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A277053
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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.
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0
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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
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OFFSET
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2,1
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COMMENTS
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It has not yet been determined if this number has a closed form.
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LINKS
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FORMULA
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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))).
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EXAMPLE
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78.5176688733806851928297599903919937600495951319589367155801108473527173126...
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MATHEMATICA
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FindRoot[Re[ProductLog[-Log[x]]^2/(x Log[x]^2 (1 + ProductLog[-Log[x]]))], {x, 78, 79},
WorkingPrecision -> 261]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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