1. Solutions to exercises. , Im(

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1 Chapter (a) 7 i (b) + i (c) i (d) 5 (e) + i Solutions to exercises (a) Re( z a z+a ) = z a z a, Im( z +a +are(z) z+a ) = aim(z) z +a +are(z) (b) i (c) (d) if n = 4k, k Z, i if n = 4k +, k Z, - if n = 4k +,k Z, i if n = 4k +,k Z (a) 5, i (b) 5 5, 5 0i (c) 0, ( ) + i ( + 9) (d) 8, 8i 4(a) e i π (b) e i π 4 (c) e i 5π 6 (d) e i π (e) e itan ( ) (f) 5 (g) 6e itan ( ) 5 (h) 4 9 eiπ 5(a) + i (b) 4i (c) (d) 6 (a) cos(log())+isin(log())=e ilog() (b) 5i = 5e i π (c) ei = e e i π (d) e φ cos( π 4 + φ) + ieφ sin( π 4 + φ) = eφ e i( π 4 +φ) 8(a) z = ±5 (b) z = ( ± i) (c) z = 5 ( ± i) (d) z = ( ± ) (e) z = 0, 0 (a) z = e i kπ, k = 0,,, 5 (b) z = e i (k+)π 4, k = 0,,, (c) z = e i (k+)π 6, k = 0,,, 5 (d) z = e i kπ, k = 0,,, and z = e i kπ, k = 0,,, z = e i π 4 and z = e i 5π 4 4 (a) open, bounded and connected

2 (b) open, not bounded, connected (c) open, bounded and connected (d) closed, bounded and connected (e) open, bounded and connected 6 (b) {z : z < } (c) {z : z = } [, ) {} (d) {} 7 E has three connected components A = {z : z < }, B = [, ) and C = {} Three ways to write E as a union of two separated sets is thus E = (A B) C = (A C) B = (C B) A 0 4 Chapter (a) 0 (b) + i 9 T (z) = ad bc (cz+d), it is nowhere zero (a) differentiable and holomorphic in C with derivative e x e iy (b) nowhere differentiable or holomorphic (c) differentiable only on {x + iy C: x = y} with derivative x, nowhere holomorphic (d) nowhere differentiable or holomorphic (e) differentiable and holomorphic in C with derivative sin(x)cosh(y) icos(x)sinh(y) (f) nowhere differentiable or holomorphic (g) differentiable only at 0 with derivative 0, nowhere holomorphic (h) differentiable only at 0 with derivative 0, nowhere holomorphic (i) differentiable only at i with derivative i, nowhere holomorphic (j) differentiable and holomorphic in C with derivative y xi = iz (k) differentiable only at 0 with derivative 0, nowhere holomorphic (l) differentiable only at 0 with derivative 0, nowhere holomorphic 6 (a) v = xy (b) v = cos(x)sinh(y) (c) v = 4xy + y (d) v = y x x x +y 8 x +y is harmonic on C \ {0} but not on C, x +y is not harmonic Chapter 9 (a) z+ z (c) z+i iz+ (+i)z (+i) (b) z sending + i to -i z+ z+ (a) the lower halfplane {w : im(w) < 0} (b) {w : Im < 0, w < } (c) {w : Re(w) <, w > } 0 One solution is iz i z i 4 ±

3 5 (a) z (b) (+i)z z +i (c) +i z 8 0,,, i, the point (,, 0) is not in the domain of the function φ 0 (,, ), {z : z i = } 6 (a) { w = } (b) { w = e} (c) { w e} (a) (b) e π (c) e π (d) cos( e e ) isin( e e ) (e) + 4i (f) 4 cos( π 8 ) + i 4 sin( π 8 ) (g) i (h) (a) i π (b) e π (c) + i π 4 C \ ( (, ] [, ) ) 6 (a) differentiable at 0, nowhere holomorphic (b) differentiable and holomorphic on C \ {, e i π, e i π } (c) differentiable and holomorphic on C \ {x + iy C: x, y = } (d) nowhere differentiable or holomorphic (e) differentiable and holomorphic on C \ {x + iy C: x, y = 0} (f) differentiable and holomorphic in C (ie entire) 7 (a) z = i (b) There is no solution (c) z = lnπ + i( π + πk), k Z (d) z = π + πk ± 4i, k Z (e) z = π + πk, k Z (f) z = πki, k Z (g) z = πk, k Z (h) z = i 8 {w : Re(w) [0, ], Im(w) ( π, π]} 40 f (z) = cz c 4 The exponentially growing spiral {w(t) = e t (cos(t) + isin(t)), t R}, tending to infinity as t tends to positive infintity and to 0 as t tends to negative infinity 4 Chapter 4 (a) 6 (b) π (c) 4 (d) 5 (a)

