Multiplicity and Branching in Complex Analysis 📂Complex Anaylsis

Multiplicity and Branching in Complex Analysis

Definition ¹

A mapping that associates the elements of $X = \mathbb{C}$ with multiple values of $Y$ is called a multifunction.
For a multifunction $g$ defined in an open set $A \subset \mathbb{C}$, if there exists at least one closed curve $\mathscr{C}$ that wraps $\alpha \in \mathbb{C}$ and lies within $A$, along which $z-\alpha$ changes by $2\pi$ continuously, and the value $g(z)$ is not the same as the original value, then $\alpha$ is called a branch point.
All neighborhoods of $\alpha$ do not contain another branch point, and the single line segment starting from the branch point $\alpha$ is called a branch cut.
Any function derived from $g$ that has only one value everywhere except on the branch cut is called a branch of $g$.

Explanation

A multifunction is a function that has multiple values. Strictly speaking, it is not a function.

Example

For example, in complex analysis, the logarithmic function is set as $\text{Log} z := \log_{\mathbb{R}} |z| + i \arg z$ and defined at all points except for $0 \in \mathbb{C}$. Here, $\log_{\mathbb{R}}$ is the logarithmic function as we originally know it, and the argument $\arg$ refers to the angle of rotation counterclockwise from the positive real axis. According to this definition, function $\text{Log}$ becomes a branch of the multifunction $\log$.

In this case, $\log$ has infinitely many function values for a given $z$ because of $\arg z = 2 n \pi + \theta_{0}$ for some $- \pi \le \theta_{0} <\pi$ and an integer $ n$. Here, the imaginary part changes every $2 \pi$ based on the semi-infinite line $\left\{ b + i0: b \in \left( -\infty , 0 \right] \right\}$, and this axis is called a branch cut. Normally, since this characteristic is unnecessary, it is limited to $n=0$, which is called the principal branch. In such cases, the argument is limited to $-\pi < \theta \le \pi$, and the representation differentiates between uppercase and lowercase as shown in $\text{Arg}$.

Furthermore, looking closely at the definition of the logarithm, one can understand that the line where the value jumps every $2 \pi$ units does not necessarily have to be $\left( -\infty , 0 \right]$. If there’s a need for a different definition or simply a desire to define it differently, it can be newly defined in any direction. However, it’s important to include the origin $O$ in any of such possibilities. This point shared by all branch cuts is called a branch point.

Just as with the classification of singularities, one might think this is just forcefully defining something that cannot be a function and playing word games. However, since the function given as an example is the logarithm, dealing with such multifunctions is a very serious matter. Without a clear understanding of branches in the complex plane, one enters a hell of feeling like they understand but not knowing for sure. It is best to study thoroughly at once without glossing over too vaguely.