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Planck's Radiation Function

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Planck

Planck's's radiation function is the function

f(x)=(15)/(pi^4)1/(x^5(e^(1/x)-1)),
(1)

which is normalized so that

int_0^inftyf(x)dx=1.
(2)

However, the function is sometimes also defined without the numerical normalization factor of 15/pi^4 (e.g., Abramowitz and Stegun 1972, p. 999).

The first and second raw moments are

mu_1^'=(30zeta(3))/(pi^4)
(3)
mu_2^'=5/(2pi^2),
(4)

where zeta(3) is Apéry's constant, but higher order raw moments do not exist since the corresponding integrals do not converge.

It has a maximum at x approx 0.201405 (OEIS A133838), where

f^'(x)=(5x-e^(1/x)(5x-1))/(x^7(e^(1/x)-1)^2)=0,
(5)

and inflection points at x approx 0.11842 (OEIS A133839) and x approx 0.283757 (OEIS A133840), where

f^('')(x)=(e^(1/x)(1+e^(1/x))+6x(e^(1/x)-1)[e^(1/x)(5x-2)-5x])/((e^(1/x)-1)^3x^9)=0.
(6)

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Planck's Radiation Function." §27.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 999, 1972.Sloane, N. J. A. Sequences A133838, A133839, A133840 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Planck's Radiation Function

Cite this as:

Weisstein, Eric W. "Planck's Radiation Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PlancksRadiationFunction.html

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