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By Robert B. Ash (Auth.)

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D) If £ is connected and z9weS, then g(w) — g(z) = In |/(u>)| — In |/(z)| + /[0(w) — 0(z)] for all continuous loga rithms g and continuous arguments Θ of/ PROOF (a) f(z) = e9(z) = eRe9{z)eilm9{z) (b) eln\f(z)\+mz) (c) effl = e^2 = / hence = \f(z)\ eilm9{z). = |/(z)| emz) = /(z). Z77/ is a continuous integer- valued function on the connected set S, hence a constant. ) Similarly e*ei = e1** = γψ-, 51 LOGARITHMS AND ARGUMENTS so -^-^—— is a continuous integer-valued function on S, hence a constant.

Now assume the result proved for n = k. Then /(*)(z + A)-/<*)(z) (k+\)\ h 2πΐ c f(w) )J rp (vi (w — z)k dw k\ Έ lf{w) X Γ [ (w __ 1 z _ /7)/>+i 1 (H; _ z)fc+i h(k + 1) I , (H; - z) fc + 2 J 31 POWER SERIES The expression in brackets is of the order of h2 as h -> 0 (form a common denominator and use the binomial theorem) so that the integral approaches 0 as h-+0. Thus the formula holds for n = k+l. 10 Corollary If/is analytic on the open set U, then/has derivatives of all orders on U. In particular, i f / h a s a primitive F on U, then (since F" exists)/is analytic on U.

Fc+1 — 6fc . Show that s Σ k=r 7. (a) s a k Abk = as+1bs+1 — arbr — £ bk+1 Aak . k=r If {bn} is bounded and the an are real and greater than 0, with oo ax > a2 > a3 > ··· —> 0, show that Σ an Abn converges. 71=1 (b) If bn = bn(z), that is, the bn are functions from S to C, *S C C, the &n are uniformly bounded on S, and the an are real and 00 decrease to 0 as in (a), show that Σ an(bn+1(z) — bn(z)) con n=1 verges uniformly on S. 38 THE ELEMENTARY THEORY (a) 7 Show that X —n converges when | z | = 1, except at the w=l single point z = 1.