Infinite radicals in the complex plane
Georgellen Schuske, W. J. Thron · Proceedings of the American Mathematical Society · 1961
is called an infinite radical. The numbers aj are the elements of the infinite radical, and the uj are called partial radicals. Throughout this paper a'12 will mean that square root whose real part is positive or zero. If a is a negative real number, a/2 will be defined to be on the positive imaginary axis. A set which is open and connected is called an open region, and an open region with part or all of its boundary is called a region. A region A is called a convergence region for an infinite radical {u2 n} if the assumption that all the elements aj lie in A is sufficient for the convergence of { u2 } . A region A is called a conditional convergence region if the assumption that all the elements aj lie in A, together with another condition which restricts the rapidity of growth of J a, J,insures the convergence of { u2. It is not difficult to prove that a necessary and sufficient condition for a sequence { Sn} of real numbers to be an infinite radical { un } is that { Sn } be a monotonically nondecreasing sequence of non-negative numbers. However, if {s.} is a sequence of complex numbers, the situation is not so simple. If Pn is the region from which Sn may be chosen, Pn CPn_lx for every n, but each Pn depends on the entire set of numbers si, S2, * * *, Sn-1, and therefore necessary and sufficient conditions are rather involved and not very meaningful geometrically. Vieta was probably the first to conceive of the idea of an infinite radical [1, p. 595]. The famous infinite product