Real and Complex Analysis (Rudin)

400 REAL AND COMPLEX ANALYSIS
Chapter 9
For another brief introduction, see [36], chap. XVI. A different proof of Plancherel's theorem is in
[33]. Group-theoretic aspects and connections with Banach algebras are discussed in [17], [19], and
[27]. For more on invariant subspaces (Sec. 9.16) see [11]; the corresponding problem in [} is
described in [27], Chap. 7.
Chapter 10
General references: [1], [4], [13], [20], [29], [31] and [37].
Sec. 10.8. Integration can also be defined over arbitrary rectifiable curves. See [13], vol. I,
Appendix C.
Sec. 10.10. The topological concept of index is applied in [29] and is fully utilized in [1]. The
computational proof of Theorem 10.10. is as in [1], p. 93.
Sec. 10.13. Cauchy published his theorem in 1825, under the additional assumption that f' is
continuous. Goursat showed that this assumption is redundant, and stated the theorem in its present
form. See [13], p. 163, for further historical remarks.
Sec. 10.16. The standard proofs of the power series representation and of the fact that f E H(O)
implies f' E H(O) proceed via the Cauchy integral formula, as here. Recently proofs have been constructed
which use the winding number but make no appeal to integration. For details see [34].
Sec. 10.25. A very elementary proof of the algebraic completeness of the complex field is in [26],
p.170.
Sec. 10.30. The proof of part (b) is as in [47].
Sec. 10.32. The open mapping theorem and the discreteness of Z(f) are topological properties of
the class of all nonconstant holomorphic functions which characterize this class up to homeomorphisms.
This is Stoilov's theorem. See [34].
Sec. 10.35. This strikingly simple and elementary proof of the global version of Cauchy's
theorem was discovered by John D. Dixon, Proc. Am. Math. Soc., vol. 29, pp. 62~26, 1971. In [1]
the proof is based on the theory of exact differentials. In the first edition of the present book it was
deduced from Runge's theorem. That approach was used earlier in [29], p. 177. There, however, it
was applied in simply connected regions only.
Chapter 11
General references: [1], Chap. 5; [20], Chap. 1.
Sec. 11.14. The reflection principle was used by H. A. Schwarz to solve problems concerning
conformal mappings of polygonal regions. See Sec. 17.6 of [13]. Further results along these lines were
obtained by Caratheodory; see [4], vol. II, pp. 88-92,.and Commentarii Mathematici He/vetici, vol. 19,
pp.263-278,1946-1947.
Secs. 11.20, 11.25. This is the principal result of the Hardy-Littlewood paper mentioned in the
reference to Sec. 7.4. The proof of the second inequality in Theorem 11.20 is as in [40], p. 23.
Sec. 11.23. The first theorems of this type are in Fatou's thesis, Series trigonometriques et series
de Taylor, Acta Math., vol. 30, pp. 335-400, 1906. This is· the first major work in which Lebesgue's
theory of integration is applied to the study of holomorphic functions.
Sec. 11.30. Part (c) is due to Herglotz, Leipziger Berichte, vol. 63, pp. 501-511, 1911.
Exercise 14. This was suggested by W. Ramey and D. Ullrich.
Chapter 12
Sec. 12.7. For further examples, see [31], pp. 176-186.
Sec. 12.11. This theorem was first proved for trigonometric series by W. H. Young (1912; q = 2,
4, 6, ... ) and F. Hausdorff (1923; 2 S q ~ (0). F. Riesz (1923) extended it to uniformly bounded