numerically satisfactory solutions

SearchAdvancedHelp

(0.001 seconds)

1—10 of 23 matching pages

1: 14.21 Definitions and Basic Properties

… ►

§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 ${𝑸}_{\nu }^{\mu }\left(z\right)$. …

2: 11.2 Definitions

… ►

§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 $z\in ℂ$ 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

… ►

§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). … ►

Table 10.25.1: Numerically satisfactory pairs of solutions of the modified Bessel’s equation.

► ►►

Pair	Region
…

►

4: 2.7 Differential Equations

… ►

§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

… ►

§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). ►

Table 9.2.1: Numerically satisfactory pairs of solutions of Airy’s equation.

► ►►

Pair	Interval or Region
…

►

…

6: 10.2 Definitions

… ►

§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 $\mathrm{\Re }\nu \ge 0$. … ►

Table 10.2.1: Numerically satisfactory pairs of solutions of Bessel’s equation.

► ►►

Pair	Interval or Region
…

►

7: 14.2 Differential Equations

… ►

§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 $\mu +\nu =-1,-2,-3,\mathrm{\dots }$, ${𝖯}_{\nu }^{-\mu }\left(x\right)$ and ${𝖯}_{\nu }^{-\mu }\left(-x\right)$ 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 ${𝑸}_{\nu }^{\mu }\left(-x\right)$ comprise a numerically satisfactory pair of solutions in the interval . …

8: 10.47 Definitions and Basic Properties

… ►

§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 $𝗃$, $𝗒$, $𝗁$, and $n$, respectively. ►For (10.47.2) numerically satisfactory pairs of solutions are ${𝗂}_n^{(1)}\left(z\right)$ and ${𝗄}_n\left(z\right)$ in the right half of the $z$-plane, and ${𝗂}_n^{(1)}\left(z\right)$ and ${𝗄}_n\left(-z\right)$ 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 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. … ►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 $\frac{1}{4}(2j-3)\pi \le \mathrm{ph}z\le \frac{1}{4}(2j+1)\pi$. …

10: 9.12 Scorer Functions

… ►

§9.12(iv) Numerically Satisfactory Solutions

… ►In $ℂ$, numerically satisfactory sets of solutions are given by …