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21: 19.2 Definitions

… ►Let $s^2(t)$ be a cubic or quartic polynomial in $t$ with simple zeros, and let $r(s,t)$ be a rational function of $s$ and $t$ containing at least one odd power of $s$. … ►Assume $1-{\sin }^2\varphi \in ℂ\setminus (-\mathrm{\infty },0]$ and $1-k^2{\sin }^2\varphi \in ℂ\setminus (-\mathrm{\infty },0]$, except that one of them may be 0, and $1-{\alpha }^2{\sin }^2\varphi \in ℂ\setminus \{0\}$. … ►The principal branch of $K\left(k\right)$ and $E\left(k\right)$ is $|\mathrm{ph}\left(1-k^2\right)|\le \pi$, that is, the branch-cuts are $(-\mathrm{\infty },-1]\cup [1,+\mathrm{\infty })$. … ►If the line segment with endpoints $x$ and $y$ lies in $ℂ\setminus (-\mathrm{\infty },0]$, then …

22: 10.74 Methods of Computation

… ►It should be noted, however, that there is a difficulty in evaluating the coefficients $A_k(\zeta )$, $B_k(\zeta )$, $C_k(\zeta )$, and $D_k(\zeta )$, from the explicit expressions (10.20.10)–(10.20.13) when $z$ is close to $1$ owing to severe cancellation. … ►If values of the Bessel functions $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$, or the other functions treated in this chapter, are needed for integer-spaced ranges of values of the order $\nu$, then a simple and powerful procedure is provided by recurrence relations typified by the first of (10.6.1). …

23: 19.9 Inequalities

… ►Simple inequalities for incomplete integrals follow directly from the defining integrals (§19.2(ii)) together with (19.6.12): … ►(19.9.15) is useful when $k^2$ and ${\sin }^2\varphi$ are both close to $1$, since the bounds are then nearly equal; otherwise (19.9.14) is preferable. …

24: 2.4 Contour Integrals

… ►Then (2.4.1) is valid in any closed sector with vertex $z=0$ and properly interior to . … ►is seen to converge absolutely at each limit, and be independent of $\sigma \in [c,\mathrm{\infty })$. … ►

Excluding $t=a$, $\mathrm{\Re }\left({\mathrm{e}}^{\mathrm{i}\theta }p(t)-{\mathrm{e}}^{\mathrm{i}\theta }p(a)\right)$ is positive when $t\in 𝒫$, and is bounded away from zero uniformly with respect to $\theta \in [{\theta }_1,{\theta }_2]$ as $t\to b$ along $𝒫$.

… ►In the commonest case the interior minimum $t_0$ of $\mathrm{\Re }\left(zp(t)\right)$ is a simple zero of $p^{\prime }(t)$. … ►Suppose that on the integration path $𝒫$ there are two simple zeros of $\partial p(\alpha ,t)/\partial t$ that coincide for a certain value $\widehat{\alpha }$ of $\alpha$. …

25: 10.21 Zeros

… ►The zeros of any cylinder function or its derivative are simple, with the possible exceptions of $z=0$ in the case of the functions, and $z=0,\pm \nu$ in the case of the derivatives. … ►All of these zeros are simple, provided that $\nu \ge -1$ in the case of $J_{\nu }^{\prime }\left(z\right)$, and $\nu \ge -\frac{1}{2}$ in the case of $Y_{\nu }^{\prime }\left(z\right)$. … ►Lastly, there are two conjugate sets, with $n$ zeros in each set, that are asymptotically close to the boundary of $𝐊$ as $n\to \mathrm{\infty }$. … ►The only other set comprises $n$ zeros that are asymptotically close to the lower boundary of $𝐊$ as $n\to \mathrm{\infty }$. … ►are simple and the asymptotic expansion of the $m$th positive zero as $m\to \mathrm{\infty }$ is given by …