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Liouville's Principle

Let F be a differential field with constant field K. For suppose that the equation (i.e. ) has a solution where G is an elementary extension of F having the same constant field K. Then there exist (note, they are in the same field as f, the integrand) and there also exist constants in K (a little sleight-of-hand with the constants, there---necessary extensions have been made to create the field G already) such that

or, in other words,

That is, All You Need is Logs (John Lennon nearly had it right).

Rob Corless
Thu Nov 23 10:59:42 PST 1995