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 & 1 & i j = 1 (f) < Im z < & (b) 1 < j 2 z & 6 j < 2 (g) & & < Re z < & (c) j z & 1 j 2 + j z + 1 j 2 < 8 (h) j Re z j < j z j (d) j z & 1 j + j z + 1 j 2 (i) Re ( iz + 2) > (e) j z & 1 j < j z j (j) j z & i j 2 + j z + i j 2 < 2 Solution (a) Circle, centre 1 + i , radius 1 . j z & 1 & i j = 1 , j ( x & 1) + i ( y & 1) j = 1 , ( x & 1) 2 + ( y & 1) 2 = 1 2 (b) Annulus with centre 3 , inner radius 1 = 2 , outer radius 1 . 1 < j 2 z & 6 j < 2 , 1 < 2 j z & 3 j < 2 , , 1 = 2 < j z & 3 j < 1 , (1 = 2) 2 < ( x & 3) 2 + y 2 < 1 2 (c) Disk, centre , radius p 3 . j x + iy & 1 j 2 + j x + iy + 1 j 2 < 8 , , ( x & 1) 2 + y 2 + ( x + 1) 2 + y 2 < 8 , x 2 + y 2 < & p 3 2 (d) Interval [ & 1 ; 1] . j z & 1 j + j z + 1 j 2 , q ( x & 1) 2 + y 2 2 & q ( x + 1) 2 + y 2 , , q ( x & 1) 2 + y 2 2 2 & 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 < & 1 ; j x & 1 j + j x + 1 j = & ( x & 1) & ( x + 1) = & 2 x as 2 ; then x < & 1 ; & 1 x 1 ; j x & 1 j + j x + 1 j = & ( x & 1) + ( x + 1) = 2 as 2 ; then & 1 x 1 ; x > 1 ; j x & 1 j + j x + 1 j = ( x & 1) + ( x + 1) = 2 x as 2 ; then x > 1 : (e) Half&plane x > 1 = 2 . j z & 1 j < j z j , j z & 1 j 2 < j z j 2 , j x + iy & 1 j 2 < j x + iy j 2 , , ( x & 1) 2 + y 2 < x 2 + y 2 , x > 1 = 2 (f) Horizontal strip, < y < & . (g) Vertical strip, & & < x < & . (h) C n R . j Re z j < j z j , j Re ( x + iy ) j 2 < j x + iy j 2 , x 2 < x 2 + y 2 , j y j > (i) Half plane y < 2 . Re ( iz + 2) > , Re ( i ( x + iy ) + 2) > , & y + 2 > , y < 2 (j) Empty set. j z & i j 2 + j z + i j 2 < 2 , j x + iy & i j 2 + j x + iy + i j 2 < 2 , , x 2 + ( y & 1) 2 + x 2 + ( y + 1) 2 < 2 , x 2 + y 2 < 3 I.1.2 Verify from the de&nitions each of the identities (a) z + w = & z + & w (b) zw = & z & w (c) j & z j = j z j (d) j z j 2 = z & z Draw sketches to illustrate (a) and (c). Solution Substitute z = x + iy and w = u + iv , and use the de&nitions. (a) z + w = ( x + iy ) + ( u + iv ) = ( x + u ) + ( y + v ) i = = ( x + u ) & ( y + v ) i = ( x & iy ) + ( u & iv ) = & z + & w: (b) zw = ( x + iy ) ( u + iv ) = ( xu & yv ) + ( xv + yu ) i = = ( xu & yv ) & ( xv + yu ) i = ( x & iy ) ( u & iv ) = & z & w: (c) j & z j = & & x + iy & & = j x & iy j = q x 2 + ( & 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 & i 2 y 2 = ( x + iy ) ( x & iy ) = z & z: 4 I.1.3 Show that the equation j z j 2 & 2 Re (& az ) + j a j 2 = & 2 represents a circle centered at a with radius & ....
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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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