Incomplete rational approximation in the complex plane

Abstract

We consider rational approximations of the form $$\left\{ {(1 + z)^{\alpha n + 1} \frac{{p_{cn} (z)}}{{q_n (z)}}} \right\}$$ in certain natural regions in the complex plane wherep cn andq n are polynomials of degreecn andn, respectively. In particular we construct natural maximal regions (as a function of α andc) where the collection of such rational functions is dense in the analytical functions. So from this point of view we have rather complete analog theorems to the results concerning incomplete polynomials on an interval. The analysis depends on an examination of the zeros and poles of the Pade approximants to (1+z)αn+1. This is effected by an asymptotic analysis of certain integrals. In this sense it mirrors the well-known results of Saff and Varga on the zeros and poles of the Pade approximant to exp. Results that, in large measure, we recover as a limiting case. In order to make the asymptotic analysis as painless as possible we prove a fairly general result on the behavior, inn, of integrals of the form $$\int_0^1 {[t(1 - t)f_z (t)]^n {\text{ }}dt,}$$ wheref z (t) is analytic inz and a polynomial int. From this we can and do analyze automatically (by computer) the limit curves and regions that we need.