This concept covers the complex power function z^alpha, defined as the multi-valued expression e^(alpha·log z) built from the multi-valued logarithm, and establishes when such powers are single-valued (alpha an integer), finitely multi-valued (alpha a reciprocal of an integer), or infinitely multi-valued. It extends this to the general principle that non-integer powers are discontinuous across a ray from the origin, requiring a branch cut and a choice of branch, with the discrepancy between values approached from either side of the cut captured by a "phase factor." It belongs to complex analysis, specifically the theory of multi-valued functions, branches, and Riemann surfaces, building on the logarithm and square-root branch constructions from earlier in the discipline.
This concept covers the complex power function z^alpha, defined as the multi-valued expression e^(alpha·log z) built from the multi-valued logarithm, and establishes when such powers are single-value…