A branch point of an analytic function is a point in the complex plane whose complex argument can be mapped from a single point in the domain to multiple points in the range. For example, consider the behavior of the point 
under the power function
 | (1) |
for complex non-integer 
, i.e., 
with 
. Writing 
and taking 
from 0 to 
gives
 |  |  | (2) |
 |  |  | (3) |
so the values of 
at 
and 
are different, despite the fact that they correspond to the same point in the domain.
Branch points whose neighborhood of values wrap around the range a finite number of times 
as 
varies from 0 to 
correspond to the point 
under functions of the form 
and are called algebraic branch points of order 
. A branch point whose neighborhood of values wraps around an infinite number of times occurs at the point 
under the function 
and is called a logarithmic branch point. Logarithmic branch points are equivalent to logarithmic singularities.
Pinch points are also called branch points.
It should be noted that the endpoints of branch cuts are not necessarily branch points.
