25 $\frac{4}{5} + j\frac{7}{5}, \frac{1}{5}\sqrt{65} \angle \tan^{-1}\frac{7}{4}$

26 (a) $-\frac{5}{12}\pi, -\frac{11}{12}\pi$ (b) $\frac{1}{2}\sqrt{3} + j\frac{3}{2}$

27 (a) $128, -\frac{1}{3}\pi$ (b) $1024, 0$ (c) $\frac{1}{16}, \frac{2\pi}{3}$

29 (a) $\frac{1}{2}\cosh 1 - j\frac{1}{2}\sqrt{3}\sinh 1$
(b) $\cosh \frac{3}{4}$

(c) $\frac{1}{2}\sinh \frac{\pi}{3} + j\frac{\sqrt{3}}{2}\cosh \frac{\pi}{3}$

(d) $\frac{1}{\sqrt{2}}$

30 (a) $\frac{1}{2}(4n + 1)\pi + j\cosh^{-1}2$

(b) $\frac{1}{2}(2n + 1)\pi + j(-1)^{n+1}\sinh^{-1}\frac{3}{4}$

(c) $\frac{1}{2}(4n + 1)\pi + j\cosh^{-1}3$

(d) $\cosh^{-1}2 + j(2n + 1)\pi$

32 $x = \frac{\tanh u \sec^2 v}{1 + \tanh^2 u \tan^2 v}$

$y = \frac{\tan v \sec^2 u}{1 + \tanh^2 u \tan^2 v}$

$\frac{2 \tanh 2}{1 + \tanh^2 2} + j \frac{\sec^2 2}{1 + \tanh^2 2}$

$= 0.9994 + j0.0366$

33 $0.1645 - j0.1214$

34 (a) $-2 + j2, -4$ (b) $-j8, -8 - j8\sqrt{3}$

(c) $117 + j44, -527 + j336$ (d) $-8, -8 + j8\sqrt{3}$

(e) $8, -8 + j8\sqrt{3}$ (f) $8, -8 - j8\sqrt{3}$

35 (a) $\frac{1}{8} \cos 4\theta + \frac{1}{2} \cos 2\theta + \frac{3}{8}$
(b) $\frac{3}{4} \sin \theta - \frac{1}{4} \sin 3\theta$

37 $2^{7/6} \angle (\frac{\pi}{12} + \frac{2}{3}k\pi), k = 0, 1, 2$

38 (a) $2^{1/4} \angle (-\frac{1}{24}\pi + \frac{1}{2}k\pi), k = 0, 1, 2, 3$

(b) $2 \angle (\frac{1}{6}\pi + \frac{2}{3}k\pi), k = 0, 1, 2$

(c) $18^{-1/3} \angle (\frac{1}{6}\pi - \frac{4}{3}k\pi), k = 0, 1, 2$

(d) $1 \angle (\frac{1}{4}\pi + \frac{1}{2}k\pi), k = 0, 1, 2, 3$

(e) $4 \angle (\frac{1}{3}\pi + \frac{8}{3}k\pi), k = 0, 1, 2$

(f) $34^{-1/4} \angle (\frac{1}{2} \tan^{-1}\frac{3}{5} - k\pi), k = 0, 1$

39 $1.455 - j0.344, 0.344 + j1.455, -1.455 + j0.344, -0.344 - j1.455$

40 $2.529 + j2.743, 0.471 + j2.257$

41 $1, -\frac{1}{2} \pm j\frac{1}{2}\sqrt{3}$

$\cos \frac{2k\pi}{n} + j \sin \frac{2k\pi}{n}, k = 1, 2, \dots, n$

(a) $j2 \cot \frac{1}{5}k\pi, k = 1, \dots, 4$

(b) $\frac{3}{2}(1 + j \cot \frac{1}{6}k\pi), k = 1, \dots, 5$

42 $5, 13$

43 (a) $x = 5$ , 一条直线

(b) 圆心 $(1, 0)$ , radius 3

(c) 圆心 $(-\frac{5}{4}, 0)$ , radius $\frac{3}{4}$

(d) half-line, $y = x - 2, x > 2$

44 圆是 $|z + 2| = 2$

直线是 $\text{Re}((3 + j) + z) = -2$

45 (a) 直线, $y = 1$

(b) Circle, centre $(0, 2)$ , radius 1

(c) Circle, centre $(0, \frac{5}{4})$ , radius $\frac{3}{4}$

(d) Circle, centre $(\sqrt{\frac{1}{3}}, 0)$ , radius $2\sqrt{\frac{1}{3}}$

(e) 等轴双曲线, $xy = 1$

(f) Ellipse, 焦点位于 $(1, 0), (0, -1)$ , through $(0, 0)$

(g) Hyperbola, 焦点位于 $(1, 0), (0, -1)$

(h) Half-line, $y = x - 2, x > 2$

(i) Half-line, $y = \sqrt{3}x - \frac{3}{2}\sqrt{3}, x < \frac{3}{2}$

(j) Circle, centre $(0, 2)$ , radius 1

46 (a) $\text{Re}[(3 + j)z] = 2$ (b) $|z + 2| = 2$

(c) $|z + 1 - j2| = 3$ (d) $\text{Re}(z^2) = 1$

47 (a) Circle, centre $(1, 0)$ , radius 2

(b) Circle, centre $(\frac{1}{2}, 0)$ , radius $\frac{3}{2}$

(c) Circle, centre $(2, 3)$ , radius 4

(d) Half-line, $y = 0, x > 0$

(e) Circle, centre $(-\frac{13}{8}, 0)$ , radius $\frac{15}{8}$

(f) Semicircle, centre $(\frac{1}{2}, -\frac{1}{2})$ , radius $\frac{1}{2}\sqrt{2}$ , through $(0, 0)$

48 $x^2 + y^2 - 4x - 2y + 1 = 0, |z - 2 - j| = 2,$

$$\arg\left(\frac{z - j}{z - 4 - j}\right) = \pm \frac{\pi}{2}$$

49 部分 $x^2 + (y - 1)^2 = 2$

50 $(x - 3)^2 + y^2 = 4$

51 (a) $u = x + y, v = y - x$

(b) $u = (x - 1)^2 - y^2, v = 2(x - 1)y$

(c) $u = x(x^2 + y^2 + 1), v = y(x^2 + y^2 - 1)$

52 $a = (j - 2)/5, b = 3(1 + 2j)/5$

56 $u^2 + v^2 = 1$

57 $100 + j100.12$

58 $\frac{8}{3} + j\frac{8}{3}$

3.7 复习题

1 (a) $12 + j9$ (b) $2 + j$ (c) $11 + j2$

(d) $7 + j24$ (e) $5$ (f) $(1 - j2)/5$

(g) $(18 + j14)/5$ (h) $\tan^{-1}(3/4) = 0.6435$

(i) $5\sqrt{5} [\cos(0.9653 + 3k\pi) + j\sin(0.9653 + 3k\pi)], k = 0, 1$

2 $x = \pm\frac{3}{2}, y = \pm 2$

3 $\frac{1}{10}(7 + j9)$