Complex Analysis
Contents
Chapter 1. Fundamental Properties of Holomorphic Functions
1. Basic denitions
2. Integration and Integral formulas
3. Some consequences of the integral formulas
5
5
6
8
Chapter 2. Runges Theorem
1. Partitions of unity
2. Smeared
MAT3400/4400 H13 Solution outline
Problem 1
1a) Obviously, since we are considering the -algebra P(X), every function from X into K is
measurable, sowe dont have to bother about measurability.
Let f (X). Since |f | f u 1X where 1X denote the function on X
MAT3000/4000 Spring 2013 Solution of Extra exercises 3 and 4
Extra exercise 3
Let (X, A, ) be a measure space, M = cfw_f : X R | f is A-measurable and E A.
We consider the measure space (E, AE , E ), where
AE = cfw_A E | A A = cfw_B A | B E A
and E : AE [
Solution to exam in MAT3400/4400, Linear analysis with applications.
Exam date Monday, December 5, 2011.
Problem 1. The n-th Fourier coecient of f for n = 0 is
0
1
(einx )dx
2
1 einx
1
inx
e dx =
=
2 0
2 in 0
1
=
(1 cos(n)
2in
1
=
(1 (1)n )
2in
f (x)en
MAT3400/4400 - Fall 2013 - Solution of Extra Exercise 9
Let a, b R, a < b and let m denote the Lebesgue measure on [a, b]. Let k : R K be a
continuous function on R = [a, b] [a, b]. For each x [a, b], we let kx : [a, b] K denote the
continuous function gi
MAT3400/4400 - Fall 2013 - Solutions of Extra Exercises 6 and 7
Extra-Exercise 6
Let p [1, ), (X, A, ) be a measure space and K denote R or C.
Let M = f : X K | f is A-measurable and Lp = cfw_f M | f
f
p
1
p
|f |p d
=
p
< , where
.
a) Assume that (X, A, )
Solutions to the exam problems in MAT3400/4400, Fall 2012
Problem 1. Using Fourier series show that if f : [0, 2] C is a C 1 -function such that
2
f (0) = f (2) and 0 f (t)dt = 0, then
2
2
|f (t)|2 dt.
|f (t)|2 dt
0
0
Describe all functions f as above su
MAT3400/4400 - Fall 2013 - Solution Extra exercise 2
We recall that J = cfw_ cfw_(a, b] | a, b R, a < b cfw_(a, ) | a R cfw_(, b] | b R,
while S denotes the collection of all subsets of R consisting of nite disjoint union of sets
belonging to J .
a) Check
Solution to exam in MAT3400/4400, Linear analysis with applications.
Exam date Monday, December 6, 2010.
Problem 1. The n-th Fourier coecient of f (x) = x2 on [, ] is (for
n = 0)
1
x2 einx dx
2
x2 einx
1
1
+
=
2
in
in
cn (f ) =
=
1
x
einx
in
in
+
xeinx
MAT3400/4400 - Fall 2013 - Solution of Extra Exercise 8
Let (X, A, ) be a measure space, K denote R or C,
M = f : X K | f is A-measurable ,
and L denote the set of all functions in M that are essentially bounded (w.r.t. ).
a) Let f M. We want to show that