Complex functions definitions
A single-valued complex function f assigns each z from a non-empty subset S \subseteq \mathbb{C} to a unique w \in \mathbb{C}:
f: S \to \mathbb{C}, \quad z \mapsto w = f(z)
However, some key complex functions cannot be neatly captured with just a single outcome for each input. The classic example is the complex logarithm: for any non-zero z, there are infinitely many w such that e^w = z.
These are multivalued functions, and to work with them rigorously, mathematicians introduce the notion of a branch: a single-valued function defined on a restricted domain, which consistently selects one value from the multivalued set for each input.
Branches require omitting some subset of the complex plane, known as a branch cut, to avoid discontinuities in the chosen value. The principal branch is a customary choice that makes computations and further theory manageable. For the logarithm, the principal branch is typically defined by restricting the argument \theta so that -\pi < \theta \leq \pi:
\mathrm{Log}\,z = \ln |z| + i \arg(z),\quad -\pi < \arg(z) \leq \pi
Representing complex functions can be done in either Cartesian or polar coordinates. Given z = x + iy, a function f has the form:
f(z) = u(x, y) + iv(x, y)
with u, v real-valued. In polar coordinates, z = r e^{i\theta}, functions take the form:
f(z) = u(r, \theta) + iv(r, \theta)
This flexibility is particularly helpful for functions with rotational symmetry or those defined via argument, such as roots or logarithms.
Moreover, the analyticity of f is determined by the Cauchy-Riemann equations for u and v, connecting these real-valued functions back to the core of complex analysis.
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