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

By R.B. Burckel

ISBN-10: 3034893744

ISBN-13: 9783034893749

ISBN-10: 3034893760

ISBN-13: 9783034893763

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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. (ii) Let ')I: [0, I] - e be a piecewise smooth curve. Show that 2: I,,(tf) f-l ,,(tf - 1)1 whenever 0 = R 10 s 1(')1) < 11 < ... < tn = I. Hint: By adding more points If the left side is not decreased, so we can suppose that all the points of discontinuity of ')I' are among the If. Then y(tf) - ,,(If - 1 ) = rttl ')I' and 1(')1) = ~-l JI(II/, - l 1')1'1. · ..

6 Vw E D(O, 1). > 0 so that D(a, R) c EE" we have 1 (e - ~:'DO < q". Then for any 0 < r < R, < 1, and so 1 0),,+1 (1 _~)"+1 e-a :r-", ~ (e - 1 ),,+1 1c~ (~) (~ = 0 with convergence uniform in such z and e. n) = ~ L=tl (L (/~~~Ll) k _ : + 1 (~)(Z - a)1c-"+1]. 9 (Cf. CAUCHY [1821], p. ) Show that if a" is positive/or all (sufficiently large) n, then lim Q,,+1 < lim an -;:;00 ~ vra.. < n - lim 1&_ co vra.. - co an . Hints: Call the last limit superior a E [0, 00]. It suffices to prove the last inequality.

It is then a theorem (and an easy one, see RUDIN [1976], p. 13(ii) below) and its length in the above sense equals Iy'l- Only this latter integral intervenes in our subsequent considerations and I chose for it its traditional name, the length of y; the above theorem is the rationale for this. f f 4S § 2.. ). After Cauchy's theorem (in a disk) is proved in Chapter V, this can easily be done byapproximating 'Y uniformly with piecewise smooth curves (whether 'Y is rectifiable in the above sense or not).

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

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