R.B. Burckel's An Introduction to Classical Complex Analysis: Vol. 1 PDF

By R.B. Burckel

ISBN-10: 3034893744

ISBN-13: 9783034893749

ISBN-10: 3034893760

ISBN-13: 9783034893763

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

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The mapping x ~ E(ix) maps IR onto the unit circle {z E 17 such that E(ii) = i and c: Izi = I}. E(C) = q{O}. Note: (ii), (iii) and (vi) say that Eis a homomorphism of the additive group C onto the multiplicative group q{O} with kernel 2-rriZ. § 3. The Complex Exponential Function 61 Proof: That E is differentiable and that = E'(z) 2.. =1 n·, 1 n. = E(z) Z .. 6. Given e E C, define a function J: C -+ C by J(z) = E(e - z). Then by (i) (and the Chain Rule), the functionJis differentiable throughout C and satisfies f' == - f Therefore the function F = J.

195-201 of SAKS [1937] or the exhaustive (but terse) treatise of TROKHIMCHUK [1964]. For a somewhat more elementary account of somewhat weaker results in this direction, see MEIER [1951], [1960] and ARSOVE [1955]. For a survey of results and history (together with a proof of the first theorem above) see GRAY and MORRIS [1978]; in TROKHIMCHUK [1962] the reader will find a longer survey and some proofs. See also TAR (1969], [1971], (1973], MEIER [1970], PONOMAREV (1970], and BERENSTEiN [1971] for recent results.

Therefore, since D(O, r) is a neighborhood of w, we have only to let z --+ win (5) to conclude the /,(w) exists and equals g(w), as claimed. 7 If the series fez) = 2::'-0 cnzn has a radius of convergence R > and /,(0) = C1 "# 0, then for some < r :S R, f is one-to-one in D(O, r). 6, + (z n 2: 2: co - w) CIS n-2 1e-1 2: W k - 1Z n -", n-2 2: n GO If(z) - f(w)1 2! 5 the series 2::'= 1 nlcnlz n- 1 has radius of convergence R, so for sufficiently small < r < R, 2::'=2 nlcnlr n- 1 < Icl 1/2 and it follows that If(z) - f(w) I > (I ClI/2)lz - wi· Remark: These same techniques show, with a little more work (see KRAFFT [I932]), that feD(O, r» is open and f- 1 is given by a power series near Co, facts which we will secure in Chapter V by other means.

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


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