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21: 4.13 Lambert $W$-Function

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Figure 4.13.2: The $W\left(z\right)$ function on the first 5 Riemann sheets. $W\left(z\right)$ maps the first Riemann sheet in the middle of the left-hand side to the region enclosed by the green curve on the right-hand side; it maps the Riemann sheet on the left-hand side to the region enclosed by the pink, green and orange curves on the right-hand side, etc. Magnify

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4.13.5_1 $${\left(\frac{W_0\left(z\right)}{z}\right)}^a={\mathrm{e}}^{-a{W}_0\left(z\right)}=\sum _{n=0}^{\mathrm{\infty }}\frac{a{\left(n+a\right)}^{n-1}}{n!}{\left(-z\right)}^n,$$ , $a\in ℂ$.

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $ℂ$: complex plane, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: integer, $a$: real or complex constant and $z$: complex variable

Notes:

See Corless et al. (1996, formula (2.36)).

Referenced by:

§4.13, Erratum (V1.1.7) for Expansion

Permalink:

http://dlmf.nist.gov/4.13.E5_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §4.13 and Ch.4

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22: Bibliography H

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Inverse virial symmetry of diatomic potential curves. J. Chem. Phys. 109 (1), pp. 11–19.

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External Links:

Document

Referenced by:

§15.18.

Encodings:

BibTeX

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A Treatise on Plane and Advanced Trigonometry. 7th edition, Cambridge University Press.

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Notes:

Reprinted by Dover Publications Inc., New York, 2004.

External Links:

MathReview, ZentralBlatt

Referenced by:

§4.16, §4.17, §4.19, §4.2(ii), §4.2(iii), §4.21(i), §4.21(ii), §4.21(iii), §4.21(iv), §4.22, §4.23(v), §4.23(vi), §4.24(i), §4.24(iii), §4.28, §4.30, §4.31, §4.35(i), §4.35(ii), §4.35(iii), §4.35(iv), §4.42(i), §4.42(ii), §4.43, §4.8(ii), Ch.4.

Encodings:

BibTeX

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23: 3.11 Approximation Techniques

… ►The error curve is shown in Figure 3.11.1. … ►In computer graphics a special type of spline is used which produces a Bézier curve. A cubic Bézier curve is defined by four points. …A complete spline results by composing several Bézier curves. A special applications area of Bézier curves is mathematical typography and the design of type fonts. …