By Robert B. Ash (Auth.)
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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. By continuity, g is nonzero in a neighborhood of z 0 , contradicting the fact that z0 is a limit point of zeros.
Z-+Z0 g(z) Z-+Z0 g (Z) 4. I f / i s analytic on the open connected set U and | / | is constant on £/, show t h a t / i s constant on I/. 5. L e t / b e analytic on C, and assume that/(z) is real valued for all z such that I z | = 1. Show that fis constant chapter 2 THE GENERAL CAUCHY THEOREM In this chapter we consider two basic questions. First, for a given open set U, we try to determine which closed paths γ in U have the property that \yj(z)dz = 0 for every/analytic on U. Second, we try to characterize those open sets U having the property that \yj{z)dz = 0 for all closed paths y in U and all functions / analytic on U.
6, we may choose the mesh so that one of the squares is centered at a particular z e A; then η(γ, z) = 1, z G AC C — U, contradicting (2). 4. 9. /' (4) implies (5): If/ is analytic and never 0 on U, -j- is analytic on U, hence has a primitive on U. 5, / h a s an analytic logarithm.