ALL 15 CHAPTERS COVERED
SOLUTIONS MANUAL
,Chapter 1
1.1 (2/3)10 = 0.0173 yd; 6(2/3)10 = 0.104 yd (compared to a total of 5 yd)
1.3 5/9 1.4 9/11 1.5 7/12
1.6 11/18 1.7 5/27 1.8 25/36
1.9 6/7 1.10 15/26 1.11 19/28
1.13 $1646.99 1.15 Blank area = 1
1.16 At x = 1: 1/(1 + r); at x = 0: r/(1 + r); maximum escape at x = 0 is 1/2.
2.1 1 2.2 1/2 2.3 0
2.4 ∞ 2.5 0 2.6 ∞
2.7 e2 2.8 0 2.9 1
4.1 an = 1/2n → 0; Sn = 1 − 1/2n → 1; Rn = 1/2n → 0
4.2 an = 1/5n−1 → 0; Sn = (5/4)(1 − 1/5n ) → 5/4; Rn = 1/(4 · 5n−1 ) → 0
4.3 an = (−1/2)n−1 → 0; Sn = (2/3)[1 − (−1/2)n ] → 2/3; Rn = (2/3)(−1/2)n → 0
4.4 an = 1/3n → 0; Sn = (1/2)(1 − 1/3n ) → 1/2; Rn = 1/(2 · 3n ) → 0
4.5 an = (3/4)n−1 → 0; Sn = 4[1 − (3/4)n ] → 4; Rn = 4(3/4)n → 0
1 1 1
4.6 an = → 0; Sn = 1 − → 1; Rn = →0
n(n + 1) n+1 n+1
(−1)n+1 (−1)n
1 1
4.7 an = (−1)n+1 + → 0 ; Sn = 1 + → 1; Rn = →0
n n+1 n+1 n+1
5.1 D 5.2 Test further 5.3 Test further
5.4 D 5.5 D 5.6 Test further
5.7 Test further 5.8 Test further
5.9 D 5.10 D
6.5 (a) D 6.5 (b) D
R∞
Note: In the following answers, I= an dn; ρ = test ratio.
6.7 D, I = ∞ 6.8 D, I = ∞ 6.9 C, I = 0
6.10 C, I = π/6 6.11 C, I = 0 6.12 C, I = 0
6.13 D, I = ∞ 6.14 D, I = ∞ 6.18 D, ρ = 2
6.19 C, ρ = 3/4 6.20 C, ρ = 0 6.21 D, ρ = 5/4
6.22 C, ρ = 0 6.23 D, ρ = ∞ 6.24 D, ρ = 9/8
6.25 C, ρ = 0 6.26 C, ρ = (e/3)3 6.27 D, ρ =P100
6.28 C, ρ =P 4/27 6.29 D, ρ =P2 6.31 D, cf. P n−1
6.32 D, cf. n−1 6.33 C, cf. 2−n 6.34 C, cf. n−2
P −2 P −1/2
6.35 C, cf. n 6.36 D, cf. n
1
,Chapter 1 2
7.1 C 7.2 D 7.3 C 7.4 C
7.5 C 7.6 D 7.7 C 7.8 C
P −1
9.1 D, cf. n 9.2 D, an 6→ 0 P −1
9.3 C, I =P0 9.4 D, I = ∞, or cf. n
9.5 C, cf. n−2 9.6 C, ρ = 1/4
9.7 D, ρ = 4/3 9.8 C, ρ = 1/5
9.9 D, ρ = e 9.10 D, an 6→
P 0 −2
D, I = ∞, or cf.P n−1
P
9.11 9.12 C, cf. n
9.13 C, I = 0, or cf. n−2 9.14 C, alt.Pser.
9.15 D, ρ = ∞, an 6→ 0 9.16 C, cf. n−2
9.17 C, ρ = 1/27 9.18 C, alt. ser.
9.19 C 9.20 C
9.21 C, ρ = 1/2
9.22 (a) C (b) D (c) k > e
10.1 |x| < √ 1 10.2 |x| < 3/2 10.3 |x| ≤ 1
10.4 |x| ≤ 2 10.5 All x 10.6 All x
10.7 −1 ≤ x < 1 10.8 −1 < x ≤ 1 10.9 |x| < 1
10.10 |x| ≤ 1 10.11 −5 ≤ x < 5 10.12 |x| < 1/2
10.13 −1 < x ≤ 1 10.14 |x| < 3 10.15 −1 < x < 5
10.16 −1 < x < 3 10.17 −2 < x ≤ 0 10.18 −3/4 ≤ x ≤ −1/4
10.19 |x| < 3 10.20 All x 10.21 0 ≤ x √≤1
10.22 No x 10.23 x > 2 or x < −4 10.24 |x| < 5/2
10.25 nπ − π/6 < x < nπ + π/6
(−1)n (2n − 1)!!
−1/2 −1/2
13.4 = 1; =
0 n (2n)!!
Answers to part (b), Problems 5 to 19:
∞ n+2 ∞
X x X 1/2 n+1
13.5 − 13.6 x (see Example 2)
1
n 0
n
∞ ∞
(−1)n x2n
X X −1/2
13.7 13.8 (−x2 )n (see Problem 13.4)
0
(2n + 1)! 0
n
∞ ∞
X X (−1)n x4n+2
13.9 1 + 2 xn 13.10
1 0
(2n + 1)!
