problem1) Z0 is removable if and only if limit from z to z0|f(z)| exists.
problem2) find out the integral:
problem3) Prove
1)
dx =π/(sin?(aπ)) ,0
2)
dx=π/(n sin?(aπ)),0
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:R^{2}→R^{2} given by f(x,y)=(e^{x} cosy, e^{x} sin y) satisfies:
The Jacobian of f is nonsingular of all (x,y).
f is not 1-1.