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21: 36.2 Catastrophes and Canonical Integrals

… ►Special cases: $K=1$, fold catastrophe; $K=2$, cusp catastrophe; $K=3$, swallowtail catastrophe. … ►

§36.2(ii) Special Cases

… ►Addendum: For further special cases see §36.2(iv) … ►

§36.2(iv) Addendum to 36.2(ii) Special Cases

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22: 18.21 Hahn Class: Interrelations

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§18.21(ii) Limit Relations and Special Cases

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23: 16.23 Mathematical Applications

… ►Many combinatorial identities, especially ones involving binomial and related coefficients, are special cases of hypergeometric identities. …

24: 27.7 Lambert Series as Generating Functions

… ►Again with , special cases of (27.7.2) include: …

25: 19.6 Special Cases

§19.6 Special Cases

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26: 34.5 Basic Properties: $\mathit{6}j$ Symbol

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§34.5(i) Special Cases

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27: 18.12 Generating Functions

… ►See §18.18(vii) for Poisson kernels; these are special cases of bilateral generating functions.

28: 6.14 Integrals

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6.14.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}{E}_1\left(t\right)dt=\frac{1}{a}\ln \left(1+a\right),$$ $\mathrm{\Re }a>-1$,

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\int$: integral, $\ln z$: principal branch of logarithm function and $\mathrm{\Re }$: real part

A&S Ref:

5.1.34 (specialcase of)

Permalink:

http://dlmf.nist.gov/6.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.14(i), §6.14 and Ch.6

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29: 7.8 Inequalities

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7.8.8 $x>0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm and $x$: real variable

Source:

Pólya (1949, (1.5))

Referenced by:

§7.8, Erratum (V1.0.17) for Equation (7.8.8)

Permalink:

http://dlmf.nist.gov/7.8.E8

Encodings:

pMML, png, TeX

Addition (effective with 1.0.17):

This inequality was added. It is Pólya (1949, (1.5)). Note that it is a specialcase of (8.10.11).

Suggested by Roberto Iacono

See also:

Annotations for §7.8 and Ch.7

30: 13.6 Relations to Other Functions

… ►Special cases are the error functions … ►Note that (13.6.11_1) and (13.6.11_2) are special cases of (13.11.1) and (13.11.2), respectively … ►Special cases of §13.6(iv) are as follows. …