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Re: Wolfram, meet Stefan and Boltzmann

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  • Subject: [mg117285] Re: Wolfram, meet Stefan and Boltzmann
  • From: AES <siegman at>
  • Date: Mon, 14 Mar 2011 05:58:08 -0500 (EST)
  • References: <ili6f5$kk2$>

Responses to all:

Many of you will recognize that the integral in question
   Integrate[x^3/(Exp[x] - 1), {x, 0, Infinity}]

arises in the derivation of the Stefan-Boltzmann constant.  

I was reviewing this derivation in Wikipedia. Wikipedia's analytical 
evaluation of the integral itself, given in an Appendix to the 
Stefan-Boltzmann article, takes a couple of screens of text and a dozen 
intermediate formulas, with excursions into contour integration and 
infinite series expansions.  

I became curious as to whether Mathematica could "just do" this integral 
so I opened Mathematica, typed in the line above, and evaluated it.  I 
was surprised at how long it took the first time, although reevaluation 
of the same cell was essentially instantaneous.

So I shut down Mathematica; re-Opened it; and repeated the evaluation 
using Timing[-].  Did this several times, and found that from a "cold 
start", that is, when I Quit and then re-Open Mathematica, it always 
takes 18 or 19 seconds to do the first evaluation.

My Mac is a 2007 or 2007 vintage MacBook with hardware overview given 
below, running Snow Leopard in a generally vanilla setup (only about 60 
GB occupied on the 160 GB HD).  

So, I'm still a bit curious:  What's Mathematica doing all that time? -- 
loading a lot of other stuff?  actually going through some lengthy 
algorithm to evaluate the integral?   ???

Hardware Overview:

  Model Name:  MacBook
  Model Identifier:  MacBook2,1
  Processor Name: Intel Core 2 Duo
  Processor Speed:   2.16 GHz
  Number Of Processors: 1
  Total Number Of Cores:   2
  L2 Cache: 4 MB
  Memory:   1 GB
  Bus Speed:   667 MHz
  Boot ROM Version:  MB21.00A5.B07
  SMC Version (system): 1.17f0

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