Re: Wolfram, meet Stefan and Boltzmann
- To: mathgroup at smc.vnet.net
- Subject: [mg117285] Re: Wolfram, meet Stefan and Boltzmann
- From: AES <siegman at stanford.edu>
- Date: Mon, 14 Mar 2011 05:58:08 -0500 (EST)
- References: <ili6f5$kk2$1@smc.vnet.net>
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