Re: Wolfram, meet Stefan and Boltzmann

• To: mathgroup at smc.vnet.net
• Subject: [mg117435] Re: Wolfram, meet Stefan and Boltzmann
• From: SigmundV <sigmundv at gmail.com>
• Date: Fri, 18 Mar 2011 06:01:46 -0500 (EST)
• References: <ilq6v8\$bm4\$1@smc.vnet.net> <ilsrcu\$svv\$1@smc.vnet.net>

```Very weird timings AES is getting. On my cheap Dell Inspiron 1545 (4
GB RAM, Intel Pentium Dual Core CPU) running Ubuntu 10.10 64-bit I get

In[1]:= AbsoluteTiming[Integrate[x^3/(Exp[x] - 1), {x, 0, Infinity}]]
Out[1]= {2.659503, \[Pi]^4/15}

with

In[2]:= \$Version
Out[2]= "8.0 for Linux x86 (64-bit) (November 7, 2010)"

This is on the high end of timings posted here, but no surprise given
my low-end hardware. The timings posted by others using similar
hardware to AES's points to his specific MacBook Pro being at fault.

It also astonished me that AES is not familiar with the term
'antiderivative'. The derivative of the antiderivative is the function
itself.

/Sigmund

On Mar 17, 11:30 am, AES <sieg... at stanford.edu> wrote:
> 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  fo=
r
> 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