Math 609 Jerry L. Kazdan
April 22, 2004 1:30-2:50
Exam 2
Directions This exam has two parts, Part A is short answer (35 points) while Part B has
traditional problems (60 points). All contour integrals are assumed to be in the positive sense
(counterclockwise).
Short Answer Problems [ 5 points each] (35 points total)
For A1–A5 let f(z) be holomorphic for 0 <|z|<∞. What can you say about f(z) if you are told
the following? Briefly justify your assertions.
A1. |z2f(z)|<5.
A2. |f(z)| → ∞ as |z| → 0.
A3. f(1
n) = 1 + (−1)n,n= 1,2,....
A4. |f(z)| ≤ |z|+ 1 and f(1
n) = 0, n= 1,2,....
A5. |f(z)| ≤ |f(3)|for |z−3|<2.
A6. Evaluate 1
2πi I|z−1|=2
e2z
z2dz .
A7. Describe the singularities of ϕ(z) := 1−cos(z5)
sin3zat z= 0 and at z=π.
Traditional Problems [10 points each] (60 points total)
B1. Let g(z) be holomorphic in the disk {|z| ≤ 3}with |g(z)| ≤ 7 on the circle {|z|= 3}. Find
some explicit upper bound for |g′(z)|in the disk {|z| ≤ 1}.
B2. Let f(z) = u+iv be holomorphic at z0=x0+iy0and f′(z0)6= 0 . Show that the level curves
of uand vthrough z0intersect orthogonally.
[You may use without (the simple) proof that if
i). h(x, y) = const is a level curve of the smooth real-valued
function h(x, y) and
ii). the gradient ∇h(x0, y0)6= 0 at a point on this curve,
then ∇h(x0, y0) is orthogonal to the tangent line of hat (x0, y0).]
h(x,y) = const.
grad h
1