∞ n n ∞
X (−1) x X (−1)n x4n+1
13.11 13.12
0
(2n + 1)! 0
(2n)!(4n + 1)
∞ n 2n+1 ∞
X (−1) x X x2n+1
13.13 13.14
0
n!(2n + 1) 0
2n + 1
∞
x2n+1
X −1/2
13.15 (−1)n
0
n 2n + 1
∞ 2n ∞
X x X xn
13.16 13.17 2
0
(2n)! n
oddn
∞
X (−1)n x2n+1 ∞
X −1/2 x2n+1
13.18 13.19
0
(2n + 1)(2n + 1)! 0
n 2n + 1
2 3 5 6
13.20 x + x + x /3 − x /30 − x /90 · · ·
13.21 x2 + 2x4 /3 + 17x6 /45 · · ·
13.22 1 + 2x + 5x2 /2 + 8x3 /3 + 65x4 /24 · · ·
13.23 1 − x + x3 − x4 + x6 · · ·
,Chapter 1 3
13.24 1 + x2 /2! + 5x4 /4! + 61x6 /6! · · ·
13.25 1 − x + x2 /3 − x4 /45 · · ·
13.26 1 + x2 /4 + 7x4 /96 + 139x6 /5760 · · ·
13.27 1 + x + x2 /2 − x4 /8 − x5 /15 · · ·
13.28 x − x2 /2 + x3 /6 − x5 /12 · · ·
13.29 1 + x/2 − 3x2 /8 + 17x3 /48 · · ·
13.30 1 − x + x2 /2 − x3 /2 + 3x4 /8 − 3x5 /8 · · ·
13.31 1 − x2 /2 − x3 /2 − x4 /4 − x5 /24 · · ·
13.32 x + x2 /2 − x3 /6 − x4 /12 · · ·
13.33 1 + x3 /6 + x4 /6 + 19x5 /120 + 19x6 /120 · · ·
13.34 x − x2 + x3 − 13x4 /12 + 5x5 /4 · · ·
13.35 1 + x2 /3! + 7x4 /(3 · 5!) + 31x6 /(3 · 7!) · · ·
13.36 u2 /2 + u4 /12 + u6 /20 · · ·
13.37 −(x2 /2 + x4 /12 + x6 /45 · · · )
13.38 e(1 − x2 /2 + x4 /6 · · · )
4
13.39 1 − (x − π/2)2 /2! + (x − π/2) /4! · · ·
3
13.40 1 − (x − 1) + (x − 1)2 − (x − 1) · · ·
13.41 e [1 + (x − 3) + (x − 3) /2! + (x − 3)3 /3! · · · ]
3 2
2
13.42 −1 + (x − π) /2! − (x − π)4 /4! · · ·
13.43 −[(x − π/2) + (x − π/2)3 /3 + 2(x − π/2)5 /15 · · · ]
13.44 5 + (x − 25)/10 − (x − 25)2 /103 + (x − 25)3 /(5 · 104 ) · · ·
14.6 Error < (1/2)(0.1)2 ÷ (1 − 0.1) < 0.0056
14.7 Error < (3/8)(1/4)2 ÷ (1 − 41 ) = 1/32
14.8 For x < 0, error < (1/64)(1/2)4 < 0.001
For x > 0, error < 0.001 ÷ (1 − 12 ) = 0.002
1
14.9 Term n + 1 is an+1 = (n+1)(n+2) , so Rn = (n + 2)an+1 .
14.10 S4 = 0.3052, error < 0.0021 (cf. S = 1 − ln 2 = 0.307)
15.1 −x4 /24 − x5 /30 · · · ≃ −3.376 × 10−16
15.2 x8 /3 − 14x12 /45 · · · ≃ 1.433 × 10−16
15.3 x5 /15 − 2x7 /45 · · · ≃ 6.667 × 10−17
15.4 x3 /3 + 5x4 /6 · · · ≃ 1.430 × 10−11
15.5 0 15.6 12 15.7 10!
15.8 1/2 15.9 −1/6 15.10 −1
15.11 4 15.12 1/3 15.13 −1
15.14 t − t3 /3, error < 10−6 15.15 23 t3/2 − 52 t5/2 , error < 17 10−7
15.16 e2 − 1 15.17 √cos π2 = 0
15.18 ln 2 15.19 2
15.20 (a) 1/8 (b) 5e (c) 9/4
15.21 (a) 0.397117 (b) 0.937548 (c) 1.291286
15.22 (a) π 4 /90 (b) 1.202057 (c) 2.612375
15.23 (a) 1/2 (b) 1/6 (c) 1/3 (d) −1/2
15.24 (a) −π (b) 0 (c) −1
(d) 0 (e) 0 (f) 0
15.27 (a) 1 − vc = 1.3 × 10−5 , or v = 0.999987c
(b) 1 − vc = 5.2 × 10−7
(c) 1 − vc = 2.1 × 10−10
(d) 1 − vc = 1.3 × 10−11
15.28 mc2 + 12 mv 2
15.29 (a) F/W = θ + θ3 /3 · · ·
(b) F/W = x/l + x3 /(2l3 ) + 3x5 /(8l5 ) · · ·
,Chapter 1 4
15.30 (a) T = F (5/x + x/40 − x3 /16000 · · · )
(b) T = 12 (F/θ)(1 + θ2 /6 + 7θ4 /360 · · · )
15.31 (a) finite (b) infinite
16.1 (c) overhang: 2 3 10 100
books needed: 32 228 2.7 × 108 4 × 1086
P −3/2
16.4 C, ρ = 0 16.5 D, an 6→ P 0 −1 16.6 C, cf. n
16.7 D, I = ∞ 16.8 D, cf. n 16.9 −1 ≤ x < 1
16.10 |x| < 4 16.11 |x| ≤ 1 16.12 |x| < 5
16.13 −5 < x ≤ 1
16.14 1 − x2 /2 + x3 /2 − 5x4 /12 · · ·
16.15 −x2 /6 − x4 /180 − x6 /2835 · · ·
16.16 1 − x/2 + 3x2 /8 − 11x3 /48 + 19x4 /128 · · ·
16.17 1 + x2 /2 + x4 /4 + 7x6 /48 · · ·
16.18 x − x3 /3 + x5 /5 − x7 /7 · · ·
16.19 −(x − π) + (x − π)3 /3! − (x − π)5 /5! · · ·
16.20 2 + (x − 8)/12 − (x − 8)2 /(25 · 32 ) + 5(x − 8)3 /(28 · 34 ) · · ·
16.21 e[1 + (x − 1) + (x − 1)2 /2! + (x − 1)3 /3! · · · ]
16.22 arc tan 1 = π/4 16.23 1 − (sinπ)/π = 1
16.24 eln 3 − 1 = 2 16.25 −2
16.26 −1/3 16.27 2/3
16.28 1 16.29 6!
