Integration in finite terms: a method for determining regular fields for the Risch algorithm
The Risch procedure for deciding if a purely transcendental elementary function y(x) has an elementary integral requires, as part of the initial data, that one be given a tower of fields \(C(x)=F_ 0\subset F_ 1\subset...\subset F_ n\) such that y is in \(F_ n\) and each \(F_ i\) is a regular extension of \(F_{i-1}\). In contrast to this, when one wishes to integrate an elementary function, one is usually presented with a formula for y(x). To apply the Risch algorithm one must decide if y(x) lies in such a tower and if so construct one. It is easy to construct some elementary tower containing y, but this tower may not be regular. Therefore there are two questions to be asked: 1) Can this tower be fixed, i.e. made regular, and 2) If not, is there some other way to unravel y to construct a regular tower? The author attempts to answer the first question, but there is a serious mistake that invalidates his conclusions. The algorithm proposed in Theorem 3 yields incorrect results when applied to \({\mathbb{C}}(x, \exp(x), \log(x \exp(x)), \exp((1/2)\log(x\exp(x))))\). This stems from the incorrect statement on p. 140, 1.9-10: ``However, the last equation above shows that u cannot depend on \(\theta_{i-1},... ,\theta_{k+1},...\)'' In fact, the equation only shows that \underbar{u'} cannot depend on these quantities. The construction of \(\theta_ 0\) is therefore invalid. The reviewer has communicated with the author who agrees with this analysis.
scientific article; zbMATH DE number 3931147