Algebraic values of analytic functions

Algebraic values of analytic functions

Given an analytic function of one complex variable f , we investigate the arithmetic nature of the values of f at algebraic points. A typical question is whether f(α) is a transcendental number for each algebraic number α. Since there exist transcendental entire functions f such that f (t)(α) ∈ Q[α] for any t ≥ 0 and any algebraic number α, one needs to restrict the situation by adding hypothes...

Analytic Functions and Boundary Values

Analytic functions in tubes T B = R+iB in C, where B 1⁄2 R, that generalize the Hardy H spaces are studied. The associated function M¤ of Komatsu, which is de ̄ned with the aid of certain sequences of positive real numbers, is involved in the generalizing bound of the L norm. Fourier-Laplace and Cauchy integral representations are obtained for these functions. These representations can be used t...

On the values of a class of analytic functions at algebraic points

We introduce a class of analytic functions of number theoretic interest, namely stammering functions. It has been recently proved that these functions take transcendental values at some algebraic points. In the present paper, we establish a general transcendence criterion that extends these results. Another aim is to underline the main difficulties arising from the use of the Schmidt Subspace T...

Boundary Values of Continuous Analytic Functions

Let U, K, and C denote the open unit disc, the closed unit disc, and the unit circumference, respectively. Let A be the set of all complex-valued functions which are defined and continuous on K and analytic in U. The theorem proved in this paper states that if £ is a closed subset of C, of (Lebesgue) measure zero, and if <p is a continuous function defined on £, then there exists a function /£....

Algebraic Independence of Values of Elliptic Functions