16.30 (b) For N = 130, 10.5821 < ζ(1.1) < 10.5868
16.31 (a) 10430 terms. For N = 200, 100.5755 < ζ(1.01) < 100.5803
16.31 (b) 2.66 × 1086 terms. For N = 15, 1.6905 < S < 1.6952
200 86
16.31 (c) ee = 103.1382×10 terms. For N = 40, 38.4048 < S < 38.4088
,Chapter 2
x y r θ
√
4.1 1 1 √2 π/4
4.2 −1 1
√ 2 3π/4
4.3 1
√ − 3 2 −π/3
4.4 − 3 1 2 5π/6
4.5 0 2 2 π/2
4.6 0 −4 4 −π/2
4.7 −1 0 1 π
4.8 3 0 √3 0
4.9 −2 2 2√2 3π/4
4.10 √2 −2 2 2 −π/4
4.11 3 1√ 2 π/6
4.12 −2 −2 3 4 −2π/3
4.13 √0 −1
√ 1 3π/2
4.14 2 2 2 π/4
4.15 −1 0 1 −π or π
4.16 5 0 √5 0
4.17 1 −1 2 −π/4
4.18 0 3 3 π/2
4.19 4.69 1.71 5 20◦ = 0.35
4.20 −2.39 −6.58 7 −110◦ = −1.92
√
5.1 1/2 −1/2 1/√2 −π/4
5.2 −1/2 −1/2 1/ 2 −3π/4 or 5π/4
5.3 1 0 1 0
5.4 0 2
√ 2 π/2
5.5 2 2 3 4 π/3
5.6 −1 0 √1 π
5.7 7/5 −1/5 2 −8.13◦ = −0.14
5.8 1.6 −2.7 3.14 −59.3◦ = −1.04
5.9 −10.4 22.7 p 25 2 = 114.6◦
5.10 −25/17 19/17 58/17 142.8◦ = 2.49
5.11 17 −12 20.8 −35.2◦ = −0.615
5.12 2.65 1.41 3 28◦ = 0.49
5.13 1.55 4.76 5 2π/5
5.14 1.27 −2.5 2.8 −1.1 = −63◦
5.15 21/29 −20/29 1 −43.6◦ = −0.76
5.16 1.53 −1.29 2 −40◦ = −0.698
5.17 −7.35 −10.9 13.1 −124◦ = −2.16
5.18 −0.94 −0.36 1 201◦ or −159◦,
3.51 or −2.77
5
,Chapter 2 6
5.19 (2 + 3i)/13; (x − yi)/(x2 + y 2 )
5.20 (−5 + 12i)/169; (x2 − y 2 − 2ixy)/(x2 + y 2 )2
5.21 (1 + i)/6; (x + 1 − iy)/[(x + 1)2 + y 2 ]
5.22 (1 + 2i)/10; [x − i(y − 1)]/[x2 + (y − 1)2 ]
5.23 (−6 − 3i)/5; (1 − x2 − y 2 + 2yi)/[(1 − x)2 + y 2 ]
5.24 (−5 − 12i)/13; (x2 − y 2 + 2ixy)/(x
p
2
+ y2)
5.26 1√ 5.27 13/2 5.28 1
5.29 5 5 5.30 3/2 5.31 1
5.32 169 5.33 5 5.34 1
5.35 x = −4, y = 3 5.36 x = −1/2, y = 3
5.37 x=y=0 5.38 x = −7, y = 2
5.39 x = y = any real number 5.40 x = 0, y = 3
5.41 x = 1, y = −1 5.42 x = −1/7, y = −10/7
5.43 (x, y) = (0, 0), or (1, 1), or (−1, 1)
5.44 x = 0, y = −2
5.45 x = 0, any real y; or y = 0, any real x
5.46 y = −x √
5.47 (x, y) = (−1, 0), (1/2, ± 3/2)
5.48 x = 36/13, y = 2/13
5.49 x = 1/2, y = 0
5.50 x = 0, y ≥ 0
5.51 Circle, center at origin, radius = 2
5.52 y axis
5.53 Circle, center at (1, 0), r = 1
5.54 Disk, center at (1, 0), r = 1
5.55 Line y = 5/2
5.56 Positive y axis
5.57 Hyperbola, x2 − y 2 = 4
5.58 Half plane, x > 2
5.59 Circle, center at (0, −3), r = 4
5.60 Circle, center at (1, −1), r = 2
5.61 Half plane, y < 0
5.62 Ellipse, foci at (1, 0) and (−1, 0), semi-major axis = 4
5.63 The coordinate axes
5.64 Straight lines, y = ± x
5.67 v = (4t2 + 1)−1 , a = 4(4t2 + 1)−3/2
5.68 Motion around circle r = 1, with v = 2, a = 4
√ √
6.2 D, ρ = 2 6.3 C, ρ = 1/ 2 6.4 D, |an | √
= 1 6→ 0
6.5 D 6.6 C 6.7 D, ρ = √ 2
6.8 D, |an | = 1 6→ 0 6.9 C 6.10 C, ρ = p2/2
6.11 C, ρ = 1/5 6.12 C 6.13 C, ρ = 2/5
7.1 All z 7.2 |z| < 1 7.3 All z
7.4 |z| < 1 7.5 |z| < 2 7.6 |z| < 1/3
7.7 All z 7.8 All z 7.9 |z| < 1
7.10 |z| < 1 7.11 |z| < 27 7.12 |z| < 4
7.13 |z − i| < 1 √ 7.14 |z − 2i| < 1 7.15 |z − (2 − i)| < 2
7.16 |z + (i − 3)| < 1/ 2
8.3 See Problem 17.30.
,Chapter 2 7
√
9.1 (1 − i)/ √2 9.2 i 9.3 −9i
9.4 −e(1 + i 3)/2 9.5 −1 √ 9.6 1
9.7 3e2 9.8 − 3 + i 9.9 −2i √
9.10 −2 9.11 −1 −i 9.12 −2 − 2i 3
9.13 −4√ + 4i 9.14 64 9.15 2i − 4
9.16 −2 3 − 2i 9.17 −(1 + i)/4 9.18 1
9.19 16 9.20 i √ 9.21 1
9.22 −i 9.23 ( 3 +√i)/4 9.24 4i
9.25 −1√
9.26 (1 + i 3)/2 9.29 1
3
9.30 e 9.31 5 9.32 3e2
3
9.33 2e 9.34 4/e 9.35 21√
9.36 4 9.37 1 9.38 1/ 2
√ √
10.1 1, (−1 ± i 3)/2 10.2 3, 3(−1 ± i 3)/2
10.3 ±1, ±i √ 10.4 ±2, ± 2i √
10.5 ±1,
√ (±1 ± √ i 3)/2 10.6 ±2, ±1 ± i 3 √
10.7 ± 2, ±i 2, ±1 ± i 10.8 ±1, ±i, (±1 ± i)/ 2
10.9 1, 0.309 ± 0.951i, −0.809 ± 0.588i
10.10 2, 0.618 ±√1.902i, −1.618 ± 1.176i √
10.11 −2, 1 ± i 3 10.12 −1, 1 ± i√ 3 /2
10.13 ±1 ± i √ √ 2
10.14 (±1 ± i)/
10.15 ±2i, ± 3 ± i 10.16 ±i, (± 3 ± i)/2
10.17 −1, 0.809√ ± 0.588i, −0.309 ± 0.951i √
10.18 ±(1 +√i)/ 2 10.19 −i,√(± 3 + i)/2
10.20 2i, ±√ 3 − i 10.21 ±( 3 + i)
10.22 r = 2, θ = 45◦ + 120◦ n: √ 1 + i, −1.366 + 0.366i,
√ 0.366 − 1.366i
10.23 r = 2, θ = 30◦ + 90◦ n: ±( 3 + i), ± 1 − i 3
10.24 r =√1, θ = 30◦ + 45◦ n:√
±( 3√+ i)/2, ± 1 − i 3 /2, ± (0.259 + 0966i), ±(0.966 − 0.259i)
10.25 r = 10 2, θ = 45◦ + 72◦ n: 0.758(1 + i), −0.487 + 0.955i,
−1.059 − 0.168i, −0.168 − 1.059i, 0.955 − 0.487i
10.26 r = 1, θ = 18◦ + 72◦ n : i, ±0.951 + 0.309i, ±0.588 − 0.809i
10.28 cos 3θ = cos3 θ − 3 cos θ sin2 θ
sin 3θ = 3 cos2 θ sin θ − sin3 θ
√
11.3 3(1 − i)/ 2 11.4 −8 11.5 1 + i 11.6 13/5
11.7 3i/5 11.8 −41/9 11.9 4i/3 11.10 −1
12.20 cosh 3z = cosh3 z + 3 cosh z sinh2 z, sinh 3z = 3 cosh2 z sinh z + sinh3 z
∞ ∞
X x2n+1 X x2n
12.22 sinh x = , cosh x =
n=0
(2n + 1)! n=0
(2n)!
