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By R.B. Burckel

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Example text

RADO [1931] and VIZZINI [1932]. 44 was extended to metric spaces by TIETZE [1914] and is often called the "Tietze extension theorem"; this designation has even come to be applied to the euclidean space version of the theorem. 44 with C compact can already be found in PHRAGMEN [1900]. 41 Chapter II (Complex) Derivative and (Curvilinear) Integrals This short chapter is comprised of very basic and very easy material, probably familiar to most readers in one form or other. Even the reader with only modest experience can probably supply his own proofs for most of these results as quickly as he can read mine!

X 4-2 (2k)! + (-1)2"X4n ] (4n)! ]X4n - 2. 1 < 1 ~--=-: (4n - 2)! co [ x 2 + n~2 1] (4n)! - (4n _ 2)! x 4n-2 Therefore C(V3) < -t. Since C(O) = Re E(O) = 1, we have that C[O, V3] is a connected subset of IR which contains positives and negatives and hence zero; that is, the compact set [0, V3] r. C-l(O) is not void. T/2 = mineO, V3] r. C-1(0). T/2) = O. From (8) and (6) we get (9) S(17/2) = ± 1. However by (5) and the definition of 17 (10) S' = C > 0 in [0,17/2), § 3. The Complex Exponential Function 63 so that S is strictly increasing in [0,71'/2].

T. WHYBURN [1932]. 29. I9(ii) is a famous result of CARATHEoDORY [1911] (p. 2(0): if S c: IRk, then every element of co Sis a convex combination of k + 1 or fewer points of S. The proof in the text is due to STElNlTZ [1913] (p. 153). 35 comes from HmNS [1962] (p. 39 is from SAKS and ZYGMUND [1971] (p. 210). See also DE VITO [1957]. 36 is due to HAUSDORFF (p. 351 of his book [1914]) and supersedes a less tractable one which Caratheodory had made earlier. 44 (due to F. Riesz) in DIEUDONNE [1969] (pp.

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