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31: 10.17 Asymptotic Expansions for Large Argument

… ►where $𝒱$ denotes the variational operator (2.3.6), and the paths of variation are subject to the condition that $|\mathrm{\Im }t|$ changes monotonically. …

32: Bibliography B

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A monotonicity property involving ${}_3{}^{}F_2^{}$ and comparisons of the classical approximations of elliptical arc length. SIAM J. Math. Anal. 32 (2), pp. 403–419.

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External Links:

ISSN 1095-7154, Document, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§19.9(i), §19.9(i).

Encodings:

BibTeX

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33: Bibliography M

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Higher monotonicity properties of certain Sturm-Liouville functions. V. Proc. Roy. Soc. Edinburgh Sect. A 77 (1-2), pp. 23–37.

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External Links:

MathReview (G. Sansone), ZentralBlatt

Referenced by:

§10.21(ii), (9.11.4), §9.11(iii).

Encodings:

BibTeX

…

34: 1.6 Vectors and Vector-Valued Functions

… ►A path ${𝐜}_1(t)$, $t\in [a,b]$, is a reparametrization of $𝐜(t^{\prime })$, $t^{\prime }\in [a^{\prime },b^{\prime }]$, if ${𝐜}_1(t)=𝐜(t^{\prime })$ and $t^{\prime }=h(t)$ with $h(t)$ differentiable and monotonic. …

35: 3.11 Approximation Techniques

… ►Since $L_0=1$, $L_n$ is a monotonically increasing function of $n$, and (for example) $L_{1000}=4.07\mathrm{\dots }$, this means that in practice the gain in replacing a truncated Chebyshev-series expansion by the corresponding minimax polynomial approximation is hardly worthwhile. …

36: 19.29 Reduction of General Elliptic Integrals

… ►Lastly, define $Q(t^2)=f+g{t}^2+h{t}^4$ and assume $Q(t^2)$ is positive and monotonic for . …

37: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Let $T$ be the self adjoint extension of a formally self-adjoint differential operator $ℒ$ of the form (1.18.28) on an unbounded interval $X\subset ℝ$, which we will take as $X=[0,+\mathrm{\infty })$, and assume that $q(x)\to 0$ monotonically as $x\to \mathrm{\infty }$, and that the eigenfunctions are non-vanishing but bounded in this same limit. …