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11: 32.3 Graphics

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►►

Figure 32.3.8: $u_k(x;\frac{1}{2})$ for $-12\le x\le 4$ with $k=0.47442$, $0.47443$. The two graphs are indistinguishable when $x$ exceeds $-2.2$, approximately. … Magnify

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12: Bibliography O

… ►

Asymptotics and Special Functions. A. K. Peters, Wellesley, MA.

ⓘ

Notes:

Reprint, with corrections, of original Academic Press edition, 1974

External Links:

ISBN 1-56881-069-5, MathReview, ZentralBlatt

Referenced by:

§1.10(iv), §1.13(viii), §1.13(i), §1.13(ii), §1.13(iv), §1.17(i), §1.4(v), §1.4(ii), §1.4(v), §1.8(i), §1.9(vi), §10.12, §10.14, §10.15, §10.16, §10.17(i), §10.17(iii), §10.17(iv), §10.19(ii), §10.19(ii), §10.2(ii), §10.20(i), §10.20(ii), §10.20(ii), §10.21(i), §10.21(vi), §10.25(ii), §10.27, §10.31, §10.34, §10.37, §10.4, §10.40(i), §10.40(ii), §10.40(iii), §10.40(iii), §10.41(ii), §10.41(iii), §10.41(iv), §10.41(v), §10.42, §10.6(i), §10.8, §10.9(i), §10.9(iii), Ch.10, §11.10(i), (11.11.10), (11.11.11), (11.11.14), (11.11.15), (11.11.16), (11.11.17), (11.11.18), (11.11.19), (11.11.8), §11.11(iii), §11.2(i), §11.2(ii), §11.2(iii), §11.6(i), §11.9(iii), Ch.11, §12.10(vi), §12.2(i), Ch.12, §13.1, §13.14(i), §13.14(ii), §13.14(v), §13.2(i), §13.2(v), §13.20(i), §13.20(ii), §13.20(v), §13.21(i), §13.21(iv), §13.22, §13.7(i), §13.8(i), Ch.13, §14.1, §14.12(ii), §14.15(iii), §14.15(iii), §14.15(iii), §14.16(ii), §14.17(iii), §14.18(ii), §14.2(i), §14.2(ii), §14.2(iii), §14.2(iv), §14.20(vii), §14.20(vii), §14.21(i), §14.21(ii), §14.21(iii), §14.23, §14.24, §14.25, §14.25, §14.26, §14.28(ii), §14.3(ii), §14.3(i), §14.3(ii), §14.3(iii), §14.3(iv), §14.7(i), §14.7(ii), §14.7(iv), §14.8(i), §14.8(iii), §14.9(i), §14.9(ii), §14.9(iii), §14.9(iv), Ch.14, §15.1, §15.10(i), §15.11(i), §15.11(ii), §15.12(iii), §15.2(ii), §15.8(i), Ch.15, §16.2(v), §18.15(iii), §18.15(v), §18.16(v), §2.1(i), §2.1(ii), §2.1(iii), §2.1(iii), §2.1(v), §2.10(ii), §2.10(iii), §2.10(iv), §2.10(i), §2.10(ii), §2.10(iii), §2.10(iv), §2.10(iv), §2.11(i), §2.11(i), §2.11(vi), §2.11(vi), §2.2, §2.2, §2.3(i), §2.3(ii), §2.3(iii), §2.3(iii), §2.3(v), §2.4(i), §2.4(i), §2.4(ii), §2.4(ii), §2.4(iii), §2.4(iii), §2.4(iv), §2.4(iv), §2.4(v), §2.4(vi), §2.7(iii), §2.7(i), §2.7(i), §2.7(ii), §2.7(ii), §2.7(iii), §2.7(iv), §2.8(i), §2.8(ii), §2.8(ii), §2.8(iii), §2.8(iii), §2.8(iv), §2.8(iv), Ch.2, §24.12(i), §24.17(i), §24.9, §26.7(iv), §3.9(vi), §4.13, §5.11(i), §5.11(i), §5.11(ii), §5.11(iii), §5.12, §5.17, §5.2(i), §5.2(ii), §5.4(i), §5.4(ii), §5.5(i), §5.5(ii), §5.5(iii), §5.7(ii), §5.8, §5.9(i), §5.9(ii), Ch.5, §6.12(i), §6.12(i), §6.2(i), §6.2(ii), §6.4, §6.5, §6.6, Ch.6, §7.12(i), §7.12(iii), §7.2(i), §7.2(ii), §7.2(iii), §7.20(i), §7.5, Ch.7, §8.10, §8.11(i), §8.11(i), §8.2(i), §8.2(ii), Ch.8, (9.10.11), (9.10.12), (9.10.17), (9.10.22), §9.10(iv), §9.10(vi), §9.10(viii), (9.12.1), (9.12.10), (9.12.25), (9.12.27), (9.12.3), (9.12.4), (9.12.5), (9.12.6), (9.12.7), (9.12.8), (9.12.9), §9.12(i), §9.12(iii), §9.12(viii), (9.2.1), (9.2.10), (9.2.11), (9.2.12), (9.2.13), (9.2.14), (9.2.15), (9.2.2), (9.2.3), (9.2.4), (9.2.5), (9.2.6), (9.2.7), (9.2.8), (9.2.9), Table 9.2.1, §9.2(i), §9.2(ii), §9.2(iii), §9.2(iv), §9.2(v), (9.4.1), (9.4.2), (9.4.3), (9.4.4), §9.4, (9.5.1), (9.5.2), (9.5.4), §9.5(i), §9.5(ii), (9.6.10), (9.6.11), (9.6.12), (9.6.13), (9.6.14), (9.6.15), (9.6.16), (9.6.17), (9.6.18), (9.6.19), (9.6.2), (9.6.20), (9.6.3), (9.6.4), (9.6.5), (9.6.6), (9.6.7), (9.6.8), (9.6.9), §9.6(i), §9.6(ii), (9.7.10), (9.7.11), (9.7.12), (9.7.13), (9.7.14), (9.7.15), (9.7.2), (9.7.3), (9.7.4), (9.7.5), (9.7.6), (9.7.7), (9.7.8), (9.7.9), Table 9.7.1, §9.7(i), §9.7(ii), §9.7(iii), §9.7(iv), (9.8.1), (9.8.14), (9.8.18), (9.8.2), (9.8.3), (9.8.4), (9.8.5), (9.8.6), (9.8.7), (9.8.8), §9.8(i), §9.8(i), §9.8(ii), §9.8(iii), (9.9.1), (9.9.2), §9.9(i), §9.9(ii), Ch.9, Profile Frank W. J. Olver, Notations P, Notations Q, Notations V, F. W. J. Olver (1974).

