Download e-book for iPad: An Introduction to Classical Complex Analysis: Vol. 1 by R.B. Burckel

By R.B. Burckel

ISBN-10: 3034893744

ISBN-13: 9783034893749

ISBN-10: 3034893760

ISBN-13: 9783034893763

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Additional resources for An Introduction to Classical Complex Analysis: Vol. 1

Example text

311 #30001. For one-to-one functions even weaker conditions suffice to ensure analyticity. See pages 5, 6 of MONTEL [1933b] and the references there, as well as MENCHOFF [1936], [1937] and BRODOVICH [1970], [1974]. Chapter 1 of MARKUSCHEWITSCH [1976J is also devoted to these questions and contains a proof of the first theorem above. 1 a commitment is made that has profound historical significance. The theory to be erected here did not spring fully-armed from the head of Zeus, but condensed gradually out of the primordial v~pors.

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]. Therefore S(7I'/2) > S(O) (9) we get (11) S(7I'/2) = 0 and from = 1. Putting (8) and (11) together: (12) i) = C{7I'/2) + E(~ whence E(271'i) @ [E(~ i) is{7I'/2) r = = i, 1 + 271'i) = E{z)E(271'i} = E(z) Vz E C. Now suppose z = x + iy (x, y E IR) is such that E(z) = (13) E(z 1. Then I = IE(z) I = IE(x)E(iy) I = E(x) IE(iy) I = E{x), since E{IR) c (0, (0) and IE(iy) I = I by (6).

1 + t(z If(zl - ZI» - f(ZI)] dt l + t(z - Zl» - f(zl)l. the desired conclusion is now immediate. 12 For a subset S of e and a bounded function f: S _ w/(8) e let = sup{lf(z) - f(w) I: z, WE S, Iz - wi s 8} for each 8 ~ O. The function Wf: [0, ex» _ [0, ex» so defined is called the oscillation of f or, in case f is continuous on S, the modulus of (uniform) continuity ofJ. 13 (i) Let S be a subset ofe andf: S _ e a boundedfunction. Show that Wf is continuous at 0 if and only iff is uniformly continuous on s.

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An Introduction to Classical Complex Analysis: Vol. 1 by R.B. Burckel


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