Planck Radiation Function

Author

Eric W. Weisstein
September 26, 2007

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/PlancksRadiationFunction.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/PlancksRadiationFunction.html.

©2007 Wolfram Research, Inc. except for portions noted otherwise

Definition

Planck[x_]:=15/Pi^4/(x^5(Exp[1/x]-1))

Plot

<<Utilities`Typesetting`

Plot[Planck[x],{x,0,2},PlotStyle->Red,AxesLabel->TraditionalForm/@{x,Planck[x]/.Exp[y_]:>Exp[InlineFraction@@Through[{Numerator,Denominator}[y]]]}]

Normalization

Integrate[Planck[x],{x,0,Infinity}]

Raw Moments

Integrate[x^n Planck[x],x]

Integrate[x^n Planck[x],{x,0,Infinity}]

Table[Integrate[x^n Planck[x],{x,0,Infinity}],{n,0,4}]

Maximum

eqn=Numerator[Together[Planck'[x]]]

Maximize[{Planck[x],x>0},x]

N[%,20]

Solve[eqn==0,x]

FindRoot[eqn==0,{x,.2},WorkingPrecision->20]

Reduce[eqn==0&&x>0,x]

N[%]

Inflection Points

<<Utilities`Plot`

PlotRoots[Planck''[x],{x,.01,1},WorkingPrecision->20]

Reduce[FullSimplify[Planck''[x]]==0&&x>0,x]