ABSTRACT
The power with positive integer order n, is an example (Section 5.1) of a single-valued function, since to each z corresponds only one zn; the corresponding points for successive n lie on a logarithmic spiral (Section 5.2). The inverse function (Section 5.3), namely the root of order n, is n-valued, since for each z there are n distinct possible values of n
√ z;
the corresponding points lie at the vertices of a regular polygon (Section 5.4) with n sides, leading to a number of trigonometric identities (Section 5.5). There are also many-valued functions, like the logarithm (Section 5.6), such that to each value of z corresponds an infinite number of values of log z. The function logarithm appears in the definition (Section 5.7) of za complex power z with complex exponent a that generalizes both the ordinary nth power and nth root that correspond respectively to a= n and a= 1/n. It can be shown that the behavior of the complex power za at the origin z → 0 and infinity z → ∞ depends only on the modulus of the base and (Section 5.8) on the real part of the exponent, and not on the imaginary part; this is a particular case of the exponential of a power having different asymptotic behavior as z → ∞ in different angular sectors (Section 5.9). Selecting of the “correct” branch of a function, and taking into account “jump conditions” between branches, may be necessary to arrive at correct results; it is also an indication that the problem in question has several solutions that may or may not be equivalent for a specific purpose.
