A277053 Decimal expansion of real zero x between 78 and 79 of the derivative of the function plotting the invariant points for the exponential function of the form x^y = y.

7, 8, 5, 1, 7, 6, 6, 8, 8, 7, 3, 3, 8, 0, 6, 8, 5, 1, 9, 2, 8, 2, 9, 7, 5, 9, 9, 9, 0, 3, 9, 1, 9, 9, 3, 7, 6, 0, 0, 4, 9, 5, 9, 5, 1, 3, 1, 9, 5, 8, 9, 3, 6, 7, 1, 5, 5, 8, 0, 1, 1, 0, 8, 4, 7, 3, 5, 2, 7, 1, 7, 3, 1, 2, 6, 0, 6, 7, 6, 3, 0, 0, 6, 4, 2, 6, 8, 9, 0, 6, 0, 7, 5, 1, 8, 8, 1, 6, 1, 7, 7, 8, 2, 3, 9, 7, 2, 2, 3, 9, 1, 7, 7, 4, 3, 0, 2, 7, 7, 7, 7, 5, 8, 2, 4, 0, 4, 0, 9, 3

Offset: 2

Author

David D. Acker, Sep 26 2016

Keywords

• nonn
• cons

Comments

It has not yet been determined if this number has a closed form.

Examples

			78.5176688733806851928297599903919937600495951319589367155801108473527173126...
		

Crossrefs

Cf. A042972, A073229.

Programs

Mathematica

FindRoot[Re[ProductLog[-Log[x]]^2/(x Log[x]^2 (1 + ProductLog[-Log[x]]))], {x, 78, 79},
WorkingPrecision -> 261]

Formula

The derivative x^y = y, or y = -ProductLog(-log(x))/log(x) when solved for y, is the function in which this value is a root. The derivative is (ProductLog(-log(x)))^2/(x*(log(x))^2*(1+ProductLog(-log(x)))).