Cv 201008 Complex Analysis Qualifier Exam, August 2010

8/13/2019 Cv 201008 Complex Analysis Qualifier Exam, August 2010

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QUALIFYING EXAM in ANALYSIS

Department of Mathematics

University of Wisconsin-Madison

Wednesday, August 18, 2010

Version for emphasis in complex analysis (Math 722)

Instructions:

Do six of the nine questions. To facilitate grading, please use a separate packet of paper for each

question. To receive credit on a problem, you must show your work and justify your conclusions.

The Doctoral Exam Committee proofreads the qualifying exams as carefully as possible. Neverthe-less, this exam may contain typographical errors. If you have any doubts about the interpretationof a problem, please consult with the proctor. If you are convinced that a problem has beenstated incorrectly, mention this to the proctor and indicate your interpretation in your solution.In any case, never interpret a problem in such a way that it becomes trivial.

Notation used on the Analysis exams:

(1) Q, RandCdenote the fields of rational, real and complex numbers, respectively.

(2) IfE Rn is a Lebesgue measurable set, thenE denotes its Lebesgue measure.

(3) If is a positive measure on a set X, and iffis a complex valued measurable function onX, then for 1 p


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Problem 1.

LetD Rd, d 2 be a compact convex set with smooth boundary D so that the originbelongs to the interior ofD. For every y D let (x) [0, ) be the angle between the

position vector x and the outer normal vector n

(x). Let d be the surface area of the unitsphere in Rd. Compute1

d

D

cos((x))

|x|d1 d(x)

where d denotes surface measure on D.Provide complete justifications for your computation. Does a reasonable interpretation of

your result hold true ifd= 1?

Problem 2.

Let

sN(x) =

Nn=1

(1)nx3n

n2/3.

Prove thatsN(x) converges to a limit s(x) on [0, 1] and that there is a constant Cso thatfor all N 1 the inequality

supx[0,1]

|sN(x) s(x)| CN2/3

holds.

Problem 3. Let Kbe a continuous function on the unit square Q= [0, 1] [0, 1] with theproperty that |K(x, y)| < 1 for all (x, y) Q. Show that there is a continuous function gdefined on [0, 1] so that

g(x) +

10

K(x, y)g(y)dy= ex

1 +x2, 0 x 1.

Problem 4. For each part (i), (ii), (iii) either prove the statement or give a counterexample.

(i) (3pts.) Let Pbe a polynomial of two variables (not identically zero) and let E ={(x, y) R2 :P(x, y) = 0}.

Prove or disprove: Eis a Lebesgue measurable set of measure zero.

(ii) (4pts.) Let En [0, 1] be Lebesgue measurable sets, with meas(En) n1. Let

E= limsup En, i.e. the set of all x [0, 1] which belong to En for infinitely many n.Prove or disprove: Eis a Lebesgue measurable set of measure zero.(iii) (3pts.) Let f and fn belong to L

1(R) L(R), for every n N. Suppose thatlimn

R

|fn(x) f(x)|2dx= 0.

Prove or disprove: limnR

fn(x)dx=R

f(x)dx.


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Problem 5. We say that a sequence {fn}nNof real-valued measurable functions defined on[0, 1] is uniformly integrable if for every >0 there is a >0 so that supnN |

E

fndx| < for all measurable subsetsE [0, 1] with measure at most .

Prove: If fn : [0, 1] R is a uniformly integrable sequence and fn(x) converges to f(x)

almost everywhere then

limn

10

fn(x) dx=

10

f(x) dx.

Problem 6. LetI= [0, 1], and define for f L2(I) the Fourier coefficients as

fk = 10

f(t)e2ikt dt.

(i) Let G be the set of all L2(I) functions with the property that |fk| |k|3/5 for allk Z. Prove thatG is a compact subset ofL2(I).

(ii) Let Ebe the set of all L2(I) functions with the property that

k |

fk|

5/3 1010. Is E

a compact subset ofL2

(I)?Problem 7C.

(i) (5pts.) Letk be a positive integer and let A 0. Identify explicitly the class of entireholomorphic functions F : C Cwith the property 2

0

|F(rei)|2d Ar2k.

(ii) (5pts.) Letfbe a function holomorphic in the open unit disk D= {z :|z| 0 and let

f(z) =n=0

zn

(n!)a.

Show that fis holomorphic in C and that there are constants A, B so that

|f(z)| AeB|z|1/a

for all z C.

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Cv 201008 Complex Analysis Qualifier Exam, August 2010