Chapter 7

Section 7.1

1. (a) $ w + z = 1 + 3i $

(c) $ 3w - 2z = 13 - 26i $

(e) $ zw = 22 + 29i $

(f) $ \displaystyle{\frac{z}{w} = -\frac{34}{25} + \frac{13}{25}i}$

(g) $ |z| = \sqrt{53} $

(i) $ \Re(z - w) = -5 $

(j) $ \Im(3z + w) = 17 $

2. (a) $ \displaystyle{\Re\left(\frac{1}{i}\right) = 0}$, $ \displaystyle{\Im\left(\frac{1}{i}\right) = -1}$

(c) $ \displaystyle{\Re\left(\frac{3-4i}{-2+3i}\right) = -\frac{18}{13}}$, $ \displaystyle{\Im\left(\frac{3-4i}{-2-3i}\right) = -\frac{1}{13}}$

3. (a) $ z = 3i $

(c) $ \displaystyle{z = -\frac{1}{2\sqrt{2}} - \frac{1}{2\sqrt{2}}i}$

4. (a) $ |z| = 1$, $ \mathrm{Arg}(z) = -\dfrac{\pi}{2} $

(c) $ |z| = \sqrt{2} $, $ \mathrm{Arg}(z) = \dfrac{\pi}{4} $

(e) $ |z| = 4 $, $ \mathrm{Arg}(z) = \dfrac{\pi}{3} $

5. (a) $ \left|w^2\right| = 9 $, $ \mathrm{Arg}\left(w^2\right) = \dfrac{\pi}{3} $, $ \displaystyle{w^2 = \frac{9}{2} + \frac{9\sqrt{3}}{2}i}$

(c) $ |wz| = 6 $, $ \mathrm{Arg}(wz) = -\dfrac{\pi}{6} $, $ \displaystyle{wz = 3\sqrt{3} - 3i}$

(e) $ \left|\dfrac{z}{w^2}\right| = \dfrac{2}{9} $, $ \mathrm{Arg}\left(\dfrac{z}{w^2}\right) = -\dfrac{2\pi}{3} $, $ \displaystyle{\frac{z}{w^2} = -\frac{1}{9} - \frac{1}{3\sqrt{3}}i}$

(f) $ \left|w^5\right| = 243 $, $ \mathrm{Arg}\left(w^5\right) = \dfrac{5\pi}{6} $, $ \displaystyle{w^5 = -\frac{243\sqrt{3}}{2} + \frac{243}{2}i}$

6. If $ z = \dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i $, then the roots are $ 1, z, z^2, z^3, z^4 $, and $ z^5 $.

8. (c) The square roots of $ 1 + \sqrt{3}i $ are $ \sqrt{\dfrac{3}{2}} + \dfrac{1}{\sqrt{2}}i $ and $ -\sqrt{\dfrac{3}{2}} - \dfrac{1}{\sqrt{2}}i $. The square roots of $ -9 $ are $ 3i $ and $ -3i $.

Section 7.2

1. (a) $ \displaystyle{\lim_{n \to \infty}z_n = -1 + \frac{1}{2}i}$

(c) $ \displaystyle{\lim_{n \to \infty}z_n = 3}$

2. (a) $ \displaystyle{\lim_{z \to i}(4z^3 - 6z + 3) = 3 - 10i}$

(c) $ \displaystyle{\lim_{w \to 3i}\frac{w^2 + 9}{w - 3i} = 6i}$

3. (a) $ f'(z) = 6z - 30z^4 $

(c) $ \displaystyle{f'(z) = -2z(z - 4i)e^{-z^2} + e^{-z^2}}$

5. (a) The result follows from \[ e^z = e^{x + yi} = e^xe^{yi} = e^x(\cos(y) + i\sin(y)). \]

(b) $ \left|e^z\right| = e^x $, $ \mathrm{arg}\left(e^z\right) = y $

11. (a) For all $ z $, \[ \sin(z) = \sum_{n=0}^\infty \frac{(-1)^nz^{2n+1}}{(2n+1)!} = z - \frac{z^3}{3!} + \frac{z^5}{5!} - \frac{z^7}{7!} + \frac{z^9}{9!} - \cdots . \]

(b) For all $ z $, \[ \cos(z) = \sum_{n=0}^\infty \frac{(-1)^nz^{2n}}{2n!} = 1 - \frac{z^2}{2} + \frac{z^4}{4!} - \frac{z^6}{6!} + \frac{z^8}{8!} - \cdots . \]

15. (b) Hint: Consider what happens if $ z = x + yi $ where $ y $ is not between $ -\pi $ and $ \pi $.

