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problem1) Z0 is removable if and only if limit from z to z0|f(z)| exists.

problem2) find out the integral: 1065_integration_3.jpg

problem3) Prove 

1)

    2490_integration_2.jpg     dx =π/(sin?(aπ))            ,0

2)

1985_integration_1.jpg  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: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.

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