A branch cut is a curve (with ends possibly open, closed, or half-open) in the complex plane across which an analyticmultivalued function is discontinuous.
For convenience, branch cuts are often taken as lines or line segments. Branch cuts
(even those consisting of curves) are also known as cut lines (Arfken 1985, p. 397),
slits (Kahan 1987), or branch lines.
For example, consider the function which maps each complex number to a well-defined number . Its inverse function , on the other hand, maps, for
example, the value
to . While a unique principal
value can be chosen for such functions (in this case, the principal
square root is the positive one), the choices cannot be made continuous over
the whole complex plane. Instead, lines of discontinuity
must occur. The most common approach for dealing with these discontinuities is the
adoption of so-called branch cuts. In general, branch cuts are not unique, but are
instead chosen by convention to give simple analytic properties (Kahan 1987). Some
functions have a relatively simple branch cut structure, while branch cuts for other
functions are extremely complicated.