Section 7.3

2. (a) $ f'(t) = -2\sin(2t) + 2i\cos(2t) $

(c) $ \displaystyle{g'(t) = \frac{1}{2}\cos\left(\frac{t}{2}\right) - \frac{1}{2}i\sin\left(\frac{t}{2}\right)}$

(d) $ z'(t) = -2\mathrm{sech}(2t)\tanh(2t) + 2i\mathrm{sech}^2(2t) $

(e) $ f'(t) = 2 + 2it $

(f) $ g'(t) = 2t + 4it^3 $

(g) $ \displaystyle{z'(t) = 3ie^{it}}$

(h) $ \displaystyle{h'(t) = 3ite^{it} + 3e^{it}}$

(i) $ \displaystyle{z'(t) = \frac{6}{t}ie^{2it} - \frac{3}{t^2}e^{2it}}$

3. (a) Velocity $ = -\sqrt{3} + i $

Speed $ = 2 $

Acceleration $ = -2 - 2\sqrt{3}i $

(c) Velocity $ = \mathrm{sech}^2(3) - i\mathrm{sech}(3)\tanh(3) = 0.009866 - 0.098837i $, rounded to six decimal places

Speed $ = 0.099328 $, rounded to six decimal places

Acceleration $ = -2\mathrm{sech}^2(3)\tanh(3) + i(\mathrm{sech}(3)\tanh^2(3) - \mathrm{sech}^3(3)) = -0.019634 + 0.097368i $, rounded to six decimal places

(e) Velocity $ = -5 $

Speed $ = 5 $

Acceleration $ = -5i $

4. (a) $ \displaystyle{\int_0^4 (2t + it)dt = 16 + 8i}$

(c) $ \displaystyle{\int_0^{\frac{\pi}{2}}(-3\sin(2t) + it^3)dt = -3 + \frac{\pi^4}{64}i}$

(e) $ \displaystyle{\int_0^\pi 2te^{3it}dt = -\frac{4}{9} + \frac{2\pi}{3}i}$

5. $ z(t) = (1 + \sin(t)) + i(2 - \cos(t)) $

6. (a) $ \displaystyle{t = \frac{1}{16}s_0\sin(\alpha)}$

(b) $ \displaystyle{R = \frac{s_0^2\sin(\alpha)\cos(\alpha)}{16} = \frac{s_0^2\sin(2\alpha)}{32}}$

7. (a) Maximum range $ = 703.125 $ feet

(b) If $ \alpha = \dfrac{\pi}{6} $, the range is $ \dfrac{5625\sqrt{3}}{16} $ feet and the projectile strikes the ground after $ 4.6875 $ seconds.

If $ \alpha = \dfrac{\pi}{3} $, the range is $ \dfrac{5625\sqrt{3}}{16} $ feet and the projectile strikes the ground after $ \dfrac{75\sqrt{3}}{16} $ seconds.

10. (b) $ \displaystyle{\int_0^\infty e^{-t}dt = \varphi(0) = 1}$

$ \displaystyle{\int_0^\infty te^{-t}dt = \frac{\varphi'(0)}{i} = 1}$

$ \displaystyle{\int_0^\infty t^2e^{-t}dt = \frac{\varphi''(0)}{i^2} = 2}$

$ \displaystyle{\int_0^\infty t^3e^{-t}dt = \frac{\varphi'''(0)}{i^3} = 6}$

$ \displaystyle{\int_0^\infty t^4e^{-t}dt = \frac{\varphi''''(0)}{i^4} = 24}$

11. $ \displaystyle{\varphi(\lambda) = \sqrt{2\pi}e^{-\frac{\lambda^2}{2}}}$

$ \displaystyle{\int_{-\infty}^\infty e^{-\frac{t^2}{2}}dt = \varphi(0) = \sqrt{2\pi}}$

$ \displaystyle{\int_{-\infty}^\infty te^{-\frac{t^2}{2}}dt = \frac{\varphi'(0)}{i} = 0}$

$ \displaystyle{\int_{-\infty}^\infty t^2e^{-\frac{t^2}{2}}dt = \frac{\varphi''(0)}{i^2} = \sqrt{2\pi}}$

$ \displaystyle{\int_{-\infty}^\infty t^3e^{-\frac{t^2}{2}}dt = \frac{\varphi'''(0)}{i^3} = 0}$

$ \displaystyle{\int_{-\infty}^\infty t^4e^{-\frac{t^2}{2}}dt = \frac{\varphi''''(0)}{i^4} = 3\sqrt{2\pi}}$

Section 7.4

1. (b)

3. The distance from the sun to Halley's comet at aphelion is $ 35.2 $ astronomical units.

5. (a) $ 0.9996x^2 + y^2 + 0.04x - 0.9992 = 0 $

(b) $ 583 $ million miles

(c) $ 66,500 $ miles per hour

7. (a) $ y_1(t) = e^t $, $ y_2(t) = e^{-t} $

(c) $ y(t) = 3e^t - e^{-t} $