Planck's Radiation Function

$\begin{figure}\begin{center}\BoxedEPSF{Planck.epsf}\end{center}\end{figure}$

The function

$\begin{displaymath} f(x)={1\over x^5(e^{1/x}-1)}. \end{displaymath}$

It has a Maximum at $x\approx 0.201405$ , where

$\begin{displaymath} f'(x)={5x-e^{1/x}(5x-1)\over x^7(e^{1/x}-1)^2}=0, \end{displaymath}$

and inflection points at $x\approx 0.11842$ and $x\approx 0.283757$ , where

$\begin{displaymath} f''(x)={e^{1/x}(1+e^{1/x})+6x(e^{1/x}-1)[e^{1/x}(5x-2)-5x]\over (e^{1/x}-1)^3 x^9}=0. \end{displaymath}$

(1)

References

Abramowitz, M. and Stegun, C. 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.