in terms of Airy functions
(0.010 seconds)
31—40 of 52 matching pages
31: 1.17 Integral and Series Representations of the Dirac Delta
Airy Functions (§9.2)
… ►In the language of physics and applied mathematics, these equations indicate the normalizations chosen for these non- improper eigenfunctions of the differential operators (with derivatives respect to spatial co-ordinates) which generate them; the normalizations (1.17.12_1) and (1.17.12_2) are explicitly derived in Friedman (1990, Ch. 4), the others follow similarly. Equations (1.17.12_1) through (1.17.16) may re-interpreted as spectral representations of completeness relations, expressed in terms of Dirac delta distributions, as discussed in §1.18(v), and §1.18(vi) Further mathematical underpinnings are referenced in §1.17(iv). …32: 12.11 Zeros
§12.11(i) Distribution of Real Zeros
… ►For large negative values of the real zeros of , , , and can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). …Here , denoting the th negative zero of the function (see §9.9(i)). … ►where , denoting the th negative zero of the function and … ►For further information, including associated functions, see Olver (1959).33: 9.10 Integrals
§9.10(i) Indefinite Integrals
… ►§9.10(ii) Asymptotic Approximations
… ►§9.10(iv) Definite Integrals
… ►§9.10(v) Laplace Transforms
… ►§9.10(vi) Mellin Transform
…34: 9.9 Zeros
§9.9(ii) Relation to Modulus and Phase
… ►§9.9(iii) Derivatives With Respect to
… ►§9.9(iv) Asymptotic Expansions
… ►§9.9(v) Tables
…35: Guide to Searching the DLMF
Terms, Phrases and Expressions
… ►a textual word, a number, or a math symbol.
Single-letter terms
36: 10.21 Zeros
37: 28.8 Asymptotic Expansions for Large
38: Bibliography S
39: 3.3 Interpolation
§3.3(i) Lagrange Interpolation
… ►With an error term the Lagrange interpolation formula for is given by … ►This represents the Lagrange interpolation polynomial in terms of divided differences: … ►Example
►To compute the first negative zero of the Airy function (§9.2). …40: Errata
§4.13 has been enlarged. The Lambert -function is multi-valued and we use the notation , , for the branches. The original two solutions are identified via and .
Other changes are the introduction of the Wright -function and tree -function in (4.13.1_2) and (4.13.1_3), simplification formulas (4.13.3_1) and (4.13.3_2), explicit representation (4.13.4_1) for , additional Maclaurin series (4.13.5_1) and (4.13.5_2), an explicit expansion about the branch point at in (4.13.9_1), extending the number of terms in asymptotic expansions (4.13.10) and (4.13.11), and including several integrals and integral representations for Lambert -functions in the end of the section.
In the line just below (18.15.4), it was previously stated “is less than twice the first neglected term in absolute value.” It now states “is less than twice the first neglected term in absolute value, in which one has to take .”
Reported by Gergő Nemes on 2019-02-08
Bounds have been sharpened. The second paragraph now reads, “The th error term is bounded in magnitude by the first neglected term multiplied by where for (9.7.7) and for (9.7.8), provided that in the first case and in the second case.” Previously it read, “In (9.7.7) and (9.7.8) the th error term is bounded in magnitude by the first neglected term multiplied by where for (9.7.7) and for (9.7.8), provided that in both cases.” In Equation (9.7.16)
the bounds on the right-hand sides have been sharpened. The factors , , were originally given by , , respectively.
The validity constraint was added. Additionally, specific source citations are now given in the metadata for all equations in Chapter 9 Airy and Related Functions.
A short paragraph dealing with asymptotic approximations that are expressed in terms of two or more Poincaré asymptotic expansions has been added below (2.1.16).