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A133838
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Decimal expansion of the value at which Planck's radiation function achieves its maximum.
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4
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2, 0, 1, 4, 0, 5, 2, 3, 5, 2, 7, 2, 6, 4, 2, 1, 8, 0, 6, 1, 5, 6, 6, 2, 6, 4, 3, 6, 5, 9, 0, 2, 7, 9, 9, 6, 0, 2, 8, 9, 3, 5, 7, 3, 7, 9, 5, 9, 3, 5, 1, 1, 4, 3, 9, 5, 7, 4, 1, 4, 6, 5, 8, 3, 2, 1, 9, 0, 2, 9, 4, 7, 6, 9, 7, 4, 9, 5, 1, 7, 7, 6, 0, 4, 6, 0, 6, 3, 2, 8, 4, 8, 1, 5, 6, 7, 7, 1, 8, 4, 7, 1, 9, 8, 3
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OFFSET
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0,1
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COMMENTS
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Consider the density of the radiation function (in wavelength form) B(lambda) = 2*h*c^2/{lambda^5*[exp(h*c/(kB*lambda*T))-1]}, where h is Planck's constant, c the speed of light, kB the Boltzmann constant, T the absolute temperature, and lambda the wavelength. Searching the maximum, we set the first derivative dB/dlambda to zero, then substitute x=lambda*T/(h*c/kB). The equation becomes 5+(1/x-5)*exp(1/x)=0 and the solution x is this constant here. - R. J. Mathar, Jan 30 2014
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LINKS
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EXAMPLE
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0.20140523527264218061... = 1/4.96511..
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MATHEMATICA
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RealDigits[ x /. FindRoot[5x - E^(1/x)*(5x - 1), {x, 1/5}, WorkingPrecision -> 105]][[1]] (* or *)
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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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