07 - ASSIGNMENT 7 SOLUTIONS MAT 572 A FALL 2007 Problem 1...

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ASSIGNMENT 7 · SOLUTIONS MAT 572 A · FALL 2007 Problem 1 ( cf. [1, Exercise III.3 #17]) . Show that if G is open and connected, f : G C is analytic, and f ( G ) is a subset of a circle, then f is constant. Proof. Let Γ be a circle in C with f ( G ) Γ. Since G is connected, it suffices to show f is locally constant. So fix z 0 G and an open ball B = B ± ( z 0 ); by shrinking ± if necessary we may assume that f ( B ) is not all of Γ. (For any γ 0 6 = f ( z 0 ) in Γ, the set f - 1 ( γ 0 ) is closed and doesn’t contain z 0 .) So we may choose a M¨obius transformation T : C C which takes Γ to R and which takes f ( B ) to R . But then T f : B R is analytic and real-valued, so constant by the Cauchy-Riemann equations (see Exercise III.2.14). Since T is invertible, it follows that f is also constant on B . ± Problem 2 ( cf. [1, Exercise IV.3 #1]) . Let f be entire, and suppose there are constants M and R > 0 and an integer n 1 such that | f ( z ) | ≤ M | z | n
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07 - ASSIGNMENT 7 SOLUTIONS MAT 572 A FALL 2007 Problem 1...

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