12.23 cos x, | cos x|
12.24 cosh x p
12.25 sin x cosh y − i cos x sinh y, sin2 x + sinh2 y
12.26 cosh 2 cos 3 − i sinh 2 sin 3 = −3.725 − 0.512i, 3.760
12.27 sin 4 cosh 3 + i cos 4 sinh 3 = −7.62 − 6.55i, 10.05
12.28 tanh 1 = 0.762 12.29 1 √
12.30 −i 12.31 (3 + 5i 3)/8
12.32 −4i/3 12.33 i tanh 1 = 0.762i
12.34 i sinh(π/2) = 2.301i 12.35 − cosh 2 = −3.76
12.36 i cosh 1 = 1.543i 12.37 cosh π
,Chapter 2 8
14.1 1 + iπ 14.2 −iπ/2 or 3πi/2
14.3 Ln 2 + iπ/6 14.4 (1/2) Ln 2 + 3πi/4
14.5 Ln 2 + 5iπ/4 14.6 −iπ/4 or√ 7πi/4
14.7 iπ/2 14.8 −1, (1 ± i 3)/2
2
14.9 e−π 14.10 e−π /4
14.11 cos(Ln 2) + i sin(Ln 2) = 0.769 + 0.639i
14.12 −ie−π/2
14.13 1/e
14.14 2e−π/2 [i cos(Ln 2) − sin(Ln 2)] = 0.3198i − 0.2657
14.15 e−π sinh 1 = 0.0249
14.16 e−π/3 = 0.351 √
√ −3π/4 i(Ln 2 +3π/4)
14.17 2e e = −0.121 + 0.057i
14.18 −1 14.19 −5/4
14.20 1 14.21 −1
14.22 −1/2 14.23 eπ/2 = 4.81
√
15.1 π/2 + 2nπ ± i Ln 2 + 3 = π/2 + 2nπ ± 1.317i
15.2 π/2 + nπ + (i Ln 3)/2
15.3 i(±π/3 + 2nπ)
15.4 i(2nπ
+ π/6), i(2nπ + 5π/6) √
15.5 ± π/2 + 2nπ − i Ln 3 + 8 = ±[π/2 + 2nπ − 1.76i]
15.6 i(nπ − π/4) √
15.7 π/2 + nπ − i Ln( 2 − 1) = π/2 + nπ + 0.881i
15.8 π/2 + 2nπ ± i Ln 3
15.9 i(π/3 + nπ)
15.10 2nπ ± i Ln 2
15.11 i(2nπ + π/4), i(2nπ + 3π/4)
15.12 i(2nπ ± π/6)
15.13 i(π + 2nπ)
15.14 2nπ + i Ln 2, (2n + 1)π − i Ln 2
15.15 nπ + 3π/8 + i Ln 2)/4
15.16 (Ln 2)/4 + i(nπ + 5π/8)
2
16.2 Motion around circle |z| = √ 5; v = 5ω,
√ a = 5ω √ .
16.3 Motion
√ around circle |z| = 2; v = 2, a = 2.
16.4 v = 13, a = 0
16.5 v = |z1 − z2 |, a = 0 √
16.6 (a) Series: 3 − 2i (b) Series: 2(1√+ i 3)
Parallel: 5 + i Parallel: i 3
16.7 (a) Series: 1 + 2i (b) Series: 5 + 5i
Parallel: 3(3 − i)/5 Parallel: 1.6 + 1.2i
R − i(ωCR2 + ω 3 L2 C − ωL) / (ωCR)2 + (ω 2 LC − 1)2 ; this
16.8
R2 C
L 1
simplifies to if ω 2 = 1− , that is, at resonance.
RC LC L
r
R R2 1 √
16.9 (a) ω = ± 2 + (b) ω = 1/ LC
2L 4Lr LC
1 1 1 √
16.10 (a) ω = − ± 2 2
+ (b) ω = 1/ LC
2RC 4R C LC
16.12 (1 + r4 − 2r2 cos θ)−1
, Chapter 2 9
√
17.1 −1 √ 17.2 ( 3 + i)/2
17.3 r = 2, θ = 45◦ + 72◦ n : 1 + i, −0.642 + 1.260i, −1.397 − 0.221i,
−0.221 − 1.397i, 1.260 − 0.642i
17.4 i cosh 1 = 1.54i 17.5 i
2 2 2
17.6 −e−π = −5.17 × 10−5 or −e−π · e±2nπ
17.7 eπ/2 = 4.81 or eπ/2 · e±2nπ
17.8 √−1 17.9 π/2 ± 2nπ
17.10 3 − √ 2 17.11 i
17.12 −1 ± 2 17.13 x = 0, y = 4
17.14 Circle with center (0, 2), radius 1
17.15 |z| < 1/e 17.16 y < −2
2
17.26 1 17.27 (c) e−2(x−t)
2
√
2
a + b2
17.28 1 + sinh2 b 17.29 −1 ± i 3 /2
2ab
∞
X xn 2n/2 cos nπ/4
17.30 ex cos x =
n=0
n!
∞ n n/2
X x 2 sin nπ/4
ex sin x =
n=0
n!
