# numerically satisfactory solutions

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##### 1: 14.21 Definitions and Basic Properties
###### §14.21(ii) NumericallySatisfactorySolutions
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 $\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)$. …
##### 2: 11.2 Definitions
###### §11.2(iii) NumericallySatisfactorySolutions
When $z=x$, $0, and $\Re\nu\geq 0$, numerically satisfactory general solutions of (11.2.7) are given by … When $z\in\mathbb{C}$ and $\Re\nu\geq 0$, numerically satisfactory general solutions of (11.2.7) are given by … When $\Re\nu\geq 0$, numerically satisfactory general solutions of (11.2.9) are given by …(11.2.17) applies when $|\operatorname{ph}z|\leq\tfrac{1}{2}\pi$ with $z$ bounded away from the origin.
##### 3: 10.25 Definitions
###### §10.25(iii) NumericallySatisfactory Pairs of Solutions
Table 10.25.1 lists numerically satisfactory pairs of solutions2.7(iv)) of (10.25.1). …
##### 4: 2.7 Differential Equations
###### §2.7(iv) NumericallySatisfactorySolutions
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 $\tfrac{1}{2}\pi$ out of phase.
##### 5: 9.2 Differential Equation
###### §9.2(iii) NumericallySatisfactory 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
###### §10.2(iii) NumericallySatisfactory Pairs of Solutions
Table 10.2.1 lists numerically satisfactory pairs of solutions2.7(iv)) of (10.2.1) for the stated intervals or regions in the case $\Re\nu\geq 0$. …
##### 7: 14.2 Differential Equations
###### §14.2(iii) NumericallySatisfactorySolutions
Hence they comprise a numerically satisfactory pair of solutions2.7(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. …
##### 8: 10.47 Definitions and Basic Properties
###### §10.47(iii) NumericallySatisfactory 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 ${\mathsf{i}^{(1)}_{n}}\left(z\right)$ 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. …
##### 9: 12.2 Differential Equations
All solutions are entire functions of $z$ and entire functions of $a$ or $\nu$. For real values of $z$ $(=x)$, numerically satisfactory pairs of solutions2.7(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. … 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 $\tfrac{1}{4}(2j-3)\pi\leq\operatorname{ph}z\leq\tfrac{1}{4}(2j+1)\pi$. …
##### 10: 9.12 Scorer Functions
###### §9.12(iv) NumericallySatisfactorySolutions
In $\mathbb{C}$, numerically satisfactory sets of solutions are given by …