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11—20 of 48 matching pages

11: 34.1 Special Notation

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12: 1.3 Determinants, Linear Operators, and Spectral Expansions

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1.3.2

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Symbols:

$det$: determinant, $j$: integer and $k$: integer

Permalink:

http://dlmf.nist.gov/1.3.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

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13: 30.9 Asymptotic Approximations and Expansions

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$$2^{14}{\beta }_3=-33q^5-1594q^3-5621q+128m^2(37q^3+167q)-2048m^{4}q.$$

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$2^9{c}_3=-33q^5-114q^3-37q+2m^2(23q^3+25q)-13m^{4}q.$

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14: 28.11 Expansions in Series of Mathieu Functions

… ►See Meixner and Schäfke (1954, §2.28), and for expansions in the case of the exceptional values of $q$ see Meixner et al. (1980, p. 33). …

15: 29.7 Asymptotic Expansions

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29.7.7 $${\tau }_3=\frac{p}{2^{14}}({(1+k^2)}^4(33p^4+410p^2+405)-24k^2{(1+k^2)}^2(7p^4+90p^2+95)+16k^4(9p^4+130p^2+173)),$$

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Symbols:

$p$: nonnegative integer, $k$: real parameter and ${\tau }_j$: coefficients

Permalink:

http://dlmf.nist.gov/29.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

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16: Bibliography R

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A non-negative representation of the linearization coefficients of the product of Jacobi polynomials. Canad. J. Math. 33 (4), pp. 915–928.

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External Links:

ISSN 0008-414X, Document, Link, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.18(v).

Encodings:

BibTeX

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Complex Coordinates in the Theory of Atomic and Molecular Structure and Dynamics. Annual Review of Physical Chemistry 33, pp. 223–255.

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Referenced by:

§1.18(viii).

Encodings:

BibTeX

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Diffraction of plane radio waves by a parabolic cylinder. Calculation of shadows behind hills. Bell System Tech. J. 33, pp. 417–504.

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External Links:

MathReview (J. Shmoys)

Referenced by:

§12.17.

Encodings:

BibTeX

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17: 23.21 Physical Applications

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18: 28.26 Asymptotic Approximations for Large $q$

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28.26.4 $${\mathrm{Fc}}_m(z,h)\sim 1+\frac{s}{8h{\cosh }^{2}z}+\frac{1}{2^{11}{h}^2}\left(\frac{s^4+86s^2+105}{{\cosh }^{4}z}-\frac{s^4+22s^2+57}{{\cosh }^{2}z}\right)+\frac{1}{2^{14}{h}^3}\left(-\frac{s^5+14s^3+33s}{{\cosh }^{2}z}-\frac{2s^5+124s^3+1122s}{{\cosh }^{4}z}+\frac{3s^5+290s^3+1627s}{{\cosh }^{6}z}\right)+\mathrm{\cdots },$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\cosh z$: hyperbolic cosine function, $m$: integer, $h$: parameter and $z$: complex variable

A&S Ref:

20.9.9 (in slightly different notation)

Referenced by:

§28.26(i)

Permalink:

http://dlmf.nist.gov/28.26.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

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19: 18.39 Applications in the Physical Sciences

… ►The spectrum is mixed, as in §1.18(viii), the positive energy, non-$L^2$, scattering states are the subject of Chapter 33. … ►Namely for fixed $l$ the infinite set labeled by $p$ describe only the $L^2$ bound states for that single $l$, omitting the continuum briefly mentioned below, and which is the subject of Chapter 33, and so an unusual example of the mixed spectra of §1.18(viii). … ►This is also the normalization and notation of Chapter 33 for $Z=1$ , and the notation of Weinberg (2013, Chapter 2). … ►As the scattering eigenfunctions of Chapter 33, are not OP’s, their further discussion is deferred to §18.39(iv), where discretized representations of these scattering states are introduced, Laguerre and Pollaczek OP’s then playing a key role. … ►The Coulomb–Pollaczek polynomials provide alternate representations of the positive energy Coulomb wave functions of Chapter 33. …

20: 26.6 Other Lattice Path Numbers

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Table 26.6.1: Delannoy numbers $D(m,n)$.

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$m$	$n$
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9	1	19	181	1159	5641	22363	75517	2 24143	5 98417	14 62563	33 17445
10	1	21	221	1561	8361	36365	1 34245	4 33905	12 56465	33 17445	80 97453

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