Math 336 𝔉𝔦𝔫𝔞𝔩𝔈𝔵𝔞𝔪, 8:30 am, June 4 , 2012

Name: One notebook-size page of notes is allowed (each side may be used).

1. Suppose that f is an entire function such that

|f(z)| ≤ a|z|^{m} + b

for all z. Then prove that f is a polynomial of degree at most m.

Final 2

2. Is there an analytic function f that maps |z| < 1 into |z| < 1 such that f(\frac{1}{2}) = \frac{3}{4}, f(\frac{1}{4}) = \frac{1}{3}?

Final 3

3. Suppose that u is a harmonic function and v is its conjugate harmonic function. Prove that u^{3} − 3uv^{2} is harmonic.

Final 4

4. Prove that there is no entire function f with f(\frac{i}{n}) = \frac{1}{n+1} for all n. Hint: show that this implies that f(z) = \frac{z}{z+i}.

Final 5

5. (a) Prove that if ∏(1 + a_{k}) converges then ∏ |1 + a_{k}| converges.

(b) Give an example to show that convergence of ∏ |1 + a_{k}| does not imply convegence of ∏(1 + a_{k}).

Final 6

6. Find a function, h(x, y), harmonic in {y > 0}, such that

h(x, y) = \begin{cases}1 & if 0 < x < 3, y = 0, \\ 0 & if x > 3, y = 0, \\ 0 & if x < 0, y = 0 \end{cases}

Is this function unique?

Final 7

7. Let f and g be entire functions so that satisfy f^{2} + g^{2} = 1. Prove that there is an entire function h so that f = cos(h), g = sin(h).

Final 8

8. Does there exist a function f, which is analytic in a neighborhood of 0, for which:

(a) f(\frac{1}{n}) = f(−\frac{1}{n}) = \frac{1}{n^{2}} for all large integers n?

(b) f(\frac{1}{n}) = f(−\frac{1}{n}) = \frac{1}{n^{3}} for all large integers n? In each case, either give an example or prove no such example exists.