problem4) Prove Rouche’s Theorem (original)
Suppose f,g are both analytic inside a simple closed contour c if ∀ z∈c, |f(z) |<|g(z) | then f,g have the same number of zeros inside contourc, counting multiplicities.
problem5)Prove that if f is a 1-1 analytic function in some domain, then f'(z)≠0 anywhere in D.
problem6) Show that if f:R→R is such thatf' (x)≠0 ∀x∈R, then must be 1-1 function.
problem7) Show f:R2→R2 given by f(x,y)=(ex cosy, ex sin y) satisfies:
The Jacobian of f is nonsingular of all (x,y).
f is not 1-1.