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Question: complete 2 611 14 17 18 25 27...

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complete 2, 6,11, 14, 17, 18, 25, 27

For each of the sets in Exercises 1 to 8, (a) describe the interior and the boundary, (b)state whether the set is open or closed or neither open nor closed, (c) state whether the interior of the set is connected (if it has an interior). 3. C={z = x + iy: x2 < y} 4. D -{z: Re(a2) 4) 9. Let a and B be complex numbers with0. Describe the set of points az + B as z varies over (a) the first quadran,-x + iy: x > 0 and y >0); (b) the upper half-plane, {z = x + iy: y > 0}; (c) the disc {z: Iz! < R}. Show that in each case the resulting set is open and connected. (Hint: First investigate the set ) 10. Describe the set of points2 as z varies over the second quadrant: {z»x + ijy x <0and y >0. Show that this is an open, connected set (Hint: Use the polar representation of z) 11. A set S in the plane is bounded if there is a positive number M such that lz M for all z in S; otherwise, S is unbounded. In Exercises 1 to 8, six of the given sets are unbounded. Find them 1.3 Subsets of the Plane 29 12. Which, if any, of the sets given in Exercises 1 to 8 contains oo? 13. (a) Show that the union of two nonempty open sets is open. Do the same, replacing open with “closed. Do the same replacing “union with intersection. (b) Repeat part (a) replacing two with finitely many. a domain. Re z <). Show that both 2, and 22 are domains but 2,2 is not. 14. Let D, and D2 be domains with a nonempty intersection. Show that Di u D2 is 15. Let Ω|-|z: 1<\리< 2 and Rez >-卦and Ω,-{z: 1<\리< 2 and 16. Let D be a domain and let p and q be points of D. Show that there is a polygonal curve joining p to q whose line segments are either horizontal or vertical (both types can be used). (Hint: Replace a slanting segment by (perhaps many) horizontal and vertical segments, see Fig. 1.18.)

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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