Serge Ballif
MATH 502 Homework 7
March 21, 2008
(1)(The inverse function theorem) Let U and V be open subsets of and let f : U V be a holomorphic bijection. Show that f is a homeomorphism (that is, show that the inverse function f 1 is continuou
MATH 502: REAL AND COMPLEX ANALYSIS SPRING 2004 A.Katok PROBLEM SET #4: February 18 LEBESGUE MEASURE Due on Wednesday 22504 13. Let be a nonatomic (complete) LebesgueStieltjes measure on [0, 1]. prove that there exists a continuous map h : [0, 1] [0, 1
MATH 502: REAL AND COMPLEX ANALYSIS SPRING 2004 A.Katok PROBLEM SET # 7 : March 18 LEBESGUE INTEGRATION Due on Wednesday 32404 In all problems the measure space is assumed to be finite. 24.We will call a function of the form i=n tn An where A1 , A2 , .
MATH 502: REAL AND COMPLEX ANALYSIS SPRING 2004 A.Katok PROBLEM SET # 6: March 3 MEASURABLE FUNCTIONS Due on Wednesday 31704 21. Find a metric in the space of mod 0 classes of Lebesgue measurable real valued functions on [0, 1] such that convergence in
MATH 502: REAL AND COMPLEX ANALYSIS SPRING 2004 A.Katok PROBLEM SET # 5 : February 26 PRODUCT MEASURES AND LEBESGUE SPACES Due on Friday, 3504 17. Consider a measurable set A [0, 1] of positive Lebesgue measure. Let A be the normalized restriction of Le
An Overview
1. Basic topology a. Topological spaces. DEFINITION 1.1. A topological space (X, T ) is a set X endowed with a collection T P(X) of subsets of X, called the topology of X, such that (1) , X T , (2) if cfw_O A T then A O T for any set A, (3) if
MATH 502: REAL AND COMPLEX ANALYSIS SPRING 2004 A.Katok PROBLEM SET # 8: March 25 SPACES OF INTEGRABLE FUNCTIONS Due on Wednesday 33104 In all problems the measure space is assumed to be finite. 28. 29. Let 1 p q . Prove that the unit ball in Lq (X, ) (
MATH 502: REAL AND COMPLEX ANALYSIS SPRING 2004 A.Katok PROBLEM SET # 9 : March 31 LEBESGUE DENSITY AND ABSOLUTE CONTINUITY Due on Wednesday 4704 32. Prove the generalization of Lebesgue Density Point Theorem for arbitrary finite Borel measure on an int
MATH 502: REAL AND COMPLEX ANALYSIS SPRING 2004 A.Katok PROBLEMS ON BANACH SPACES: F1. Consider the linear space of all polynomials with real or complex coefficients. Prove that it s impossible to introduce a norm in this space which makes it a Banach spa
MATH 502: REAL AND COMPLEX ANALYSIS SPRING 2004 A.Katok PROBLEM SET # 10: April 8 RADONNIKODYM ANS RIESZ REPRESENTATION THEOREMS Due on Wednesday April 14 36. Suppose is a Borel measure on [0, 1] such that for some constant C and every interval [a, b] [0
NOTE ON ABSTRACT RIEMANN INTEGRAL Based on the Appendix to B. Hasselblatt and A. Katok, A First Course in Dynamics, Cambridge University press, 2003. 1. Metric spaces a. Definitions. D EFINITION 1.1. If X is a set then d : X X R is said to be a metric or
MATH 502: REAL AND COMPLEX ANALYSIS SPRING 2004 A.Katok PROBLEM SET # 3: February 10 LEBESGUE MEASURES AND ALGRBRAS Due on Wednesday 21804 9. Given a number , 0 < 1 consider the transformation T of the unit interval [0, 1) onto itself: T (x) = x + if 0
THE MONOTONICITY OF THE Lp norm Some of you pointed out to a problem in an old qualifying exam which easily reduces to proving the following: T he norm f p = ( f p )1/p is non  decreasing in p.
Misha Guysinsky in his explanation deduces the statement
MATH 502: REAL AND COMPLEX ANALYSIS SPRING 2002 A.Katok SECOND MIDTERM EXAMINATION April 16 , 2004 For a perfect score you should give complete solutions of two problems from each of the two sections. SECTION 1 1.1. Prove that the set of Lebesgue density
Serge Ballif Gamma function
Complex Analysis Facts and Formulas
Spring 2008
(z) =
0
tz1 et dt
Zeta function (s) =
n=1
1 = ns
b
p
1 1  ps
Path Integral f (z) dz :=
a
f (t) (t) dt
LM Estimate 
f (z) dz M Length() for M f (z)
Serge Ballif
MATH 502 Homework 1
January 25, 2008
(1) Let f = u + iv be a holomorphic function of z = x + iy. Show that the function g = log f 2 = log(u2 + v 2 ) satisfies Laplaces equation 2g 2g + 2 =0 x2 y in a neighborhood of any point where
Serge Ballif
MATH 502 Homework 2
February 1, 2008
(1) Let a and b be complex numbers with strictly negative real part. Prove the inequality ea  eb  a  b.
Define : [0, 2] to be the straight path (t) = (1  t)a + tb from a to b. z Define f
Serge Ballif
(1) Evaluate the integral
MATH 502 Homework 3
February 8, 2008
cos z dz z around the unit circle. Deduce that
0 2
cos(cos ) cosh(sin ) d = 2.
We know that cos(z) is a holomorphic function on all of . The Cauchy integral 1 formula te
Serge Ballif
MATH 502 Homework 4
February 15, 2008
(1) Consider a continuous family {ft , t (1, 1)} of functions that are continuous on the closed unit disc U and holomorphic on the open disc U . (In other words, t ft is a continuous map from (
Serge Ballif
(1) Evaluate the integral
MATH 502 Homework 5
February 22, 2008

x2 cos x dx 4 + x4
by means of contour integration. Be sure to justify carefully all the steps in your argument.
To compute this integral, we will consider the func
Serge Ballif
MATH 502 Homework 6
February 29, 2008
(1) The function f (z) is entire, and it is known that f (z) 1 whenever z = 1 and that f (z) 10 whenever z = 10. Show that f (z) 5 whenever z = 5. What can be said about f (0)?
C
MATH 502: REAL AND COMPLEX ANALYSIS SPRING 2002 A.Katok FIRST MIDTERM EXAMINATION Wednesday March 3 , 2004 For a perfect score you should give complete solutions of two problems from each of the two sections. SECTION 1 1.1. Let X be a compact metric space
MATH 502: REAL AND COMPLEX ANALYSIS SPRING 2004 A.Katok PROBLEM SET # 1: January 14 NULLSETS and RIEMANN INTEGRATION Due on Wednesday 12104 1. Prove that the set of discontinuity points for any function f : X R on a metric space is the union of a count
MATH 502: REAL AND COMPLEX ANALYSIS SPRING 2004 A.Katok PROBLEM SET # 2: January 22 RIESZ INTEGRAL Due on Wednesday 12804 5. Consider a Riesz integral l on a compact metric space. Prove that for disjoint closed sets A1 and A2 the upper Riemann measure i
Updated: 1/3/2017 Stewart 8e
Students are responsible for knowing the concepts associated with the following homework
problems. Students wishing to check understanding may wish to do odd problems from this
list (answers in the back of the book). Instructo