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In this paper we investigate the relationship between the degree of best rational approximation to a given function f(z) and the regions in which f(z) is meromorphic. We show, for example, that if rational functions r.,(z) of respective types (n, ;.), i.e., rational functions with v free poles, converge geometrically (as n-* oo) to f(z) on a closed Jordan region E, then f(z) must be merornorphic in a region which contains E in its interior.

Regions of meromorphy determined by the degree of best rational approximation