Remarks on Analytic Continuation

The collection Jt of multi-valued analytic functions with three singular points z = a, b,c on the Riemann sphere ..., wn) ^ 0 with complex coefficients which vanishes identically when w0 is replaced by w(z) and the Wj are replaced by d w/dz. According to a theorem of Holder, the gamma function has this property, but T(z) is single-valued and has poles at the negative integers. Our result is obtained by supplementing the reasoning employed in [4] with that of Ritt and Gourin [3]. Moreover, replacing the Golod-Shafarevitch group utilized in [4] by interesting two-generator groups permits us to restrict our attention to the collection of functions with three singular points and to make a few additional observations about the branching behaviour of some of its members. Finally, we prove a theorem about the symmetric group on a countable set which yields yet another interesting function in our class Ji. Throughout this paper we replace the phrase "homogeneous linear differential equation with single-valued analytic coefficients with singularities at z = ah i = 1, . . . ,r + l, r a positive integer", by the term "linear differential equation". Now, suppose that G is an infinite group generated by r elements; of course, G is countable. Selecting r + 1 arbitrary points z = aif i = l , . . . , r + l on

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