208ss216-ex2 - Math 208, Summer II, 2016, Exam # 2 July 28,...

Math 208, Summer II, 2016, Exam # 2
July 28, 2016
1. State the Cauchy Integral Formula, being careful to include the conditions on the func-
tion f ( z ) . (10 points) Let f ( z ) be an entire function so that there is a positive number
A for which | f ( z ) | ≤ A | z | for all z ∈ C with | z | > 1 . Show that f ( z ) = az + b for some
complex numbers a and b . (10 points)
2. Find the following contour integrals. (20 points)
(a)
Z
C
e
z
( z - 1) 2
. where C is the circle | z | = 2 .
(b)
R
C
sin( z ) dz, where C is the unit circle | z | = 1 , with the positive orientation.
(c)
Z
C
z dz
z 2 - 1
, where C is the circle | z - 1 | = 1 . d the Taylor series, centered at the point
indicated, of the following functions