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 # 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
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
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
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