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Power Equation

المؤلف:  Dunn, A

المصدر:  Mathematical Bafflers. New York: Dover

الجزء والصفحة:  ...

5-6-2020

1492

Power Equation

Consider solutions to the equation

(1)

Real solutions are given by  for , together with the solution of

(2)

which is given by

{exp[-W_(-1)(-(lnx)/x)] for 1<x<e; exp[-W(-(lnx)/x)] for x>e, " src="https://mathworld.wolfram.com/images/equations/PowerEquation/NumberedEquation3.gif" style="height:82px; width:238px" />

(3)

where  is the Lambert W-function. This function is illustrated above by the blue curve.

Rational parametric solutions are given by

			(4)
			(5)

for , , ... (Dunn 1980, Pickover 2002). These solutions are shown on the plot as red dots.

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REFERENCES:

Dunn, A. Mathematical Bafflers. New York: Dover, p. 213, 1980.

Pickover, C. A. "The Gaps of Omicron." Ch. 23 in The Mathematics of Oz: Mental Gymnastics from Beyond the Edge. New York: Cambridge University Press, pp. 53-54 and 265, 2002.