Functional Analysis

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By Robert B. Ash (Auth.)

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Analytic logarithms exist under wide conditions, as we now show. 6 Theorem If/is analytic and never 0 on the convex open set U C C,/has an analytic logarithm on U. More generally, if U is an open subset of C such that Jy h(z) dz = 0 for every closed path y in U and every h analytic on U, then every /analytic and never 0 on U has an analytic logarithm on U. 53 LOGARITHMS AND ARGUMENTS PROOF f First consider the convex case. 11 yields for every closed path y in U. 5,/has an analytic logarithm. In the more general case, result follows as before.

A zero of order 1 is sometimes called a simple zero. 6 Theorem Let / b e analytic on the open connected set UC C. Suppose t h a t / has a limit point of zeros in U9 that is, there is a point z0e U and a sequence of points z n e U9 zn φ z 0 , such that z n —> z0 and /(z w ) = 0 for all n (hence/(z 0 ) = 0). Then/is identically 0 on U. PROOF 00 Expand/in a Taylor series about z 0 , say/(z) = X ÖW(Z — z0)w, | z — z01 < r. We show that all an = 0. 5,/(z) = (z — z0)m g(z), where g is analytic at z0 and g(z0) 7^ 0.

4. 7, Mi,) ί fM f'(z\ 7 Jz = *M''» - &(yfo-i))> where the integration is taken along y. Let θ(ί) be a continuous argument of /(y(0) — zo > a ^ t ^ b; note that g/y(0) is a continuous logarithm of /(y(0) — zo » ^-i < * < * , · . 2(d) - θ(α)] = /l(/o y , z 0 ) . | Finally, we consider the behavior of η(γ, ζ0) as z0 varies. 6 Theorem If y is a closed path, then η(γ, z0), regarded as a function of z 0 , is constant on each component of C — y* and 0 on the unbounded component. 13, η(γ, ·) is analytic, hence continuous on C — y*.

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