# numerically satisfactory solutions

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##### 1: 14.21 Definitions and Basic Properties

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###### §14.21(ii) Numerically Satisfactory Solutions

►When $\mathrm{\Re}\nu \ge -\frac{1}{2}$ and $\mathrm{\Re}\mu \ge 0$, a numerically satisfactory pair of solutions of (14.21.1) in the half-plane $|\mathrm{ph}z|\le \frac{1}{2}\pi $ is given by ${P}_{\nu}^{-\mu}\left(z\right)$ and ${\mathit{Q}}_{\nu}^{\mu}\left(z\right)$. …##### 2: 11.2 Definitions

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###### §11.2(iii) Numerically Satisfactory Solutions

… ►When $z=x$, $$, and $\mathrm{\Re}\nu \ge 0$, numerically satisfactory general solutions of (11.2.7) are given by … ►When $z\in \u2102$ and $\mathrm{\Re}\nu \ge 0$, numerically satisfactory general solutions of (11.2.7) are given by … ►When $\mathrm{\Re}\nu \ge 0$, numerically satisfactory general solutions of (11.2.9) are given by …(11.2.17) applies when $|\mathrm{ph}z|\le \frac{1}{2}\pi $ with $z$ bounded away from the origin.##### 3: 10.25 Definitions

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###### §10.25(iii) Numerically Satisfactory Pairs of Solutions

►Table 10.25.1 lists numerically satisfactory pairs of solutions (§2.7(iv)) of (10.25.1). … ►##### 4: 2.7 Differential Equations

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###### §2.7(iv) Numerically Satisfactory Solutions

… ►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. … ►In consequence, if a differential equation has more than one singularity in the extended plane, then usually more than two standard solutions need to be chosen in order to have numerically satisfactory representations everywhere. ►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.##### 5: 9.2 Differential Equation

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###### §9.2(iii) Numerically Satisfactory Pairs 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). ► …##### 6: 10.2 Definitions

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###### §10.2(iii) Numerically Satisfactory Pairs 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 $\mathrm{\Re}\nu \ge 0$. … ►##### 7: 14.2 Differential Equations

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###### §14.2(iii) Numerically Satisfactory Solutions

… ►Hence they comprise a numerically satisfactory pair of solutions (§2.7(iv)) of (14.2.2) in the interval $$. When $\mu -\nu =0,-1,-2,\mathrm{\dots}$, or $\mu +\nu =-1,-2,-3,\mathrm{\dots}$, ${\mathsf{P}}_{\nu}^{-\mu}\left(x\right)$ and ${\mathsf{P}}_{\nu}^{-\mu}\left(-x\right)$ are linearly dependent, and in these cases either may be paired with almost any linearly independent solution to form a numerically satisfactory pair. … ►Hence they comprise a numerically satisfactory pair of solutions of (14.2.2) in the interval $$. With the same conditions, ${P}_{\nu}^{-\mu}\left(-x\right)$ and ${\mathit{Q}}_{\nu}^{\mu}\left(-x\right)$ comprise a numerically satisfactory pair of solutions in the interval $$. …##### 8: 13.2 Definitions and Basic Properties

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###### §13.2(v) Numerically Satisfactory Solutions

►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 … ► …##### 9: 10.47 Definitions and Basic Properties

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###### §10.47(iii) Numerically Satisfactory Pairs 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 ${\mathsf{i}}_{n}^{(1)}\left(z\right)$ and ${\mathsf{k}}_{n}\left(z\right)$ in the right half of the $z$-plane, and ${\mathsf{i}}_{n}^{(1)}\left(z\right)$ and ${\mathsf{k}}_{n}\left(-z\right)$ in the left half of the $z$-plane. …##### 10: 13.14 Definitions and Basic Properties

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