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Unformatted text preview: MAT 461 Assignment D - Solutions Posted November 12 (1) Assume f ( z ) is analytic on and inside the circle | z | = 4. Show that if f ( z ) = 0 for every z such that | z | = 4, then f ( z ) = 0 for every z such that | z | < 4 as well. (That is, show that if f ( z ) = 0 on the circle, then f ( z ) = 0 inside too.) By the Maximum Modulus Principle (or more specifically, the corollary on pg 171), the maximum value of | f ( z ) | in the circle | z | 4 must occur on the boundary | z | = 4. Since f ( z ) = 0 on the boundary, | f ( z ) | 0 inside the circle. This implies that f ( z ) = 0 inside the circle. Another way: Let C denote the circle and let z be a point interior to the circle. Then the conditions necessary for the Cauchy Integral Formula hold, and we can write f ( z ) = 1 2 i integraldisplay C f ( z ) z z dz Since f ( z ) = 0 on C (and z z negationslash = 0 on C ), the integral on the right is zero and so f ( z ) = 0. This works for all point inside C , so f ( z ) = 0 in the whole inside of...
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This note was uploaded on 05/11/2008 for the course MAT 461 taught by Professor Ibrahim during the Fall '07 term at ASU.
- Fall '07