The solutions are ordered according to their imaginary parts.
For , ProductLog[
z
] is real.
ProductLog[
z
] satisfies the differential equation .
ProductLog[
z
] has a branch cut discontinuity in the complex z plane running from to . ProductLog[
k
,
z
] for integer has a branch cut discontinuity from to .
in 
. 
solution. 
, ProductLog[
z
] is real. 
. 
to 
. ProductLog[
k
,
z
] for integer 
has a branch cut discontinuity from 
to 
. 