Integrals

Date: Jul 13, 2011

A Sage package for evaluating log-sine integrals

This Sage package accompanies the paper Special values of generalized log-sine integrals.

Besides the package you find a notebook which shows basic usage. If you don't have access to Sage you can still look at the attached pdf file logsine-sage-usage.pdf which illustrates the package in use.

Note that the package is preliminary and has several rough edges—please report any issues to math@arminstraub.com. Integration into the core of Sage is a hoped for possibility. read more »

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logsine.sage
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logsine-sage-usage.sws
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logsine-sage-usage.pdf
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Date: Dec 17, 2010

LsToLi: A Mathematica package for evaluating log-sine integrals

This Mathematica package accompanies the paper Special values of generalized log-sine integrals. Besides the package you find a notebook which shows basic usage. If you don't have access to Mathematica you can still look at the attached pdf file logsine-usage.pdf which illustrates the package in use.

Note that the package is preliminary and hasn't received final touches yet—please report any issues to math@arminstraub.com. read more »

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logsine.m
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logsine-usage.nb
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logsine-usage.pdf
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Date: May 13, 2009

A trick for playing a multivariate integral

For suitable functions f the integral

$$ \int_0^{\infty} \int_0^{\infty} \ldots \int_0^{\infty} \frac{dx_1 dx_2 \cdots dx_n}{(f(x_1) + f(x_2) + \ldots + f (x_n))^{k+1}} $$

is equal to

$$ \frac{1}{k!} \int_0^{\infty} t^k \left( \int_0^{\infty} e^{- t f(x)} dx \right)^n dt . $$

To see why this is true, start with the last integral and write the n-th power of the inner integral as a multiple integral over variables x1, …, xn. Then change order of integration and evaluate read more »

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