A branch point of a multiple-valued function is a point around which analytic continuation
of a local branch along a sufficiently small closed
curve encircling that point can return a different value. A single-valued analytic function on a full neighborhood
does not have a branch point there. 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.
For an irreducible fraction 
with 
, continuation of 
around the origin multiplies
its value by 
.
The original branch returns after 
turns, giving an algebraic
branch point of order 
.
For the logarithm 
, each turn adds 
, so there are infinitely many branches.
The origin is a logarithmic
branch point, also called a logarithmic
singularity.
Pinch points are also called branch points.
It should be noted that the endpoints of branch cuts are not necessarily branch points.
