MATH 116PROBLEM SET VIII
JUN LI
This set is due 5PM Wednesday, March 12, to oce 380-383Z
a. Prove that if cfw_fn is a sequence of injective holomorphic functions in such
that it converges uniformly on every compact subset of to an f : C, then
either f is
HW 5
Problem A: Write out complete statements for six theorems that have
been proven in the course. Pick the six theorems you think are most
important.
Problem B: Consider the entire function f (z) = ez . Show that f (z) =
w has infinitely many solutions
Problem A: Prove the second and fourth statements (the product rule and the chain rule) of Proposition 2.2 on page 10.
Problem B: Any 2 by 2 matrix gives a linear transformation from R^2 to itself. Identifying R^2 with the complex plane, we get a map fro
HW 8
Problem A: Suppose k 0 is an integer, and an (n = 1, 2, 3, . . .) are
nonzero complex numbers such that
X
1
< .
|an |k+1
n=1
Prove that
Y
Ek (z/an )
n=1
converges and defines an entire function that vanishes exactly at the
an .
Problem B: Prove that
HW 6
Problem A: Formulate and prove a version of lHopitals rule for
(z)
, where
holomorphic functions. Your result should relate limzz0 fg(z)
f (z0 ) = 0 = g(z0 ) and f and g are holomorphic functions, to limzz0
f 0 (z)
.
g 0 (z)
Exercises starting on pag
HW 9
Problem A: Show that if a sequence of continuous functions fn on a
compact metric space converges uniformly to a continuous function f ,
then F = cfw_fn is an equicontinuous family.
Problem B: Prove C is not biholomorphic to the disk.
Problem C: Sup
HW 4
Problem A: Let K C be compact. Suppose that A0 is the set of all
polynomials, and suppose A1 , . . . , An are sets of functions from K to
C. Let f be a function from K to C. Suppose that f can be uniformly
approximated by polynomials in functions in
Homework 5
Math 116
Problem 1 :
Write out complete statements for six theorems that have been proven in the course. Pick the six theorems you think
are most important.
Proof. Any six theorems from the book.
Problem 2 :
Consider the entire function f (z) =
Homework 8
Math 116
Problem 1 :
Suppose k 0 is an integer, and an (n = 1, 2, 3, . . .) are nonzero complex numbers such that
X
1
< .
k+1
|a
n|
n=1
Prove that
Y
Ek (z/an )
n=1
converges and defines an entire function that vanishes exactly at the an .
Proof
Homework 4
Math 116
Problem 1 :
Let K C be compact. Suppose that A0 is the set of all polynomials, and suppose A1 , ., An are sets of functions
from K to C. Let f be a function from K to C. Suppose that f can be uniformly approximated by polynomials in
fu
Homework 3
Math 116
Problem 1 :
Suppose that f (z) and g(z) are entire holomorphic functions such that g(z) never vanishes. If |f (z)| |g(z)| for all
z, prove that f is a constant multiple of g.
Proof. Since g(z) never vanishes
functions, then so is
f (z)
Homework 1
Math 116
Problem 1 :
If f and g are holomorphic in , then
a) f g is holomorphic in and (f g)0 = f 0 g + f g 0 .
b) f g is holomorphic in and (f g)0 = f 0 (g(z)g 0 (z)
Proof. a) Note that limit of a sum or difference of two complex-valued functi
Homework 7
Math 116
Problem 1 :
Give another proof of Jensens formula in the unit disc using the functions (called Blaschke factors)
(z) =
z
.
1
z
Proof. Let f be holomorphic on the open unit disk and let z1 , ., zn be the set of zeros of f inside the un
Homework 6
Math 116
Problem 1 :
Formulate and prove a version of lHopitals rule for holomorphic functions.
Proof. Suppose that f and g are holomorphic on , f (z0 ) = g(z0 ) = 0 for some z0 and f and g dont vanish on
a nbhd around z0 . Suppose also that g
Homework 9
Math 116
Problem 1 :
Show that if a sequence of continuous functions fn on a compact metric space converges uniformly to a continuous
function f , then F = cfw_fn is an equicontinuous family.
Proof. Note that since K is compact then fn and f a
SOLUTION SEVEN
a. Since f is an automorphism of the unit disk, it must be of the form:
z
f (z ) = ei
,
1 z
where D. To decide the parameters, we need the condition: f (0) =
f (1) = i, which means:
1
1
ei
= , ei
= i.
1
2
1
By these two equations, we decide
SOLUTION 1
1
2k
1.4.3. For s > 0, the solutions are: s n ei( n + n ) ; for 1 k n; So there are n
solutions for s > 0. For s = 0, there is only one solution.
