Perlen-Kolloquium: David Masser (Universität Basel)

The unlikelihood of integrability in elementary terms

Functions such as 1/(x(x−λ))^½ can always be integrated (with respect to x) in“elementary terms” involving logarithms and exponentials. But not 1/(x(x−1)(x−λ))^½ unless λ=0,1. A more interesting example is 1/((x−1+λ3)(x(x−1)(x−λ))^½), which can be done also for λ=(1+(−3)^½)/2. In 1981 James Davenport claimed that an arbitrary such algebraic f(x,λ) can be integrated for at most finitely many special complex values λ (unless it can be integrated for a general value of λ). Umberto Zannier and Idare to hope for a full proof in the next couple of years; but for now I will content myself with a general discussion of the problem together with some of the key concepts involved in settling significant special cases.

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