“Lambert's W-Function”的概念、定义、习题解题方法及翻译-数学辞典

单词 Lambert's W-Function

释义

Lambert's W-Function

The inverse of the function

(1)

also called the Omega Function. The function is implemented as the Mathematica

(WolframResearch, Champaign, IL) function ProductLog[z].

is called the Omega Constant and can be considered asort of ``Golden Ratio'' of exponentials since

(2)

giving

(3)

Lambert's -Function has the series expansion

(4)

The Lagrange Inversion Theorem gives the equivalent series expansion

(5)

where

is a Factorial. However, this series oscillates between ever larger Positive and Negative valuesfor Real

, and so cannot be used for practical numerical computation. An asymptoticFormula which yields reasonably accurate results for

where

			(7)
			(8)

(Corless et al.), correcting a typographical error in de Bruijn (1961). Another expansion due to Gosper is theDouble Sum

(9)

where

is a nonnegative Stirling Number of the First Kind and

is a first approximation which can beused to select between branches. Lambert's

-function is two-valued for

. For

, the function isdenoted

or simply

, and this is called the principal branch. For

, the function is denoted

. The Derivative of

(10)

for

. For the principal branch when

(11)

See also Iterated Exponential Constants, Omega Constant

References

de Bruijn, N. G. Asymptotic Methods in Analysis. Amsterdam, Netherlands: North-Holland, pp. 27-28, 1961.

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