of solutions of both Mathieu’s equation (28.2.1) and the modified Mathieu equation (28.20.1). … Dunster (1994a) supplies uniform asymptotic approximations for numerically satisfactory pairs of solutions of Mathieu’s equation (28.2.1). …