1.4.6(a). Cz is open and connected: since is open, choose one point x Cz , we
can nd an open disc
SOLUTION 3
1. Consider the function f (z ) = n=0 an xn , which converges for |x| < 1. So f
is a holomorphic function on the unit disc. Similarly, so is the function g (z ) =
1
1
n
n=0 bn x . Notice f ( n ) = g ( n ), so for n , we get: f (0) = g (0) = a0
SOLUTION 4
a. Consider < 1 small enough, such that: D (z ) D1 (0) and does not intersect
D (1). Now consider the area S = D1 (0) D (z ) D (0). Therefore, the function:
( )
g ( ) = f , is holomorphic in S . By Cauchys Theorem: S g ( )d = 0. And it is
z
eas
MATH 116PROBLEM SET VII
JUN LI
This set is due 5PM Wednesday, March 5, to oce 380-383Z
Page 248. 1, 3, 4, 5, 10, 11, 12, 14
a. Find an automorphism f of the unit disk D so that f (0) =
many such automorphisms are there?
1
2
and f (1) = i. How
b. Find an e
MATH 116PROBLEM SET VI
JUN LI
This set is due 5PM Wednesday, Feb. 26, to oce 380-383Z
Page 154. 6, 9.
a. Apply Rouches theorem to prove the Fundamental Theorem of Algebra.
b. Prove that sin z = z
z2
.
n2
1
n=1
c. Let D = D1 (0) be the unit disk and D = D
MATH 116PROBLEM SET II
JUN LI
This set is due 5PM Wednesday, Jan. 22, to oce 380-383Z
a. Compute the following complex line integral using Cauchy integral formula:
2 +
d,
( 2i)( + 4)
when
(1) is the circle |z | = 1 counterclockwise orientated;
(2) is the
MATH 116PROBLEM SET III
JUN LI
This set is due 5PM Wednesday, Jan. 29, to oce 380-383Z
1. Let n=0 an xn and n=0 bn xn be real power series which converge for |x| < 1.
Suppose that n=0 an xn = n=0 bn xn for x = 1/2, 1/3, , 1/m, . Prove that
an = bn for all
MATH 116PROBLEM SET V
JUN LI
This set is due 5PM Wednesday, Feb. 19, to oce 380-383Z
Page 105. 10, 14*, 17, 22.
page 108. 3,
a. Evaluate the integral
0
x
dx.
1 + x2
b. Let f be an entire function. Suppose f has a pole at . Show that f is a
polynomial.
c.
MATH 116PROBLEM SET I
JUN LI
This set is due 5PM Wednesday, Nov. 15, to oce 380-383Z
1.4. 3, 6, 8, 12, 13, 16ace, 19.
a. Let f : C be holomorphic. Let z (t) be a smooth arc. Prove that
d
f (z (t) = f (z (t) z (t).
dt
n
Use this to prove ex (cos y + i sin
SOLUTION 2
2
+
a.1. Set f (z ) = ( i)(+4) , Then f is holomorphic in a neighbourhood of B (0, 1),
2
so by Cauchy theorem, the integral =0.
a.2. Since f is singular inside only at z = 2i, the integral = 2iRes(f, 2i) =
2 +
2i (+4) =2i = 1.6 1.2i.
a.3. Sinc
SOLUTION FIVE
log x
. Consider the small semicircle centered at 0 with
x2 + a2
radius r and large semicircle centered at 0 with radius R > a. By Cauchy
Theorem,
3.8.10. Set f (x) =
r
(1)
R
)f (x)dx +
+
(
R
r
We calculate
2iResai (f ) = 2i
As for
log ai
=
SOLUTION SIX
5.6.6. By the product formula for sin z :
(1
sin z =
n=1
choose z =
1
2
z2
),
n2
:
1
),
4 n2
(1
1=
n=1
Now, we see that:
N
1
=
N
n=1 (1
4n2
,
4n2 1
n=1
1
4n2 )
Set N , we see the limit of the right hand side exists, and equals:
N
2n 2n
,
(
SOLUTION EIGHT
a. Assume that f is neither constant nor injective, then there are two points a, b
, such that f (a) = f (b). Consider g (z ) = f (z ) f (a); it has two zeros a and b.
Because g is non-constant, b must be an isolated zero of g . Therefore,
Homework 2
Math 116
Problem 1 :
Let U , V be open sets in C and assume that U V is nonempty and connected. Let f be holomorphic on U V . If
f has a holomorphic primitive on U and a holomorphic primitive on V , prove that f has a holomorphic primitive on
U