SOLUTIONS MANUAL
,Chapter 1
1.1 (2/3)10 = 0.0173 yd; 6(2/3)10 = 0.104 yd (compared to a total of 5 yd)
1.3 5/9 1.4 9/11 1.5 7/12
1.6 11/18 1.7 5/27 1.8 25/36
1.9 6/7 1.10 15/26 1.11 19/28
1.13 $1646.99 1.15 Blank area = 1
1.16 At x = 1: 1/(1 + r); at x = 0: r/(1 + r); maximum escape at x = 0 is 1/2.
2.1 1 2.2 1/2 2.3 0
2.4 ∞ 2.5 0 2.6 ∞
2.7 e2 2.8 0 2.9 1
4.1 an = 1/2n → 0; Sn = 1 − 1/2n → 1; Rn = 1/2n → 0
4.2 an = 1/5n−1 → 0; Sn = (5/4)(1 − 1/5n ) → 5/4; Rn = 1/(4 · 5n−1 ) → 0
4.3 an = (−1/2)n−1 → 0; Sn = (2/3)[1 − (−1/2)n ] → 2/3; Rn = (2/3)(−1/2)n → 0
4.4 an = 1/3n → 0; Sn = (1/2)(1 − 1/3n ) → 1/2; Rn = 1/(2 · 3n ) → 0
4.5 an = (3/4)n−1 → 0; Sn = 4[1 − (3/4)n ] → 4; Rn = 4(3/4)n → 0
1 1 1
4.6 an = → 0; Sn = 1 − → 1; Rn = →0
n(n + 1) n+1 n+1
(−1)n+1 (−1)n
1 1
4.7 an = (−1)n+1 + → 0 ; Sn = 1 + → 1; Rn = →0
n n+1 n+1 n+1
5.1 D 5.2 Test further 5.3 Test further
5.4 D 5.5 D 5.6 Test further
5.7 Test further 5.8 Test further
5.9 D 5.10 D
6.5 (a) D 6.5 (b) D
R∞
Note: In the following answers, I= an dn; ρ = test ratio.
6.7 D, I = ∞ 6.8 D, I = ∞ 6.9 C, I = 0
6.10 C, I = π/6 6.11 C, I = 0 6.12 C, I = 0
6.13 D, I = ∞ 6.14 D, I = ∞ 6.18 D, ρ = 2
6.19 C, ρ = 3/4 6.20 C, ρ = 0 6.21 D, ρ = 5/4
6.22 C, ρ = 0 6.23 D, ρ = ∞ 6.24 D, ρ = 9/8
6.25 C, ρ = 0 6.26 C, ρ = (e/3)3 6.27 D, ρ =P100
6.28 C, ρ =P 4/27 6.29 D, ρ =P2 6.31 D, cf. P n−1
6.32 D, cf. n−1 6.33 C, cf. 2−n 6.34 C, cf. n−2
P −2 P −1/2
6.35 C, cf. n 6.36 D, cf. n
1
,Chapter 1 2
7.1 C 7.2 D 7.3 C 7.4 C
7.5 C 7.6 D 7.7 C 7.8 C
P −1
9.1 D, cf. n 9.2 D, an 6→ 0 P −1
9.3 C, I =P0 9.4 D, I = ∞, or cf. n
9.5 C, cf. n−2 9.6 C, ρ = 1/4
9.7 D, ρ = 4/3 9.8 C, ρ = 1/5
9.9 D, ρ = e 9.10 D, an 6→
P 0 −2
D, I = ∞, or cf.P n−1
P
9.11 9.12 C, cf. n
9.13 C, I = 0, or cf. n−2 9.14 C, alt.Pser.
9.15 D, ρ = ∞, an 6→ 0 9.16 C, cf. n−2
9.17 C, ρ = 1/27 9.18 C, alt. ser.
9.19 C 9.20 C
9.21 C, ρ = 1/2
9.22 (a) C (b) D (c) k > e
10.1 |x| < √ 1 10.2 |x| < 3/2 10.3 |x| ≤ 1
10.4 |x| ≤ 2 10.5 All x 10.6 All x
10.7 −1 ≤ x < 1 10.8 −1 < x ≤ 1 10.9 |x| < 1
10.10 |x| ≤ 1 10.11 −5 ≤ x < 5 10.12 |x| < 1/2
10.13 −1 < x ≤ 1 10.14 |x| < 3 10.15 −1 < x < 5
10.16 −1 < x < 3 10.17 −2 < x ≤ 0 10.18 −3/4 ≤ x ≤ −1/4
10.19 |x| < 3 10.20 All x 10.21 0 ≤ x √≤1
10.22 No x 10.23 x > 2 or x < −4 10.24 |x| < 5/2
10.25 nπ − π/6 < x < nπ + π/6
(−1)n (2n − 1)!!
−1/2 −1/2
13.4 = 1; =
0 n (2n)!!
Answers to part (b), Problems 5 to 19:
∞ n+2 ∞
X x X 1/2 n+1
13.5 − 13.6 x (see Example 2)
1
n 0
n
∞ ∞
(−1)n x2n
X X −1/2
13.7 13.8 (−x2 )n (see Problem 13.4)
0
(2n + 1)! 0
n
∞ ∞
X X (−1)n x4n+2
13.9 1 + 2 xn 13.10
1 0
(2n + 1)!
