Implications of Stahl's theorems to holomorphic embedding part I: Theoretical convergence

What is known as Stahl's Theorem in power engineering circles is used to justify a convergence guarantee of the Holomorphic Embedding Method (HEM) as it applies to power-flow (PF) problem. In this two-part paper, we examine in more detail the implications of Stahl's theorems to both theoretical and numerical convergence for a wider range of problems to which these theorems are now being applied. We show that the difference between Stahl's extremal domain and the function's domain is responsible for theoretical nonconvergence and that the fundamental cause of numerical nonconvergence is the magnitude of logarithmic capacity of the branch cut, a concept central to understanding nonconvergence. We introduce theorems using the necessary mathematical parlance and then translate the language to show its implications to convergence of nonlinear problems in general and specifically to the PF problem. We show that, among other possibilities, the existence of Chebotarev points, which are embedding specific, are a possible theoretical impediment to convergence. The theoretical foundation of Part I is necessary for understanding the numerical behavior of HEM discussed in Part II.