### Differential Equations to Difference Equations

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)

Planet Aphelion
Mercury 0.47
Venus 0.73
Earth 1.02
Mars 1.65
Jupiter 5.47
Saturn 10.17
Uranus 20.22
Neptune 30.40
Pluto 46.67

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