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Question: complete 1727...

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complete 17,27

17. Fix a nonzero complex number zo.Show that the set D obtained from the plane by deleting the ray (tzo:0 <t< co is a domain. 18. An open set D is star-shaped if there is some point p in D with the property that the line segment from p to z lies in D for each z in D. (a) Show that the disc {z: lz -zol <r is star-shaped. (b) Show that any convex set is star- shaped. 19. Determine which of the following sets are star-shaped: (b) D={z = x + iy: x > 0 and Izi > 1} (d) D=(z=x+iy: x > 0 and [x > y + 1 or x > 1-y]} 20. Show that each star-shaped set is connected. Chapter 1 The Complex Plane Separation of a Point and Convex Set* Let C be a closed convex set and zo a point not in C. It is a fact that there is a point p in C with r-lzo-pl < Izo-gl for all q in C. (This last statement requires a bit of proof, but let us assume its validity.) 21. Show that the only point of C in the disc |zo- zl r is the point p. 22. Let L be the perpendicular bisector of the line segment from zo to p. Show that no point of C lies on L or in the half-plane, determined by L, which contains zo. 23. Conclude from Exercise 22 that L separates zo from C: zo and C lie in the two open half-planes determined by L, but not in the same open half-plane. 24. Show that each closed convex set is the intersection of all the closed half-planes that contain it. Topological Properties 25. Show that the boundary of any set D is itself a closed set. 26. Show that if pe D, then p is either an interior point of D or a boundary point 27. Show that a set D coincides with its boundary if and only if D is closed and D 28. Show that if D is a set and E is a closed set containing D, then E must contain 29. Show that if D is a set and S is an open set that is a subset of D, then S must be 30. Let C be a bounded closed convex set and let D be the complement of C. Show of D has no interior points. the boundary of D composed entirely of interior points of D that D is a domain.

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