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integrate x^3/(exp(x)-1) between 0 and infinity

asked 2017-06-08 12:56:33 -0500

epimetheus gravatar image

updated 2017-06-08 13:08:12 -0500

If I type

integrate(x^3/(exp(x)-1),x,0,infinity) I get -1/15pi^4 + limit(-1/4x^4 + x^3log(-e^x + 1) + 3x^2dilog(e^x) - 6xpolylog(3, e^x) + 6polylog(4, e^x), x, +Infinity, minus)

The command numerical_integral(x^3/(exp(x)-1),0,infinity) gives 6.4939394075

I have two questions :

  1. How do I evaluate the limit ?
  2. The correct answer is pi^4/15(=6.49393940226683) : why SageMath does not give it with symbolic integration ?

Thanks

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Comments

This seems closely related to https://groups.google.com/forum/#!top...

kcrisman gravatar imagekcrisman ( 2017-06-08 13:02:20 -0500 )edit

3 answers

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answered 2017-06-08 13:56:10 -0500

eric_g gravatar image

updated 2017-06-08 15:17:47 -0500

For this type of computation, it is worth trying with some algorithm different from the default one (Maxima):

sage: integrate(x^3/(exp(x)-1),x,0,infinity, algorithm='giac')
1/15*pi^4

EDIT: the above works only for Sage 8.0.beta5 and higher versions (after the ticket https://trac.sagemath.org/ticket/22891 has been merged). This will therefore be available in the next stable release of Sage (8.0).

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answered 2017-06-08 15:27:32 -0500

tmonteil gravatar image

To complement Eric's answer, you will can use giac integration algorithm from Sage version 8.0.beta4, so you should either install the latest development version, or wait for the official 8.0 version. Also, it is not enough to install giac package, you should also install the giacpy_sage package.

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Will the latter be standard as a package too in 8.0?

kcrisman gravatar imagekcrisman ( 2017-06-09 13:34:00 -0500 )edit
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answered 2017-06-08 14:49:42 -0500

epimetheus gravatar image

updated 2017-06-08 14:51:15 -0500

Before proceeding, I have installed the giac package 1.2.3.25

integral(x^3/(exp(x)-1),x,0,infinity, algorithm='giac')

returns

ValueError: Unknown algorithm: giac

even after restarting SageMath

What would you advice ?

Thank you.

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Either wait for the release of Sage 8.0 (quite soon I think) or install the latest development version of Sage, since it works only for Sage >= 8.0.beta5.

eric_g gravatar imageeric_g ( 2017-06-08 15:24:16 -0500 )edit

if is_package_installed("giac") is True, then i guess giac("integrate(x^3/(exp(x)-1), x, 0, inf)") will also work in v.7.6.

mforets gravatar imagemforets ( 2017-06-08 16:59:38 -0500 )edit

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Asked: 2017-06-08 12:56:33 -0500

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Last updated: Jun 08