with respect to order (ν-zeros)

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1: 10.21 Zeros

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§10.21(vii) Asymptotic Expansions for Large Order

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§10.21(viii) Uniform Asymptotic Approximations for Large Order

… ►Figures 10.21.1, 10.21.3, and 10.21.5 plot the actual zeros for $n=1,5$, and $10$, respectively. … ►Figures 10.21.2, 10.21.4, and 10.21.6 plot the actual zeros for $n=1,5$, and $10$, respectively. … ►

§10.21(xiv) $\nu$-Zeros

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2: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►For $𝒟(T)$ we can take $C^2(X)$, with appropriate boundary conditions, and with compact support if $X$ is bounded, which space is dense in $L^2\left(X\right)$, and for $X$ unbounded require that possible non-$L^2$ eigenfunctions of (1.18.28), with real eigenvalues, are non-zero but bounded on open intervals, including $\pm \mathrm{\infty }$. … ►The implicit boundary conditions taken here are that the ${\varphi }_n(x)$ and ${\varphi }_n^{\prime }(x)$ vanish as $x\to \pm \mathrm{\infty }$, which in this case is equivalent to requiring ${\varphi }_n(x)\in L^2\left(X\right)$, see Pauling and Wilson (1985, pp. 67–82) for a discussion of this latter point. … ►The Fourier cosine and sine transform pairs (1.14.9) & (1.14.11) and (1.14.10) & (1.14.12) can be easily obtained from (1.18.57) as for $\nu =\pm \frac{1}{2}$ the Bessel functions reduce to the trigonometric functions, see (10.16.1). … ►Unlike in the example in the paragraph above, in 3-dimensions a “dip below zero, or a potential well” in $V(r)$ does not always correspond to the existence of a discrete part of the spectrum. … ►The materials developed here follow from the extensions of the Sturm–Liouville theory of second order ODEs as developed by Weyl, to include the limit point and limit circle singular cases. …

3: 3.8 Nonlinear Equations

… ►Bisection of this interval is used to decide where at least one zero is located. … ►has $n$ zeros in $ℂ$, counting each zero according to its multiplicity. … ►The zeros are $\pm 1$ and $\pm \mathrm{i}$. … ►For describing the distribution of complex zeros of solutions of linear homogeneous second-order differential equations by methods based on the Liouville–Green (WKB) approximation, see Segura (2013). … ►Starting this iteration in the neighborhood of one of the four zeros $\pm 1,\pm \mathrm{i}$, sequences $\{z_n\}$ are generated that converge to these zeros. …

4: 18.16 Zeros

§18.16 Zeros

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§18.16(iii) Ultraspherical, Legendre and Chebyshev

… ►Arrange them in decreasing order: …where $a_m$ is the $m$th negative zero of $\mathrm{Ai}\left(x\right)$ (§9.9(i)), , and as $n\to \mathrm{\infty }$ with $m$ fixed … ►Lastly, in view of (18.7.19) and (18.7.20), results for the zeros of $L_n^{(\pm \frac{1}{2})}\left(x\right)$ lead immediately to results for the zeros of $H_n\left(x\right)$. …