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Table of Integrals

A. Dieckmann, Physikalisches Institut der Uni Bonn

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This integral table contains hundreds of expressions: indefinite and definite integrals of elliptic integrals, of square roots, arcustangents and a few more exotic functions. Most of them are not found in Gradsteyn-Ryzhik.

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Sometimes m, n, k denote real parameters and are restricted mostly to 0<{m, n, k}<1, at times they represent natural numbers.

Results may be valid outside of the given region of parameters, but should always be checked numerically!

Definite Integrals:

Substitute   DefInt_5.gif   and the Feynman-Hibbs Integral can be calculated with Mathematica:

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To see a nice cancellation of singularities at work plot the next expression around c = negative Integer:

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this is a special case of the next integral below (m = -1 / 2).

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( Z stands for J or Y; in case a = n π, the sum is zero );

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in the following expressions (∫ f(x)/(a x^2 + b x + c ) dx)   we abbreviate s = DefInt_424.gif:

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the values at integer n can be found approximately by setting n near to an integer DefInt_428.gif.

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in the following expressions (∫ f(x)/(a x^4 + b x^2 + c ) dx)   we abbreviate s = DefInt_453.gif:

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Master formula of Boros and Moll:

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Here the result is a threefold sum shown in Mathematica syntax:
KSubsets[aList, k] is in Package DiscreteMath`Combinatorica` and gives a list of all subsets with k elements of aList .
For n=3 the sum is DefInt_661.gif.
<<DiscreteMath`Combinatorica`

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