For a fixed complex number, show that if
and
are large integers, then the equation
has exactly
solutions in the horizontal strip
.
The hint says to considerand sketch the image of the boundary of
. Keep in mind
.
I do not see how to prove this now. I need some help on this one. Thank you.
I'd like to try and tackle this if I may although not via Rouche': First, it's easy to solve for the roots ofin terms of the Lambert-W function:
and keep in mind the W-function is infinitely valued and looks a bit like the log multi-sheet so we should actually write it as:
whereand the zeros
as the black dots. As you can see, the contour is enclosing three zeros. The lower-left is the image of
over that contour. On the brown contour we have:
and if
is large, the expression is dominated by
giving rise to the circular contours around the origin but note the remaining contours complete the circuit around the origin giving rise to a winding number of
. This is primarily due I think to the blue contour over which we have
and since the
term is small, the real part is dominated by
and if this is positive, then the circular contour complete the circuit around the origin
times as shown by the lower left plot. However, in the case of
and
as the second set shows, then
which causes a detour around the origin and gives rise to a winding number of
. This is reflected in the top right plot showing only two zeros enclosed by the contour . . . I'm thinkin' B at best.
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