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Re: Wolfram, meet Stefan and Boltzmann

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  • Subject: [mg117389] Re: Wolfram, meet Stefan and Boltzmann
  • From: AES <siegman at>
  • Date: Thu, 17 Mar 2011 06:30:40 -0500 (EST)
  • References: <ilq6v8$bm4$>

David Lichtbau notes:

> (3) Find the antiderivative. It is
> In[22]:= InputForm[Integrate[x^3/(Exp[x] - 1), x]]
> Out[22]//InputForm=
> -x^4/4 + x^3*Log[1 - E^x] + 3*x^2*PolyLog[2, E^x] -
>    6*x*PolyLog[3, E^x] + 6*PolyLog[4, E^x]

The original integral shown above is used to evaluate the 
Stefan-Boltzmann constant sigma in the expression P/A = sigma T^4  for 
total power per unit area radiated by a blackbody surface at temperature 
T.  Physically it represents integrating over all the frequencies or 
wavelengths coming from the blackbody radiator.

I'm not personally familiar with the term "antiderivative" (nor the term 
"PolyLog" for that matter); but if it means in essence "indefinite 
integral" then this antiderivative might be used to evaluate the total 
P/A coming from a _frequency_ or _spectrally_ filtered blackbody source 
using a flat-topped passband filter.

Hmm -- wonder if the thermodynamic or blackbody-radiation communities 
know about that?

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