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31: 4.2 Definitions

… ►This is an analytic function of $z$ on $ℂ\setminus (-\mathrm{\infty },0]$, and is two-valued and discontinuous on the cut shown in Figure 4.2.1, unless $a\in \mathbb{Z}$. …

32: 33.2 Definitions and Basic Properties

… ► $F_{\mathrm{\ell }}(\eta ,\rho )$ is a real and analytic function of $\rho$ on the open interval , and also an analytic function of $\eta$ when . …

33: Mathematical Introduction

… ►This is because $𝐅$ is akin to the notation used for Bessel functions (§10.2(ii)), inasmuch as $𝐅$ is an entire function of each of its parameters $a$, $b$, and $c$:​ this results in fewer restrictions and simpler equations. … ► ►►►►

$ℂ$	complex plane (excluding infinity).
…
$(a,b)$	open interval in $ℝ$, or open straight-line segment joining $a$ and $b$ in $ℂ$.
$[a,b]$	closed interval in $ℝ$, or closed straight-line segment joining $a$ and $b$ in $ℂ$.

► ►►

$(a,b]$ or $[a,b)$	half-closed intervals.
…

… ►This means that the variable $x$ ranges from 0 to 1 in intervals of 0. …

34: 2.6 Distributional Methods

… ►Let $f(t)$ be locally integrable on $[0,\mathrm{\infty })$. …Since $f(t)$ is locally integrable on $[0,\mathrm{\infty })$, it defines a distribution by … ►if $\alpha \in (0,1)$ in (2.6.9), or … ►We again assume $f(t)$ is locally integrable on $[0,\mathrm{\infty })$ and satisfies (2.6.9). … ►of two locally integrable functions on $[0,\mathrm{\infty })$, (2.6.33) can be written …

35: 3.8 Nonlinear Equations

… ►If with , then the interval $[a,b]$ contains one or more zeros of $f$. …All zeros of $f$ in the original interval $[a,b]$ can be computed to any predetermined accuracy. … ►The convergence is linear, and again more than one zero may occur in the original interval $[x_0,x_1]$. … ►There is no guaranteed convergence: the first approximation $x_2$ may be outside $[x_0,x_1]$. … ►Suppose $f(z)$ also depends on a parameter $\alpha$, denoted by $f(z,\alpha )$. …

36: 18.15 Asymptotic Approximations

… ►For an asymptotic expansion of $P_n^{(\alpha ,\beta )}\left(z\right)$ as $n\to \mathrm{\infty }$ that holds uniformly for complex $z$ bounded away from $[-1,1]$, see Elliott (1971). …

37: 13.8 Asymptotic Approximations for Large Parameters

… ►as $b\to \mathrm{\infty }$, uniformly in compact $\lambda$-intervals of $(0,\mathrm{\infty })$ and compact real $a$-intervals. For the parabolic cylinder function $U$ see §12.2, and for an extension to an asymptotic expansion see Temme (1978). … ►(13.8.8) holds uniformly with respect to $x\in [0,\mathrm{\infty })$. … ►For an extension to an asymptotic expansion complete with error bounds see Temme (1990b), and for related results see §13.21(i). … ►where $C_{\nu }(a,\zeta )=\cos \left(\pi a\right)J_{\nu }\left(\zeta \right)+\sin \left(\pi a\right)Y_{\nu }\left(\zeta \right)$ and …

38: 1.14 Integral Transforms

… ►If $x^{\sigma -1}f(x)$ is integrable on $(0,\mathrm{\infty })$ for all $\sigma$ in , then the integral (1.14.32) converges and $ℳf\left(s\right)$ is an analytic function of $s$ in the vertical strip . …

39: 13.31 Approximations

… ►Luke (1969b, pp. 35 and 25) provides Chebyshev-series expansions of $M(a,b,x)$ and $U(a,b,x)$ that include the intervals $0\le x\le \alpha$ and , respectively, where $\alpha$ is an arbitrary positive constant. …

40: 29.9 Stability

… ►If $\nu$ is not an integer, then (29.2.1) is unstable iff $h\le a_{\nu }^0\left(k^2\right)$ or $h$ lies in one of the closed intervals with endpoints $a_{\nu }^m\left(k^2\right)$ and $b_{\nu }^m\left(k^2\right)$, $m=1,2,\mathrm{\dots }$. …