∞ n n ∞
X (−1) x X (−1)n x4n+1
13.11 13.12
0
(2n + 1)! 0
(2n)!(4n + 1)
∞ n 2n+1 ∞
X (−1) x X x2n+1
13.13 13.14
0
n!(2n + 1) 0
2n + 1
∞
x2n+1
X −1/2
13.15 (−1)n
0
n 2n + 1
∞ 2n ∞
X x X xn
13.16 13.17 2
0
(2n)! n
oddn
∞
X (−1)n x2n+1 ∞
X −1/2 x2n+1
13.18 13.19
0
(2n + 1)(2n + 1)! 0
n 2n + 1
2 3 5 6
13.20 x + x + x /3 − x /30 − x /90 · · ·
13.21 x2 + 2x4 /3 + 17x6 /45 · · ·
13.22 1 + 2x + 5x2 /2 + 8x3 /3 + 65x4 /24 · · ·
13.23 1 − x + x3 − x4 + x6 · · ·
,Chapter 1 3
13.24 1 + x2 /2! + 5x4 /4! + 61x6 /6! · · ·
13.25 1 − x + x2 /3 − x4 /45 · · ·
13.26 1 + x2 /4 + 7x4 /96 + 139x6 /5760 · · ·
13.27 1 + x + x2 /2 − x4 /8 − x5 /15 · · ·
13.28 x − x2 /2 + x3 /6 − x5 /12 · · ·
13.29 1 + x/2 − 3x2 /8 + 17x3 /48 · · ·
13.30 1 − x + x2 /2 − x3 /2 + 3x4 /8 − 3x5 /8 · · ·
13.31 1 − x2 /2 − x3 /2 − x4 /4 − x5 /24 · · ·
13.32 x + x2 /2 − x3 /6 − x4 /12 · · ·
13.33 1 + x3 /6 + x4 /6 + 19x5 /120 + 19x6 /120 · · ·
13.34 x − x2 + x3 − 13x4 /12 + 5x5 /4 · · ·
13.35 1 + x2 /3! + 7x4 /(3 · 5!) + 31x6 /(3 · 7!) · · ·
13.36 u2 /2 + u4 /12 + u6 /20 · · ·
13.37 −(x2 /2 + x4 /12 + x6 /45 · · · )
13.38 e(1 − x2 /2 + x4 /6 · · · )
4
13.39 1 − (x − π/2)2 /2! + (x − π/2) /4! · · ·
3
13.40 1 − (x − 1) + (x − 1)2 − (x − 1) · · ·
13.41 e [1 + (x − 3) + (x − 3) /2! + (x − 3)3 /3! · · · ]
3 2
2
13.42 −1 + (x − π) /2! − (x − π)4 /4! · · ·
13.43 −[(x − π/2) + (x − π/2)3 /3 + 2(x − π/2)5 /15 · · · ]
13.44 5 + (x − 25)/10 − (x − 25)2 /103 + (x − 25)3 /(5 · 104 ) · · ·
14.6 Error < (1/2)(0.1)2 ÷ (1 − 0.1) < 0.0056
14.7 Error < (3/8)(1/4)2 ÷ (1 − 41 ) = 1/32
14.8 For x < 0, error < (1/64)(1/2)4 < 0.001
For x > 0, error < 0.001 ÷ (1 − 12 ) = 0.002
1
14.9 Term n + 1 is an+1 = (n+1)(n+2) , so Rn = (n + 2)an+1 .
14.10 S4 = 0.3052, error < 0.0021 (cf. S = 1 − ln 2 = 0.307)
15.1 −x4 /24 − x5 /30 · · · ≃ −3.376 × 10−16
15.2 x8 /3 − 14x12 /45 · · · ≃ 1.433 × 10−16
15.3 x5 /15 − 2x7 /45 · · · ≃ 6.667 × 10−17
15.4 x3 /3 + 5x4 /6 · · · ≃ 1.430 × 10−11
15.5 0 15.6 12 15.7 10!
15.8 1/2 15.9 −1/6 15.10 −1
15.11 4 15.12 1/3 15.13 −1
15.14 t − t3 /3, error < 10−6 15.15 23 t3/2 − 52 t5/2 , error < 17 10−7
15.16 e2 − 1 15.17 √cos π2 = 0
15.18 ln 2 15.19 2
15.20 (a) 1/8 (b) 5e (c) 9/4
15.21 (a) 0.397117 (b) 0.937548 (c) 1.291286
15.22 (a) π 4 /90 (b) 1.202057 (c) 2.612375
15.23 (a) 1/2 (b) 1/6 (c) 1/3 (d) −1/2
15.24 (a) −π (b) 0 (c) −1
(d) 0 (e) 0 (f) 0
15.27 (a) 1 − vc = 1.3 × 10−5 , or v = 0.999987c
(b) 1 − vc = 5.2 × 10−7
(c) 1 − vc = 2.1 × 10−10
(d) 1 − vc = 1.3 × 10−11
15.28 mc2 + 12 mv 2
15.29 (a) F/W = θ + θ3 /3 · · ·
(b) F/W = x/l + x3 /(2l3 ) + 3x5 /(8l5 ) · · ·
,Chapter 1 4
15.30 (a) T = F (5/x + x/40 − x3 /16000 · · · )
(b) T = 12 (F/θ)(1 + θ2 /6 + 7θ4 /360 · · · )
15.31 (a) finite (b) infinite
16.1 (c) overhang: 2 3 10 100
books needed: 32 228 2.7 × 108 4 × 1086
P −3/2
16.4 C, ρ = 0 16.5 D, an 6→ P 0 −1 16.6 C, cf. n
16.7 D, I = ∞ 16.8 D, cf. n 16.9 −1 ≤ x < 1
16.10 |x| < 4 16.11 |x| ≤ 1 16.12 |x| < 5
16.13 −5 < x ≤ 1
16.14 1 − x2 /2 + x3 /2 − 5x4 /12 · · ·
16.15 −x2 /6 − x4 /180 − x6 /2835 · · ·
16.16 1 − x/2 + 3x2 /8 − 11x3 /48 + 19x4 /128 · · ·
16.17 1 + x2 /2 + x4 /4 + 7x6 /48 · · ·
16.18 x − x3 /3 + x5 /5 − x7 /7 · · ·
16.19 −(x − π) + (x − π)3 /3! − (x − π)5 /5! · · ·
16.20 2 + (x − 8)/12 − (x − 8)2 /(25 · 32 ) + 5(x − 8)3 /(28 · 34 ) · · ·
16.21 e[1 + (x − 1) + (x − 1)2 /2! + (x − 1)3 /3! · · · ]
16.22 arc tan 1 = π/4 16.23 1 − (sinπ)/π = 1
16.24 eln 3 − 1 = 2 16.25 −2
16.26 −1/3 16.27 2/3
16.28 1 16.29 6!
