Chapter_1

# Chapter_1 - IV 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18...

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Unformatted text preview: IV 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 X X X X X X X 2 X X X X 3 4 5 X 6 X X X 7 8 X 1 I.1.1 Identify and sketch the set of points satisfying. (a) j z &amp; 1 &amp; i j = 1 (f) &lt; Im z &lt; &amp; (b) 1 &lt; j 2 z &amp; 6 j &lt; 2 (g) &amp; &amp; &lt; Re z &lt; &amp; (c) j z &amp; 1 j 2 + j z + 1 j 2 &lt; 8 (h) j Re z j &lt; j z j (d) j z &amp; 1 j + j z + 1 j 2 (i) Re ( iz + 2) &gt; (e) j z &amp; 1 j &lt; j z j (j) j z &amp; i j 2 + j z + i j 2 &lt; 2 Solution (a) Circle, centre 1 + i , radius 1 . j z &amp; 1 &amp; i j = 1 , j ( x &amp; 1) + i ( y &amp; 1) j = 1 , ( x &amp; 1) 2 + ( y &amp; 1) 2 = 1 2 (b) Annulus with centre 3 , inner radius 1 = 2 , outer radius 1 . 1 &lt; j 2 z &amp; 6 j &lt; 2 , 1 &lt; 2 j z &amp; 3 j &lt; 2 , , 1 = 2 &lt; j z &amp; 3 j &lt; 1 , (1 = 2) 2 &lt; ( x &amp; 3) 2 + y 2 &lt; 1 2 (c) Disk, centre , radius p 3 . j x + iy &amp; 1 j 2 + j x + iy + 1 j 2 &lt; 8 , , ( x &amp; 1) 2 + y 2 + ( x + 1) 2 + y 2 &lt; 8 , x 2 + y 2 &lt; &amp; p 3 2 (d) Interval [ &amp; 1 ; 1] . j z &amp; 1 j + j z + 1 j 2 , q ( x &amp; 1) 2 + y 2 2 &amp; q ( x + 1) 2 + y 2 , , q ( x &amp; 1) 2 + y 2 2 2 &amp; q ( x + 1) 2 + y 2 2 , , q ( x + 1) 2 + y 2 x + 1 , q ( x + 1) 2 + y 2 2 ( x + 1) 2 , y = 0 Now, take y = 0 in the inequality, and compute the three interval 2 x &lt; &amp; 1 ; j x &amp; 1 j + j x + 1 j = &amp; ( x &amp; 1) &amp; ( x + 1) = &amp; 2 x as 2 ; then x &lt; &amp; 1 ; &amp; 1 x 1 ; j x &amp; 1 j + j x + 1 j = &amp; ( x &amp; 1) + ( x + 1) = 2 as 2 ; then &amp; 1 x 1 ; x &gt; 1 ; j x &amp; 1 j + j x + 1 j = ( x &amp; 1) + ( x + 1) = 2 x as 2 ; then x &gt; 1 : (e) Half&amp;plane x &gt; 1 = 2 . j z &amp; 1 j &lt; j z j , j z &amp; 1 j 2 &lt; j z j 2 , j x + iy &amp; 1 j 2 &lt; j x + iy j 2 , , ( x &amp; 1) 2 + y 2 &lt; x 2 + y 2 , x &gt; 1 = 2 (f) Horizontal strip, &lt; y &lt; &amp; . (g) Vertical strip, &amp; &amp; &lt; x &lt; &amp; . (h) C n R . j Re z j &lt; j z j , j Re ( x + iy ) j 2 &lt; j x + iy j 2 , x 2 &lt; x 2 + y 2 , j y j &gt; (i) Half plane y &lt; 2 . Re ( iz + 2) &gt; , Re ( i ( x + iy ) + 2) &gt; , &amp; y + 2 &gt; , y &lt; 2 (j) Empty set. j z &amp; i j 2 + j z + i j 2 &lt; 2 , j x + iy &amp; i j 2 + j x + iy + i j 2 &lt; 2 , , x 2 + ( y &amp; 1) 2 + x 2 + ( y + 1) 2 &lt; 2 , x 2 + y 2 &lt; 3 I.1.2 Verify from the de&amp;nitions each of the identities (a) z + w = &amp; z + &amp; w (b) zw = &amp; z &amp; w (c) j &amp; z j = j z j (d) j z j 2 = z &amp; z Draw sketches to illustrate (a) and (c). Solution Substitute z = x + iy and w = u + iv , and use the de&amp;nitions. (a) z + w = ( x + iy ) + ( u + iv ) = ( x + u ) + ( y + v ) i = = ( x + u ) &amp; ( y + v ) i = ( x &amp; iy ) + ( u &amp; iv ) = &amp; z + &amp; w: (b) zw = ( x + iy ) ( u + iv ) = ( xu &amp; yv ) + ( xv + yu ) i = = ( xu &amp; yv ) &amp; ( xv + yu ) i = ( x &amp; iy ) ( u &amp; iv ) = &amp; z &amp; w: (c) j &amp; z j = &amp; &amp; x + iy &amp; &amp; = j x &amp; iy j = q x 2 + ( &amp; y ) 2 = p x 2 + y 2 = j x + iy j = j z j : (d) j z j 2 = j x + iy j 2 = p x 2 + y 2 2 = x 2 + y 2 = = x 2 &amp; i 2 y 2 = ( x + iy ) ( x &amp; iy ) = z &amp; z: 4 I.1.3 Show that the equation j z j 2 &amp; 2 Re (&amp; az ) + j a j 2 = &amp; 2 represents a circle centered at a with radius &amp; ....
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## This note was uploaded on 11/13/2010 for the course MATH 132 taught by Professor Grossman during the Spring '08 term at UCLA.

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Chapter_1 - IV 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18...

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