To verify the radius of convergence of the series
(4.13.6) map the plane of onto the plane of
via , where
. Then is
analytic at , and its nearest singularities to the origin are
located at .
Figure 4.13.1 was produced at NIST.
: nonprincipal branch of Lambert -function and
: principal branch of Lambert -function
On the -interval there is one real solution, and it is
nonnegative and increasing. On the -interval there are two real
solutions, one increasing and the other decreasing. We call the solution for
which the principal branch and
denote it by . The other solution is denoted by
. See Figure 4.13.1.