A punctured disk is an open disk with its center removed. In the complex plane, the punctured disk of radius 
centered at 
is
 |
The punctured unit disk is 
.
Unlike an open disk, a punctured disk is not simply connected. For a counterclockwise circle 
of radius 
centered at 
, with 
,
 |
Consequently, there is no single-valued function that is a holomorphic branch
of the complex natural logarithm of 
throughout the punctured disk (Lebl 2026, pp. 91-94
and 99-101).
Removing a radial segment from the puncture to the boundary instead gives a slit disk. For example, 
is simply connected
and admits a holomorphic branch
of the natural logarithm of 
.
A holomorphic function on 
has a Laurent series
about 
converging throughout that punctured disk. Its isolated
singularity at 
is classified as a removable singularity,
a pole, or an essential
singularity (Lebl 2026, pp. 103-104 and 124-125).
