Math 419 Final Exam, Fall 2009

Math 419 Final Exam, Fall 2009
Due 12/18 by 5 p.m. (to WB 221 or Dan’s mailbox)
This test is worth 100 points. Please indicate your solutions to the following problems as clearly as possible. Please be thorough – I will be awarding
partial credit as appropriate.
Also, as this exam is open-book, open-notes, I expect that every answer
will be fully justified – be sure to explain your solutions thoroughly, citing any
theorems you use and showing that you checked all appropriate conditions.
1. Let f (z) =
(a) (12 points) What are all possible values of the integral of f (z)
around a closed, simple, positively-oriented contour C?
(b) (8 points) For each value in (a), describe which contours will produce it.
2. (20 points) Let SL be the square of sidelength L centered about the origin. For example, S2 has vertices (−1, −1), (−1, 1), (1, −1), (1,
R 1). Suppose f (z) is entire with |f (z)| < |z|12 for all z. Find limL→∞ SL f (z)dz
without relying on Cauchy-Goursat, the Cauchy integral formula, or
3. Let f (z) =
z 2 −5z+6
(a) (2 points) Where are the isolated singularities of f (z)?
(b) (18 points) Write down all Taylor and Laurent series representations for f (z), indicating the domain on which each converges.
(There will be at least one Taylor and at least one Laurent.)
4. Each of the following functions has an isolated singularity at z = 0.
Determine which type of isolated singularity each one is AND
• if it is removable, define f (0) so that f (z) is analytic;
• if it is a pole, state its order and find the residue of f (z) at z = 0;
• if it is essential, decide which value (if any) is neglected from the
range in the neighborhood of z = 0.
(a) (5 points)
(b) (5 points)
(c) (5 points) e z − 5
(d) (5 points)
z 2 +1
5. Suppose u(x, y) is harmonic with harmonic conjugate v(x, y) throughout the real plane. Also suppose that u(x, y) ≤ v(x, y) ≤ 4 throughout
the real plane.
(a) (12 points) Prove that u(x, y) is constant.
(b) (8 points) Notice that u+iv reaches its maximum within any given
domain. Why does this not contradict the maximum modulus