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Homework answers / question archive / Math 4234 Exam 1 Do all problems

Math 4234 Exam 1 Do all problems

Math

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 limz-z0  f(z) = L and limz-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

         Zm-1/zn+1 < Rm+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).

 

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