Every polynomial p has the property that p n p n 1 1

(Every polynomial P has the property that
P ( n )
P ( n +1)
→ 1 since if the leading
coefficient is c
k
n
k
,
P ( n )
P ( n +1)
≈
cn
k
c ( n +1) k
= (1 -
1
n +1
)
k
→ 1.)
Exercise 15: Suppose f is a non-vanishing continuous function on
¯
D that is
holomorphic in D . Prove that if
| f ( z ) | = 1 whenever | z | = 1 ,
then f is constant.

11
Chapter 2.7, Page 67
Problem 3: Morera’s theorem states that if f is continuous on C , and
R
T
f ( z ) dz =
0 for all triangles T , then f is holomorphic in C . Naturally, we may ask if
the conclusion still holds if we replace triangles by other sets.
(a) Suppose that f is continuous on C , and
Z
C
f ( z ) dz = 0
or every circle C . Prove that f is holomorphic.
(b) More generally, let Γ be any toy contour, and F the collection of all
translates and dilates of Γ. Show that if f is continuous on C , and
Z
γ
f ( z ) dz = 0 for all γ ∈ F
then f is holomorphic. In particular, Morera’s theorem holds under
the weaker assumption that
R
T
f ( z ) dz = 0 for all equilateral triangles.
Chapter 3.8, Page 103
Exercise 1: Using Euler’s formula
sin πz =
e
iπz
- e
- iπz
2 i
,
show that the complex zeros of sin πz are exactly at the integers, and that
they are each of order 1.
Calculate the residue of 1 / sin πz at z = n ∈ Z .