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31: 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 $𝗃$, $𝗒$, $𝗁$, 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. …

32: 14.21 Definitions and Basic Properties

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

33: 23.1 Special Notation

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$𝕃$	lattice in $ℂ$.
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$G\times H$	Cartesian product of groups $G$ and $H$, that is, the set of all pairs of elements $(g,h)$ with group operation $(g_1,h_1)+(g_2,h_2)=(g_1+g_2,h_1+h_2)$.

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34: 26.14 Permutations: Order Notation

… ►As an example, $35247816$ is an element of ${𝔖}_8.$ The inversion number is the number of pairs of elements for which the larger element precedes the smaller: … ►A descent of a permutation is a pair of adjacent elements for which the first is larger than the second. …

35: 10.63 Recurrence Relations and Derivatives

… ►Let $f_{\nu }(x)$, $g_{\nu }(x)$ denote any one of the ordered pairs: …

36: 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 …

37: 28.33 Physical Applications

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Boundary-values problems arising from solution of the two-dimensional wave equation in elliptical coordinates. This yields a pair of equations of the form (28.2.1) and (28.20.1), and the appropriate solution of (28.2.1) is usually a periodic solution of integer order. See §28.33(ii).

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38: 31.7 Relations to Other Functions

… ►Other reductions of $H\mathrm{\ell }$ to a ${}_2{}^{}F_1^{}$, with at least one free parameter, exist iff the pair $(a,p)$ takes one of a finite number of values, where $q=\alpha \beta p$. …

39: 31.9 Orthogonality

… ►and the integration paths ${ℒ}_1$, ${ℒ}_2$ are Pochhammer double-loop contours encircling distinct pairs of singularities $\{0,1\}$, $\{0,a\}$, $\{1,a\}$. …

40: 3.8 Nonlinear Equations

… ►Let $z^2-sz-t$ be an approximation to the real quadratic factor of $p(z)$ that corresponds to a pair of conjugate complex zeros or to a pair of real zeros. …