Multifunctions and Branches in Complex Analysis 📂Complex Anaylsis

Multifunctions and Branches in Complex Analysis

Definition ¹

A mapping that sends an element of $X = \mathbb{C}$ to several values of $Y$ is called a multifunction.
For a multifunction $g$ defined on an open set $A \subset \mathbb{C}$, if there exists at least one closed curve $\mathscr{C}$ that encloses $\alpha \in \mathbb{C}$ and lies within $A$, such that as $z-\alpha$ keeps changing by $2\pi$ along $\mathscr{C}$ the value $g(z)$ ends up not being its original value, then $\alpha$ is called a branch point.
A single line segment starting from a branch point $\alpha$, where no neighborhood of $\alpha$ contains another branch point, is called a branch cut.
Any function built from $g$ that takes only a single value everywhere except on the branch cut is called a branch of $g$.

Explanation

A multifunction is a function that takes multiple values, and strictly speaking it is not a function.

Example

As an example, in complex analysis the logarithmic function is set as $\text{Log} z := \log_{\mathbb{R}} |z| + i \arg z$ and defined at every point except $0 \in \mathbb{C}$. Here $\log_{\mathbb{R}}$ is the logarithm we originally knew, and the argument $\arg$ refers to the rotation angle in the counterclockwise direction with respect to the positive real axis. By this definition, the function $\text{Log}$ becomes a branch of the multifunction $\log$.

In this case, since $\arg z = 2 n \pi + \theta_{0}$ for some $- \pi \le \theta_{0} <\pi$ and an integer $ n$, $\log$ takes infinitely many function values for a given $z$. Here the value of the imaginary part changes every $2 \pi$ with respect to the ray $\left\{ b + i0: b \in \left( -\infty , 0 \right] \right\}$, and such an axis is called a branch cut. Usually this property is not needed, so it is restricted to $n=0$, which is called the principal branch. In such a case the argument is restricted to $-\pi < \theta \le \pi$, and, distinguishing upper and lower case from the previous notation, it is written as $\text{Arg}$.

On the other hand, if one looks closely at this definition of the logarithm, one can see that the line along which the value jumps in units of $2 \pi$ need not necessarily be $\left( -\infty , 0 \right]$. If there is a need to define it differently, or if one simply wants to, one may newly define it in any direction. However, whatever such possibility one considers, it turns out that the origin $O$ must always be included. The point shared by all such branch cuts in this way is called a branch point.

Just as with the classification of singularities, one might get the thought that we are forcibly defining something that cannot even become a function in the first place and merely playing word games. However, since the very function taken as an example is the logarithm, handling such multifunctions is a rather serious matter, and as long as one deals with the complex plane, if the concept of branches is not clear, a hell of feeling like you know it but not knowing it exactly continues. As far as possible, let us not gloss over it vaguely but study it properly all at once.