Real and Complex Analysis (Rudin)

230 REAL AND COMPLEX ANALYSIS
and
(O:s; t :s; 2n).
(b) Show by means of (a) that every f E H(A) can be decomposed into a sumf = fl + f2' wheref1
is holomorphic outside D(O; rl) and f2 E H(D(O; r2»; the decomposition is unique if we require that
f.(z)--+ 0 as 1 z 1--+ 00.
(c) Use this decomposition to associate with eachf E H(A) its so-called" Laurent series"
""
L c"z"
which converges tofin A. Show that there is only one such series for each! Show that it converges to
funiformly on compact subsets of A.
(d) Iff E H(A) andfis bounded in A, show that the componentsf1 andf2 are also bounded.
(e) How much of the foregoing can you extend to the case r1 = O(or r2 = oo,or both)?
(f) How much of the foregoing can you extend to regions bounded by finitely many (more than
two) circles?
26 It is required to expand the function
1 1
--+--
1-z 2 3-z
""
in a series of the form L en z".
-""
How many such expansions are there? In which region is each of them valid? Find the coefficients
en explicitly for each of these expansions.
27 Suppose Q is a horizontal strip, determined by the inequalities a < y < b, say. SupposefE H(Q),
andf(z) = f(z + 1) for all z E Q. Prove thatfhas a Fourier expansion in n,
""
f(z) = L e n e 2dn .,
which converges uniformly in {z: a + £:s; y:s; b - £}, for every £ > O. Hint: The map z--+ e 2ni • convertsfto
a function in an annulus.
Find the integral formulas by means of which the coefficients en can be computed from!
28 Suppose r is a closed curve in the plane, with parameter interval [0, 2n]. Take ex ¢ P. Approximate
r uniformly by trigonometric polynomials rn. Show that Indr• (ex) = Indr• (ex) if m and n are
sufficiently large. Define this common value to be Indr (ex). Prove that the result does not depend on
the choice of {r n}; prove that Lemma 10.39 is now true for closed curves, and use this to give a
different proof of Theorem 10.40.
29 Define
1 i1 In de
f(z) = - r dr -18-.
-n re + z
n 0
Show thatf(z) = z if 1 z 1 < 1 and thatf(z) = liz if 1 z 1 ~ 1.
Thus f is not holomorphic in the unit disc, although the integrand is a holomorpbic function of
z. Note the contrast between this, on the one hand, and Theorem 10.7 and Exercise 16 on the other.
Suggestion: Compute the inner integral separately for r < 1 z 1 and for r > 1 z I.
30 Let Q be the plane minus two points, and show that some closed paths r in Q satisfy assumption
(1) of Theorem 10.35 without being null-homotopic in Q.