Real and Complex Analysis (Rudin)

ANALYTIC CONTINUATION 333
P(f1' 01) = O. Extend this to more than two functions. Is there such a theorem for some class of
functions P which is larger than the polynomials?
3 Suppose 0 is a simply connected region, and u is a real harmonic function in O. Prove that there
exists an f e H(O) such that u = Re f Show that this fails in every region which is not simply connected.
4 Suppose X is the closed unit square in the plane,fis a continuous complex function on X, andfhas
no zero in X. Prove that there is a continuous function 0 on X such thatf= e'. For what class of
spaces X (other than the above square) is this also true?
5 Prove that the transformations z--+ z + 1 and z--+ -liz generate the full modular group G. Let R
consist of all z = x + iy such that I x I < t, y > 0, and I z I > 1, plus those limit points which have
x ~ O. Prove that R is a fundamental domain of G.
6 Prove that G is also generated by the transformations cp and "', where
1
cp(z) = --,
z
z-1
",(z)=-.
z
Show that cp has period 2, '" has period 3.
7 Find the relation between composition of linear fractional transformations and matrix multiplication.
Try to use this to construct an algebraic proof of Theorem 16.l9(c) or of the first part of
Exercise 5.
8 Let E be a compact set on the real axis, of positive Lebesgue measure, let 0 be the complement of
E, relative to the plane, and define
f(z)= i -
dt
E t - z
(z eO).
Answer the following questions:
(a) Isf constant?
(b) Canfbe extended to an entire function?
(c) Does lim zf(z) exist as z--+ oo? If so, what is it?
(d) Doesfhave a holomorphic square root in O?
(e) Is the real part off bounded in O?
(f) Is the imaginary part off bounded in O?
[If" yes" in (e) or (f), give a bound.]
(0) What is L f(z) dz if y is a positively oriented circle which has E in its interior?
(h) Does there exist a bounded holomorphic function cp in 0 which is not constant?
9 Check your answers in Exercise 8 against the special case
E=[-I,I].
10 Call a compact set E in the plane removable if there are no nonconstant bounded holomorphic
functions in the complement of E.
(a) Prove that every countable compact set is removable.
(b) If E is a compact subset of the real axis, and m(E) = 0, prove that E is removable. Hint: E
can be surrounded by curves of arbitrarily small total length. Apply Cauchy's formula, as in Exercise
25, Chap 10.
(c) Suppose E is removable, 0 is a region, E c: !l,fe H(O - E), andfis bounded. Prove thatf
can be extended to a holomorphic function in O.
(d) Formulate and prove an analogue of part (b) for sets E which are not necessarily on the real
axis.
(e) Prove that no compact connected subset of the plane (with more than one point) is removable.