16.30 (b) For N = 130, 10.5821 < ζ(1.1) < 10.5868
16.31 (a) 10430 terms. For N = 200, 100.5755 < ζ(1.01) < 100.5803
16.31 (b) 2.66 × 1086 terms. For N = 15, 1.6905 < S < 1.6952
200 86
16.31 (c) ee = 103.1382×10 terms. For N = 40, 38.4048 < S < 38.4088
,Chapter 2
x y r θ
√
4.1 1 1 √2 π/4
4.2 −1 1
√ 2 3π/4
4.3 1
√ − 3 2 −π/3
4.4 − 3 1 2 5π/6
4.5 0 2 2 π/2
4.6 0 −4 4 −π/2
4.7 −1 0 1 π
4.8 3 0 √3 0
4.9 −2 2 2√2 3π/4
4.10 √2 −2 2 2 −π/4
4.11 3 1√ 2 π/6
4.12 −2 −2 3 4 −2π/3
4.13 √0 −1
√ 1 3π/2
4.14 2 2 2 π/4
4.15 −1 0 1 −π or π
4.16 5 0 √5 0
4.17 1 −1 2 −π/4
4.18 0 3 3 π/2
4.19 4.69 1.71 5 20◦ = 0.35
4.20 −2.39 −6.58 7 −110◦ = −1.92
√
5.1 1/2 −1/2 1/√2 −π/4
5.2 −1/2 −1/2 1/ 2 −3π/4 or 5π/4
5.3 1 0 1 0
5.4 0 2
√ 2 π/2
5.5 2 2 3 4 π/3
5.6 −1 0 √1 π
5.7 7/5 −1/5 2 −8.13◦ = −0.14
5.8 1.6 −2.7 3.14 −59.3◦ = −1.04
5.9 −10.4 22.7 p 25 2 = 114.6◦
5.10 −25/17 19/17 58/17 142.8◦ = 2.49
5.11 17 −12 20.8 −35.2◦ = −0.615
5.12 2.65 1.41 3 28◦ = 0.49
5.13 1.55 4.76 5 2π/5
5.14 1.27 −2.5 2.8 −1.1 = −63◦
5.15 21/29 −20/29 1 −43.6◦ = −0.76
5.16 1.53 −1.29 2 −40◦ = −0.698
5.17 −7.35 −10.9 13.1 −124◦ = −2.16
5.18 −0.94 −0.36 1 201◦ or −159◦,
3.51 or −2.77
5
,Chapter 2 6
5.19 (2 + 3i)/13; (x − yi)/(x2 + y 2 )
5.20 (−5 + 12i)/169; (x2 − y 2 − 2ixy)/(x2 + y 2 )2
5.21 (1 + i)/6; (x + 1 − iy)/[(x + 1)2 + y 2 ]
5.22 (1 + 2i)/10; [x − i(y − 1)]/[x2 + (y − 1)2 ]
5.23 (−6 − 3i)/5; (1 − x2 − y 2 + 2yi)/[(1 − x)2 + y 2 ]
5.24 (−5 − 12i)/13; (x2 − y 2 + 2ixy)/(x
p
2
+ y2)
5.26 1√ 5.27 13/2 5.28 1
5.29 5 5 5.30 3/2 5.31 1
5.32 169 5.33 5 5.34 1
5.35 x = −4, y = 3 5.36 x = −1/2, y = 3
5.37 x=y=0 5.38 x = −7, y = 2
5.39 x = y = any real number 5.40 x = 0, y = 3
5.41 x = 1, y = −1 5.42 x = −1/7, y = −10/7
5.43 (x, y) = (0, 0), or (1, 1), or (−1, 1)
5.44 x = 0, y = −2
5.45 x = 0, any real y; or y = 0, any real x
5.46 y = −x √
5.47 (x, y) = (−1, 0), (1/2, ± 3/2)
5.48 x = 36/13, y = 2/13
5.49 x = 1/2, y = 0
5.50 x = 0, y ≥ 0
5.51 Circle, center at origin, radius = 2
5.52 y axis
5.53 Circle, center at (1, 0), r = 1
5.54 Disk, center at (1, 0), r = 1
5.55 Line y = 5/2
5.56 Positive y axis
5.57 Hyperbola, x2 − y 2 = 4
5.58 Half plane, x > 2
5.59 Circle, center at (0, −3), r = 4
5.60 Circle, center at (1, −1), r = 2
5.61 Half plane, y < 0
5.62 Ellipse, foci at (1, 0) and (−1, 0), semi-major axis = 4
5.63 The coordinate axes
5.64 Straight lines, y = ± x
5.67 v = (4t2 + 1)−1 , a = 4(4t2 + 1)−3/2
5.68 Motion around circle r = 1, with v = 2, a = 4
√ √
6.2 D, ρ = 2 6.3 C, ρ = 1/ 2 6.4 D, |an | √
= 1 6→ 0
6.5 D 6.6 C 6.7 D, ρ = √ 2
6.8 D, |an | = 1 6→ 0 6.9 C 6.10 C, ρ = p2/2
6.11 C, ρ = 1/5 6.12 C 6.13 C, ρ = 2/5
7.1 All z 7.2 |z| < 1 7.3 All z
7.4 |z| < 1 7.5 |z| < 2 7.6 |z| < 1/3
7.7 All z 7.8 All z 7.9 |z| < 1
7.10 |z| < 1 7.11 |z| < 27 7.12 |z| < 4
7.13 |z − i| < 1 √ 7.14 |z − 2i| < 1 7.15 |z − (2 − i)| < 2
7.16 |z + (i − 3)| < 1/ 2
8.3 See Problem 17.30.
,Chapter 2 7
√
9.1 (1 − i)/ √2 9.2 i 9.3 −9i
9.4 −e(1 + i 3)/2 9.5 −1 √ 9.6 1
9.7 3e2 9.8 − 3 + i 9.9 −2i √
9.10 −2 9.11 −1 −i 9.12 −2 − 2i 3
9.13 −4√ + 4i 9.14 64 9.15 2i − 4
9.16 −2 3 − 2i 9.17 −(1 + i)/4 9.18 1
9.19 16 9.20 i √ 9.21 1
9.22 −i 9.23 ( 3 +√i)/4 9.24 4i
9.25 −1√
9.26 (1 + i 3)/2 9.29 1
3
9.30 e 9.31 5 9.32 3e2
3
9.33 2e 9.34 4/e 9.35 21√
9.36 4 9.37 1 9.38 1/ 2
√ √
10.1 1, (−1 ± i 3)/2 10.2 3, 3(−1 ± i 3)/2
10.3 ±1, ±i √ 10.4 ±2, ± 2i √
10.5 ±1,
√ (±1 ± √ i 3)/2 10.6 ±2, ±1 ± i 3 √
10.7 ± 2, ±i 2, ±1 ± i 10.8 ±1, ±i, (±1 ± i)/ 2
10.9 1, 0.309 ± 0.951i, −0.809 ± 0.588i
10.10 2, 0.618 ±√1.902i, −1.618 ± 1.176i √
10.11 −2, 1 ± i 3 10.12 −1, 1 ± i√ 3 /2
10.13 ±1 ± i √ √ 2
10.14 (±1 ± i)/
10.15 ±2i, ± 3 ± i 10.16 ±i, (± 3 ± i)/2
10.17 −1, 0.809√ ± 0.588i, −0.309 ± 0.951i √
10.18 ±(1 +√i)/ 2 10.19 −i,√(± 3 + i)/2
10.20 2i, ±√ 3 − i 10.21 ±( 3 + i)
10.22 r = 2, θ = 45◦ + 120◦ n: √ 1 + i, −1.366 + 0.366i,
√ 0.366 − 1.366i
10.23 r = 2, θ = 30◦ + 90◦ n: ±( 3 + i), ± 1 − i 3
10.24 r =√1, θ = 30◦ + 45◦ n:√
±( 3√+ i)/2, ± 1 − i 3 /2, ± (0.259 + 0966i), ±(0.966 − 0.259i)
10.25 r = 10 2, θ = 45◦ + 72◦ n: 0.758(1 + i), −0.487 + 0.955i,
−1.059 − 0.168i, −0.168 − 1.059i, 0.955 − 0.487i
10.26 r = 1, θ = 18◦ + 72◦ n : i, ±0.951 + 0.309i, ±0.588 − 0.809i
10.28 cos 3θ = cos3 θ − 3 cos θ sin2 θ
sin 3θ = 3 cos2 θ sin θ − sin3 θ
√
11.3 3(1 − i)/ 2 11.4 −8 11.5 1 + i 11.6 13/5
11.7 3i/5 11.8 −41/9 11.9 4i/3 11.10 −1
12.20 cosh 3z = cosh3 z + 3 cosh z sinh2 z, sinh 3z = 3 cosh2 z sinh z + sinh3 z
∞ ∞
X x2n+1 X x2n
12.22 sinh x = , cosh x =
n=0
(2n + 1)! n=0
(2n)!
