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Extra info for A Treatise on Linear Differential Equations, Volume I: Equations with Uniform Coefficients

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If we consider the γc only up to homotopy, then there is a unique way to transport the γc as c varies.

4. Suppose there exists an analytic function z(x) with z(0) = 0, z (0) < 0 such that both F (z(x)) = F (x) and G(z(x)) = G(x) in a neighborhood of x = 0, Then for all elements H of the field F generated by F and G we have H(z(x)) = H(x) considered as meromorphic functions of x about x = 0. Proof. Note first that H(z(x)) = 0 if and only if H(x) = 0. Thus we need only verify that addition, multiplication and inversion of non-zero elements of F preserve this property, which is clearly the case. Recall that L¨ uroth’s Theorem states that if k is a field, any subfield of k(x) which strictly contains k is isomorphic to k(x).

This corresponds to integrals in “closed form” (without quadratures). In an earlier paper, Prelle and Singer [50] showed that all elementary first integrals have integrating factors which are fractional powers of a rational function in x and y. The proof is similar to the Liouvillian case. There have been attempts to implement these results as symbolic integration routines. One indication of how this can be done is given in [40]. Liouvillian integrability is discussed in [3]. There are also some partial results known about Liouvillian integrability for higher order differential equations (see [2] for example).

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A Treatise on Linear Differential Equations, Volume I: Equations with Uniform Coefficients by Thomas Craig

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