Encodings:

BibTeX

…

13: 3.3 Interpolation

… ►With $x_0=-2.2$, $x_1=-2.3$, $x_2=-2.4$, we obtain ►

3.3.40 $$x=-2.2+\mathrm{1.44011\hspace{0.33em}1973}(f-\mathrm{0.09614\hspace{0.33em}53780})+\mathrm{0.08865\hspace{0.33em}85832}(f-\mathrm{0.09614\hspace{0.33em}53780})(f-\mathrm{0.02670\hspace{0.33em}63331}),$$

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Symbols:

$f(z)$: function

Permalink:

http://dlmf.nist.gov/3.3.E40

Encodings:

pMML, png, TeX

See also:

Annotations for §3.3(v), §3.3(v), §3.3 and Ch.3

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3.3.41 $$f(x)=\mathrm{0.09614\hspace{0.33em}53780}+\mathrm{0.69439\hspace{0.33em}04495}(x+2.1)-\mathrm{0.03007\hspace{0.33em}14275}(x+2.2)(x+2.3),$$

ⓘ

Symbols:

$f(z)$: function

Permalink:

http://dlmf.nist.gov/3.3.E41

Encodings:

pMML, png, TeX

See also:

Annotations for §3.3(v), §3.3(v), §3.3 and Ch.3

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14: Bibliography F

… ►

On the reversion of an asymptotic expansion and the zeros of the Airy functions. SIAM Rev. 41 (4), pp. 762–773.

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External Links:

ISSN 1095-7200, Document, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§2.2, §2.2, §3.8(v), §9.9(iv), §9.9(iv).

Encodings:

BibTeX

…

15: 16.2 Definition and Analytic Properties

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16: 2.1 Definitions and Elementary Properties

… ►For reversion see §2.2. …

17: 13.3 Recurrence Relations and Derivatives

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18: 21.7 Riemann Surfaces

… ►For more information, see Dubrovin (1981), Brieskorn and Knörrer (1986, §9.3), Belokolos et al. (1994, Chapter 2), and Mumford (1984, §2.2–2.3). …

19: 33.14 Definitions and Basic Properties

… ►

33.14.15 $${\int }_0^{\mathrm{\infty }}{\varphi }_{m,\mathrm{\ell }}(r){\varphi }_{n,\mathrm{\ell }}(r)dr={\delta }_{m,n}.$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and ${\varphi }_{n,\mathrm{\ell }}(r)$: function

Proof sketch:

Derivable from (33.14.14) by combining it with Dunkl (2003, Theorem 2.2).

Referenced by:

§13.10(vi), §13.23(v), §33.14(iv), Erratum (V1.0.24) for Equation (33.14.15), Erratum (V1.0.24) for Subsection 33.14(iv)

Permalink:

http://dlmf.nist.gov/33.14.E15

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.24):

This equation, originally written as a definite integral ${\int }_0^{\mathrm{\infty }}{\varphi }_{n,\mathrm{\ell }}^2(r)dr=1$, was rewritten more generally as an orthonormality relation.

See also:

Annotations for §33.14(iv), §33.14 and Ch.33

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20: 18.2 General Orthogonal Polynomials

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