A logarithmic branch point is a branch point around which repeated analytic continuation produces
infinitely many distinct branches. For 
, continuing a chosen branch counterclockwise
around zero adds 
on each turn, giving the values
 |
where 
,
is an initial complex argument of 
and 
is an integer. Thus, zero is a
logarithmic branch point, whereas an algebraic
branch point has a finite cycle of branches.
The Riemann surface of 
can be parametrized by 
, 
, with 
. The projection 
is the universal cover
of the punctured plane, and changing 
to 
moves between its sheets (Honda n.d., p. 70; McMullen
2026, p. 35).
Logarithmic branching concerns analytic continuation, while a logarithmic singularity describes
local asymptotic behavior. For example, 
has a logarithmic branch point at zero, since each turn
changes it by 
,
but it tends to zero along any fixed ray on a chosen branch.
