1943/01/01 by Georg Pólya · 3 citations
Mathematics · #Iterative Methods for Nonlinear Equations
paper · pdf · doi:10.1090/s0002-9904-1943-07853-6
openalex publication_date 1943/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/07
Introduction. How do the zeros of the nth derivative (n) (x) behave when n becomes very large? How does this behavior depend on the analytic nature of the function f(x) ? At first sight this question appears to be a little out of the way. In fact it is intimately connected with essential parts of the theory of functions. In the Hadamard theory of the singularities of the Taylor series we consider the sequence of the derivatives (x), '(#), "(#), for a fixed complex value of x; in the theory of quasi-analytic functions we consider the same sequence for variable real x, and we consider especially the maximum absolute value of each of its terms in a given interval ; now we consider again the same sequence (#), '(#), "(#)> ' ' ' for variable x, that may be complex or real, and we consider especially the values of x for which its terms vanish. In all three cases the main object of consideration is the connection of the chosen feature of the sequence (#)> '(*)> f n ( x )> ' ' ' with the analytic nature of the function f(x).