12.23 cos x, | cos x|
12.24 cosh x p
12.25 sin x cosh y − i cos x sinh y, sin2 x + sinh2 y
12.26 cosh 2 cos 3 − i sinh 2 sin 3 = −3.725 − 0.512i, 3.760
12.27 sin 4 cosh 3 + i cos 4 sinh 3 = −7.62 − 6.55i, 10.05
12.28 tanh 1 = 0.762 12.29 1 √
12.30 −i 12.31 (3 + 5i 3)/8
12.32 −4i/3 12.33 i tanh 1 = 0.762i
12.34 i sinh(π/2) = 2.301i 12.35 − cosh 2 = −3.76
12.36 i cosh 1 = 1.543i 12.37 cosh π
,Chapter 2 8
14.1 1 + iπ 14.2 −iπ/2 or 3πi/2
14.3 Ln 2 + iπ/6 14.4 (1/2) Ln 2 + 3πi/4
14.5 Ln 2 + 5iπ/4 14.6 −iπ/4 or√ 7πi/4
14.7 iπ/2 14.8 −1, (1 ± i 3)/2
2
14.9 e−π 14.10 e−π /4
14.11 cos(Ln 2) + i sin(Ln 2) = 0.769 + 0.639i
14.12 −ie−π/2
14.13 1/e
14.14 2e−π/2 [i cos(Ln 2) − sin(Ln 2)] = 0.3198i − 0.2657
14.15 e−π sinh 1 = 0.0249
14.16 e−π/3 = 0.351 √
√ −3π/4 i(Ln 2 +3π/4)
14.17 2e e = −0.121 + 0.057i
14.18 −1 14.19 −5/4
14.20 1 14.21 −1
14.22 −1/2 14.23 eπ/2 = 4.81
√
15.1 π/2 + 2nπ ± i Ln 2 + 3 = π/2 + 2nπ ± 1.317i
15.2 π/2 + nπ + (i Ln 3)/2
15.3 i(±π/3 + 2nπ)
15.4 i(2nπ
+ π/6), i(2nπ + 5π/6) √
15.5 ± π/2 + 2nπ − i Ln 3 + 8 = ±[π/2 + 2nπ − 1.76i]
15.6 i(nπ − π/4) √
15.7 π/2 + nπ − i Ln( 2 − 1) = π/2 + nπ + 0.881i
15.8 π/2 + 2nπ ± i Ln 3
15.9 i(π/3 + nπ)
15.10 2nπ ± i Ln 2
15.11 i(2nπ + π/4), i(2nπ + 3π/4)
15.12 i(2nπ ± π/6)
15.13 i(π + 2nπ)
15.14 2nπ + i Ln 2, (2n + 1)π − i Ln 2
15.15 nπ + 3π/8 + i Ln 2)/4
15.16 (Ln 2)/4 + i(nπ + 5π/8)
2
16.2 Motion around circle |z| = √ 5; v = 5ω,
√ a = 5ω √ .
16.3 Motion
√ around circle |z| = 2; v = 2, a = 2.
16.4 v = 13, a = 0
16.5 v = |z1 − z2 |, a = 0 √
16.6 (a) Series: 3 − 2i (b) Series: 2(1√+ i 3)
Parallel: 5 + i Parallel: i 3
16.7 (a) Series: 1 + 2i (b) Series: 5 + 5i
Parallel: 3(3 − i)/5 Parallel: 1.6 + 1.2i
R − i(ωCR2 + ω 3 L2 C − ωL) / (ωCR)2 + (ω 2 LC − 1)2 ; this
16.8
R2 C
L 1
simplifies to if ω 2 = 1− , that is, at resonance.
RC LC L
r
R R2 1 √
16.9 (a) ω = ± 2 + (b) ω = 1/ LC
2L 4Lr LC
1 1 1 √
16.10 (a) ω = − ± 2 2
+ (b) ω = 1/ LC
2RC 4R C LC
16.12 (1 + r4 − 2r2 cos θ)−1
, Chapter 2 9
√
17.1 −1 √ 17.2 ( 3 + i)/2
17.3 r = 2, θ = 45◦ + 72◦ n : 1 + i, −0.642 + 1.260i, −1.397 − 0.221i,
−0.221 − 1.397i, 1.260 − 0.642i
17.4 i cosh 1 = 1.54i 17.5 i
2 2 2
17.6 −e−π = −5.17 × 10−5 or −e−π · e±2nπ
17.7 eπ/2 = 4.81 or eπ/2 · e±2nπ
17.8 √−1 17.9 π/2 ± 2nπ
17.10 3 − √ 2 17.11 i
17.12 −1 ± 2 17.13 x = 0, y = 4
17.14 Circle with center (0, 2), radius 1
17.15 |z| < 1/e 17.16 y < −2
2
17.26 1 17.27 (c) e−2(x−t)
2
√
2
a + b2
17.28 1 + sinh2 b 17.29 −1 ± i 3 /2
2ab
∞
X xn 2n/2 cos nπ/4
17.30 ex cos x =
n=0
n!
∞ n n/2
X x 2 sin nπ/4
ex sin x =
n=0
n!
