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Liouville's Principle


Let F be a differential field with constant field K. For f in F, suppose that the equation g^'=f (i.e., g=intf) has a solution g in G, where G is an elementary extension of F having the same constant field K. Then there exist nu_0, nu_1, ..., nu_m in F and constants c_1, ..., c_m in K such that

 f=nu_0^'+sum_(i=1)^mc_i(nu_i^')/(nu_i),

In other words, such that

 intf=nu_0+sum_(i=1)^mc_ilnnu_i.

See also

Elementary Function, Risch Algorithm

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References

Geddes, K. O.; Czapor, S. R.; and Labahn, G. "Liouville's Principle." §12.4 in Algorithms for Computer Algebra. Amsterdam, Netherlands: Kluwer, pp. 523-529, 1992.

Referenced on Wolfram|Alpha

Liouville's Principle

Cite this as:

Weisstein, Eric W. "Liouville's Principle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LiouvillesPrinciple.html

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