On the convergence of infinite exponentials
Donald L. Shell · Proceedings of the American Mathematical Society · 1962
Given a sequence of complex numbers {ai}, we define a sequence of functions:a\, ay , ai , • • • .This sequence is formally represented by £(oi, c2, a%, • • • ; z), and is called an infinite exponential.If all the ai = a, as is the case in this paper, the infinite exponential is represented by E(a; z).This nomenclature follows Barrow [l].Another symbolism has been used by Thron [9].It is necessary to specify the values log an to make the sequence determinate.Euler [4] was the first to investigate seriously the convergence of the sequence E(a; z).He stated and demonstrated the results for the case where a and 2 are real.However, his demonstration was not a rigorous proof.Later proofs of these results were given independently by Seidel [8], Gravé [S], and Barrow [l].We quote the result that E(a; 1) converges when e~e ^ a ^ e11".