Branch Cuts
Roughly speaking, branch points are the points where the various sheets of a multiple valued function come together. The branches of the function are the various sheets of the function. For example, the function w = z1/2 has two branches: one where the square root comes in with a plus sign, and the other with a minus sign. A branch cut is a curve in the complex plane such that it is possible to define a single branch of a multi-valued function. Branch cuts are usually, but not always, taken between pairs of branch points.
Branch cuts allow one to work with a collection of single-valued functions, "glued" together along the branch cut instead of a multivalued function. For example, to make the function
single-valued, one makes a branch cut along the interval on the real axis, connecting the two branch points of the function. The same idea can be applied to the function √z; but in that case one has to perceive that the point at infinity is the appropriate 'other' branch point to connect to from 0, for example along the whole negative real axis.
The branch cut device may appear arbitrary (and it is); but it is very useful, for example in the theory of special functions. An invariant explanation of the branch phenomenon is developed in Riemann surface theory (of which it is historically the origin), and more generally in the ramification and monodromy theory of algebraic functions and differential equations.
Read more about this topic: Branch Point
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