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21: 35.7 Gaussian Hypergeometric Function of Matrix Argument

… ►Let $f:𝛀\to ℂ$ (a) be orthogonally invariant, so that $f(𝐓)$ is a symmetric function of $t_1,\mathrm{\dots },t_m$, the eigenvalues of the matrix argument $𝐓\in 𝛀$; (b) be analytic in $t_1,\mathrm{\dots },t_m$ in a neighborhood of $𝐓=𝟎$; (c) satisfy $f(𝟎)=1$. …

22: 2.10 Sums and Sequences

… ►in the neighborhood of each singularity $z_j$, again with ${\sigma }_j>0$. …

23: 2.11 Remainder Terms; Stokes Phenomenon

… ►In effect, (2.11.7) “corrects” (2.11.6) by introducing a term that is relatively exponentially small in the neighborhood of $\mathrm{ph}z=\pi$, is increasingly significant as $\mathrm{ph}z$ passes from $\pi$ to $\frac{3}{2}\pi$, and becomes the dominant contribution after $\mathrm{ph}z$ passes $\frac{3}{2}\pi$. …

24: 10.75 Tables

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Olver (1962) provides tables for the uniform asymptotic expansions given in §10.20(i), including $\zeta$ and ${(4\zeta /(1-x^2))}^{\frac{1}{4}}$ as functions of $x$ $(=z)$ and the coefficients $A_k(\zeta )$, $B_k(\zeta )$, $C_k(\zeta )$, $D_k(\zeta )$ as functions of $\zeta$. These enable $J_{\nu }\left(\nu x\right)$, $Y_{\nu }\left(\nu x\right)$, $J_{\nu }^{\prime }\left(\nu x\right)$, $Y_{\nu }^{\prime }\left(\nu x\right)$ to be computed to 10S when $\nu \ge 15$, except in the neighborhoods of zeros.

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25: 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 …

26: 13.14 Definitions and Basic Properties

… ►Fundamental pairs of solutions of (13.14.1) that are numerically satisfactory (§2.7(iv)) in the neighborhood of infinity are …

27: 18.15 Asymptotic Approximations

… ►The latter expansions are in terms of Bessel functions, and are uniform in complex $z$-domains not containing neighborhoods of 1. …

28: 3.5 Quadrature

… ►To avoid cancellation we try to deform the path to pass through a saddle point in such a way that the maximum contribution of the integrand is derived from the neighborhood of the saddle point. …

29: Errata

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A sentence was added just below (1.4.15) indicating that we assume that $g^{\prime }(x)\ne 0$ for all $x$ in some neighborhood of $a$ with $x\ne a$. Suggested by Svante Janson on 2023-08-21

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