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11: 18.32 OP’s with Respect to Freud Weights

… ►where $Q(x)$ is real, even, nonnegative, and continuously differentiable, where $x{Q}^{\prime }(x)$ increases for $x>0$, and $Q^{\prime }(x)\to \mathrm{\infty }$ as $x\to \mathrm{\infty }$, see Freud (1969). …

12: 24.19 Methods of Computation

… ►We list here three methods, arranged in increasing order of efficiency. …

13: 26.2 Basic Definitions

… ►A k-dimensional lattice path is a directed path composed of segments that connect vertices in ${\{0,1,2,\mathrm{\dots }\}}^k$ so that each segment increases one coordinate by exactly one unit. …

14: 28.34 Methods of Computation

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Asymptotic approximations by zeros of orthogonal polynomials of increasing degree. See Volkmer (2008). This method also applies to eigenvalues of the Whittaker–Hill equation (§28.31(i)) and eigenvalues of Lamé functions (§29.3(i)).

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15: 6.18 Methods of Computation

… ►The attainable accuracy of the asymptotic expansions can be increased considerably by exponential improvement. …

16: 8.13 Zeros

… ►As $x$ increases the positive zeros coalesce to form a double zero at ($a_n^{\ast },x_n^{\ast }$). …

17: 7.13 Zeros

… ► $\mathrm{erf}z$ has a simple zero at $z=0$, and in the first quadrant of $ℂ$ there is an infinite set of zeros $z_n=x_n+\mathrm{i}{y}_n$, $n=1,2,3,\mathrm{\dots }$, arranged in order of increasing absolute value. … ►In the sector , $\mathrm{erfc}z$ has an infinite set of zeros $z_n=x_n+\mathrm{i}{y}_n$, $n=1,2,3,\mathrm{\dots }$, arranged in order of increasing absolute value. … ►In the first quadrant of $ℂ$ $C\left(z\right)$ has an infinite set of zeros $z_n=x_n+\mathrm{i}{y}_n$, $n=1,2,3,\mathrm{\dots }$, arranged in order of increasing absolute value. …

18: 1.4 Calculus of One Variable

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§1.4(i) Monotonicity

… ►If the $\le$ sign is replaced by , then $f(x)$ is increasing (also called strictly increasing) on $I$. …Each of the preceding four cases is classified as monotonic; sometimes strictly monotonic is used for the strictly increasing or strictly decreasing cases. …

19: 10.69 Uniform Asymptotic Expansions for Large Order

… ►Accuracy in (10.69.2) and (10.69.4) can be increased by including exponentially-small contributions as in (10.67.3), (10.67.4), (10.67.7), and (10.67.8) with $x$ replaced by $\nu x$. …

20: 23.5 Special Lattices

… ►As functions of $\mathrm{\Im }{\omega }_3$, $e_1$ and $e_2$ are decreasing and $e_3$ is increasing. … ►As a function of $\mathrm{\Im }{e}_3$ the root $e_1$ is increasing. …