linear functionals

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21: 9.15 Mathematical Applications

… ►Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …

22: 32.2 Differential Equations

… ►The fifty equations can be reduced to linear equations, solved in terms of elliptic functions (Chapters 22 and 23), or reduced to one of ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$. …

23: 9.13 Generalized Airy Functions

… ►and ${𝒵}_p$ is any linear combination of the modified Bessel functions $I_p$ and ${\mathrm{e}}^{p\pi \mathrm{i}}{K}_p$ (§10.25(ii)). …

24: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►For scattering problems, the interior solution is then matched to a linear combination of a pair of Coulomb functions, $F_{\mathrm{\ell }}(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$, or $f(ϵ,\mathrm{\ell };r)$ and $h(ϵ,\mathrm{\ell };r)$, to determine the scattering $S$-matrix and also the correct normalization of the interior wave solutions; see Bloch et al. (1951). …

25: 3.7 Ordinary Differential Equations

… ►Consideration will be limited to ordinary linear second-order differential equations …

26: 19.29 Reduction of General Elliptic Integrals

… ► $I(𝐦)$ can be reduced to a linear combination of basic integrals and algebraic functions. …

27: Bibliography I

… ►

The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of $J_0(z)-i{J}_1(z)$ and of Bessel functions $J_m(z)$ of any real order $m$ . Linear Algebra Appl. 194, pp. 35–70.

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External Links:

ISSN 0024-3795, Document, MathReview (Alan L. Andrew), ZentralBlatt

Referenced by:

§10.21(ix).

Encodings:

BibTeX

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28: 10.72 Mathematical Applications

… ►Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter. …

29: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

§1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ► $V$ becomes a normed linear vector space. … ►

Bounded and Unbounded Linear Operators

… ► … ► …

30: Philip J. Davis

… ►After being asked by Milton Abramowitz to work on the project, he chose to write the Chapter “Gamma Function and Related Functions. … ►Olver had been recruited to write the Chapter “Bessel Functions of Integer Order” for A&S by Milton Abramowitz, who passed away suddenly in 1958. … ►This immediately led to discussions among some of the project members about what might be possible, and the discovery that some interactive graphics work had already been done for the NIST Matrix Market, a publicly available repository of test matrices for comparing the effectiveness of numerical linear algebra algorithms. … ►DLMF users can rotate, rescale, zoom and otherwise explore mathematical function surfaces. The surface color map can be changed from height-based to phase-based for complex valued functions, and density plots can be generated through strategic scaling. …