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Math 4234 Exam 1 Do all problems

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Math 4234

Exam 1

Do all problems. Justify your answers.

1. Determine and sketch the set of all z? C such that I m(z^-1)<1. Is this set open, closed, or neither? Is it connected? Is it bounded?

2. Suppose lim_z-z0  f(z) = L and lim_z-z0 g(z) = M. Use the € ,8 definition of limit to prove that align* lim’z—z‘0 (af(z) + Øverlineg(z)) align* exists for any constant a€ C.

3. Prove that if z? C and |z| = R>1, then

         Z^m-1/zⁿ+1 < R^m+1/Rⁿ-1

for any positive integers m, n.

4. Determine all entire functions f(z) such that

            Re(f (z)) + 2? I m(f(z)) = 3

for all z€ C.

5. Prove that cos²z + sin²z = 1 and cos²z — sin²z = cos(2z) for all z? C.

6. Let u(x, y) = xy + 3x²y — y³. Find a function ?(x, y) so that f = u + i? is entire.

7. Let f(z) = I/(z⁸ +1—i).

(a) Where is f(z) not defined? (List all points.)

(b) Compute f'(z).