4 4 (b) + i (c) + i πi 4 (a) 8πi (b) 0 (c) 0 (d) 0 5 (a) +i, +i, i, (b) πi, π, 0, πi (c) πr i, πr, 0, πr i 6 (a) e (e + cos() + isin()) (b) 0 (c) e+i i 8 9 e z0 0+0i 0 7+ln( ) + i(6 + tan ( 7 )) cosh() (c) i π 4 0 if r (0, ) (, ), π if r (, ) π 8 0 for r < a, πi for r > a 9 0 for r =, πi for r =, 0 for r = 5 πi 0 πi, πi (e ) 5 Chapter 5 (a) πi (b) πi (c) 4πi (d) 0 (a) 0 (b) πi (c) 0 (d) πi (e) 0 (f) 0 (g) i πsinh( ) (h) πie w if w <, 0 if w > (i) πi 7 4 πi(e 4 + e ) 8 Any simply connected set which does not contain the origin, for example C \ (, 0] 9 (a) ( + i)cosh( π )

5 5 (b) ( i)sinh(π) ( + i)cosh( π ) 6 Chapter 6 4(b) ie z 5 No 7 Chapter 7 (a) divergent (b) convergent (limit 0) (c) divergent (d) convergent (limit i ) (e) convergent (limit 0) (a) convergent (b) divergent (c) divergent (d) convergent 0 (a) Pointwise convergence on z <, uniform convergence on z r for r < (b) Pointwise convergence on z, uniform convergence on z (c) Pointwise convergence on Re(z) 0, uniform convergence on z r for r > 0 4 (a) k 0 ( 4)k z k (b) k 0 z k 6 k (c) k k z k 4 k 5 (a) ( ) k k 0 (b) k 0 (k)! zk ( ) k (k)! z4k ( ) k+ (k )! zk+ (c) k (d) ( ) k+ k k (k)! z k 7 (a) k 0 ( )k (z ) k, convergence radius (b) ( ) k k k (z ) k,convergence radius 0 (a) if a <, if a =, and 0 if a > (b) (c) (d) (e) (f) (g) 4 (a) exp(z ) (b) ( z) z (c) ( z) 8 Chapter 8 (a) {z C: z < }, {z C: z r} for any r < (b) C, {z C: z r} for any r

6 6 (c) {z C: z > }, {z C: r z R} for any < r R (a) ( z) (b) sinh(z) (c) + z 4 k 0 e k! (z )k ( ) k k+ zk+ 5 k 0 6 (a) +z = (z ) + 4 (z ) + 0 (z ) +, the convergence radius is (b) e z + = 4 z + 0 z + 48 z +, the convergence radius is π 0 The maximum is (attained at z = ±i), and the minimum is (attained at z = ±) One Laurent series is k 0 ( )k (z ) k, converging for z > One Laurent series is k 0 ( )k (z ) k, converging for z > 4 One Laurent series is (z + ) +, converging for z 5 sin(z) = z + 6 z z + ( ) k (k)! zk 0 (a) k 0 + z z4 + (a) (b) (c) 4 4(a) ±i, multiplicity 4 (b) kπ, k Z multiplicity (c) (k + )iπ, k Z, multiplicity (d) 0, multiplicity, and π + kπ, multiplicity ( )k ( + )z k for z <, ( ) k k+ k 0 5 k 0 k<0 ( + ( )k k+ )z k for z > 8 It is less than or equal to 9 (a) R (b) R (c) R (d) R (e) R k+ z k k<0 zk for < z < and 9 Chapter 9 (a), i, i, of order 4,, (b) kπ, k Z \ {0}, of order (c) 0 of order 4 (d) kiπ, k Z, of order (e) kiπ, k Z \ {0},of order 7 (a) 0 (b) (c) 4 9 (a) One Laurent series is k ( ) k 4 k+ (z ) k, converging for 0 < z < 4

7 7 (b) πi 8 0 (a) πi (b) 7πi (c) πi 7 (d) πi (e) πi (f) 0 (a) k 0 4 ek! (z + k ) (b) πi e! (a) (b) (c) 5 (d) e (e) 4 4 (a) +i 8 (b) πi (c) πi( cos()) (d) pii 5 (b) 0 6 (a) π for R >, 0 for R < (c) π 7 πi f(a) f(b) a b