Math 617 Final Examination December 16, 2003
Instructions In each item, either give a concrete example satisfying the
conditions or prove that no example exists (whichever is appropriate). Work
as many of the 16 items as you wish.
Your score for ncorrect items is 10min(n, 8) + 5 max(n−8,0); that is,
your first 8 correct items count 10 points each, and additional correct items
count 5 points each. (More than 12 correct items will give you extra credit:
namely, a score greater than 100.)
1. A non-empty open set Uand a holomorphic function fon Usuch that
there does not exist a holomorphic function gon Uwith the property
that the derivative g0=f.
2. A non-empty open set U, a holomorphic function fon U, and a simple
closed curve γin Usuch that Rγf(z)dz =√2.
3. A holomorphic function fon the unit disc {z∈C:|z|<1}such that
the nth derivative f(n)(0) = (n!)2for every positive integer n.
4. A non-polynomial entire function with exactly one zero.
5. A holomorphic function fon the disc {z∈C:|z|<2}such that
f(1/n)=(−1)n/n for every positive integer n.
6. A holomorphic function fon the punctured disc {z∈C:0<|z|<2}
such that R|z|=1 f(z)dz = 0, and fhas an essential singularity at the
origin.
7. A rational function fhaving a pole at 0 such that the residue of fat 0
equals 2 and the residue of the derivative f0at 0 equals 1.
8. A positive real number asuch that Z∞
−∞
1
x4+a4dx =1.
Theory of Functions of a Complex Variable I Dr. Boas
