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21: 14.21 Definitions and Basic Properties

… ►When $z$ is complex $P_{\nu }^{\pm \mu }\left(z\right)$, $Q_{\nu }^{\mu }\left(z\right)$, and ${𝑸}_{\nu }^{\mu }\left(z\right)$ are defined by (14.3.6)–(14.3.10) with $x$ replaced by $z$: the principal branches are obtained by taking the principal values of all the multivalued functions appearing in these representations when $z\in (1,\mathrm{\infty })$, and by continuity elsewhere in the $z$-plane with a cut along the interval $(-\mathrm{\infty },1]$; compare §4.2(i). …

22: 20.13 Physical Applications

… ►For $\tau =\mathrm{i}t$, with $\alpha ,t,z$ real, (20.13.1) takes the form of a real-time $t$ diffusion equation …

23: 8.6 Integral Representations

… ► $t^{a-1}$ takes its principal value where the path intersects the positive real axis, and is continuous elsewhere on the path. …

24: 8.4 Special Values

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25: 14.13 Trigonometric Expansions

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26: 15.11 Riemann’s Differential Equation

… ►Also, if any of $\alpha$, $\beta$, $\gamma$, is at infinity, then we take the corresponding limit in (15.11.1). …

27: 31.7 Relations to Other Functions

… ►Other reductions of $H\mathrm{\ell }$ to a ${}_2{}^{}F_1^{}$, with at least one free parameter, exist iff the pair $(a,p)$ takes one of a finite number of values, where $q=\alpha \beta p$. …

28: 18.15 Asymptotic Approximations

… ►When $\alpha ,\beta \in (-\frac{1}{2},\frac{1}{2})$, the error term in (18.15.1) is less than twice the first neglected term in absolute value, in which one has to take $\cos {\theta }_{n,m,\mathrm{\ell }}=1$. … ►Another expansion follows from (18.15.10) by taking $\lambda =\frac{1}{2}$; see Szegő (1975, Theorem 8.21.5). …

29: 19.29 Reduction of General Elliptic Integrals

… ►The advantages of symmetric integrals for tables of integrals and symbolic integration are illustrated by (19.29.4) and its cubic case, which replace the $8+8+12=28$ formulas in Gradshteyn and Ryzhik (2000, 3.147, 3.131, 3.152) after taking $x^2$ as the variable of integration in 3. … ►If $x=\mathrm{\infty }$, then $U$ is found by taking the limit. …

30: 4.13 Lambert $W$-Function

… ►

4.13.3_1

ⓘ

Symbols:

$W\left(z\right)$: Lambert $W$-function, $\mathrm{e}$: base of natural logarithm and $x$: real variable

Proof sketch:

Take $x=W_0\left(t\right)>-1$, with $t>-{\mathrm{e}}^{-1}$. Then $W_0\left(x{\mathrm{e}}^x\right)=W_0\left(W_0\left(t\right){\mathrm{e}}^{W_0\left(t\right)}\right)=W_0\left(t\right)=x$.

Referenced by:

§4.13, Erratum (V1.1.7) for Expansion

Permalink:

http://dlmf.nist.gov/4.13.E3_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §4.13 and Ch.4

►

4.13.3_2

ⓘ

Symbols:

$W\left(z\right)$: Lambert $W$-function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $x$: real variable

Proof sketch:

Take , with . Then $W_1\left(x{\mathrm{e}}^x-0\mathrm{i}\right)=W_1\left(W_1\left(t-0\mathrm{i}\right){\mathrm{e}}^{W_1\left(t-0\mathrm{i}\right)}-0\mathrm{i}\right)=W_1\left(t-0\mathrm{i}\right)=x$.

Referenced by:

§4.13, Erratum (V1.1.7) for Expansion

Permalink:

http://dlmf.nist.gov/4.13.E3_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §4.13 and Ch.4

…