Okay, here are the steps to find the minimum value of P:
1) Given: |z|=|w|=1 (z and w are complex numbers with unit modulus)
|z+w|=sqrt(2)
Find z and w such that these conditions are satisfied.
Possible solutions:
z = 1, w = i (or vice versa)
z = i, w = 1 (or vice versa)
2) Substitute into P = |w-\frac{4}{z}+2(1+\frac{w}{z})i|
For the cases:
z = 1, w = i: P = |-1-4+2(1+i)i| = |-5+2i| = sqrt(25+4) = 5
z = i, w = 1: P = |1-\frac{4}{i}+2(1+\frac{1}{i})i| = |-3+2i| = sqrt(9+4) = 5
3) The minimum value of P is 5.
So in summary, the minimum value of
P = |w-\frac{4}{z}+2(1+\frac{w}{z})i|
is 5.
Let me know if you have any other questions!