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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/z^{n}+1 __< __R^{m}+1/R^{n}-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 cos^{2}z + sin^{2}z = 1 and cos^{2}z — sin^{2}z = cos(2z) for all z? C.

6. Let u(x, y) = xy + 3x^{2}y — y^{3}. Find a function ?(x, y) so that f = u + i? is entire.

7. Let f(z) = I/(z^{8} +1—i).

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

(b) Compute f'(z).

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