numerical DE: Theta method

[eckiller]

Hello,

I am given the method: y_(n+1) = y_n + h f(t_n + w h, (1-w)y_n + w y_(n+1).

I am to determine the region of absolute stability;

I am also to determine for which w in [0, 1] is the method A(a) stable,
i.e., the region of absolute stability contains a sector about the negative
real axis.

I found the root of the characteristic polynomial in the complex plane to
be:

z = [1 + h k (1-w) ] / [1 - h k w]

So RAS = {hk : |z| < 1}

Can I simply that further? I.e., can I get a more explicity formula for
what the hk that satisfy z < 1 is?

Also I am at a loss on how to solve for when the method is A(a) stable.
Please help if you can. Thanks in advance.