Question

Task 204！

olomorphic. f F has a maximum value on U, then f is constant. One proof follows a line of reasoning that is typical for analytic functions, by which information nea one point can be transferred to other points by moving along a contour. Suppose Ifl has a maximun value at zo. Then the local version of the maximum modulus principle implies that f must be constan near zo. Consequently, the Taylor series of f at any point near zo must have only a constant term; all othe derivatives vanish. By joining any other point z E U to zo with a contour, we can conclude that the power series of f at every point of the contour must be constant, and so f(z) = Task 203 (E). Suppose that U is an open set in which any two points can be joined by a contour, and f u Cis holomorphic. Explain why, if A is a closed set in U, then the maximum value of Ifl on A must occur at a point of oA, if at all. Task 204 (C). Find the maximum value of lz- zl on the closed unit disk D (z:21s1) On the next page, you will see that if f is holomorphic on all of C, then even if Ifl is bounded (without assuming it has a maximum value) f must be constant! t If z E A and zo 뜨 aA, then zo has an ε neighborhood that is contained in A. what do you know if n h value at such a point of A? 43

Answer 1

a 22 1) に) 리스 ↑ 22一 2 2.2 2