Chapter 24 — Exercises

36 problems exploring Euler's formula and identity, complex exponentials, Fourier-series basics, evaluating functions by truncated series, Basel-type sums, and applied problems from electrical engineering, signal processing, probability, and physics.

Verify every transcendental value by truncating a series and bounding the error, exactly as Sections 24.6 and 24.8 do. Answers to odd-numbered problems are in appendices/answers-to-selected.md.

Difficulty tiers. ⭐ routine practice · ⭐⭐ standard skill · ⭐⭐⭐ multi-step / synthesis · ⭐⭐⭐⭐ challenge.

Part A — Euler's Formula and Identity (Sections 24.3–24.4)

A1. ⭐ Using Euler's formula $e^{i\theta}=\cos\theta+i\sin\theta$ (Section 24.3), evaluate $e^{i\theta}$ at $\theta=0$, $\theta=\pi/2$, $\theta=\pi$, and $\theta=3\pi/2$. Plot each result in the complex plane.

A2. ⭐ Starting from $e^{i\pi}=-1$ (Section 24.4), add $1$ to both sides to recover Euler's identity. Name the five constants it contains and the province of mathematics each comes from.

A3. ⭐ Compute the powers $i^1,i^2,i^3,i^4,i^5,i^6$. Explain in one sentence why this period-4 cycle is exactly what makes the Euler-formula derivation work (Section 24.3).

A4. ⭐ Verify $|e^{i\theta}|=1$ for any real $\theta$ by computing $\sqrt{\cos^2\theta+\sin^2\theta}$.

A5. ⭐ Evaluate $e^{2\pi i}$ and $e^{-i\pi}$. Confirm both lie on the unit circle and identify their location.

A6. ⭐ Write $-3$ and $2i$ in polar form $re^{i\theta}$ (Section 24.5).

A7. ⭐⭐ Using $e^{i\pi}=e^{3i\pi}=e^{-i\pi}=-1$, explain why the complex logarithm is multi-valued (Section 24.4) and write all values of $\ln(-1)$.

A8. ⭐⭐ Use Euler's formula to prove de Moivre's formula $(\cos\theta+i\sin\theta)^n=\cos(n\theta)+i\sin(n\theta)$ (Section 24.5). Then expand $(\cos\theta+i\sin\theta)^3$ and match real parts to derive $\cos 3\theta=4\cos^3\theta-3\cos\theta$.

Part B — Complex Exponentials and Trig Identities (Section 24.5)

B1. ⭐ Using $\cos\theta=\dfrac{e^{i\theta}+e^{-i\theta}}{2}$ and $\sin\theta=\dfrac{e^{i\theta}-e^{-i\theta}}{2i}$, confirm both formulas at $\theta=0$.

B2. ⭐ Compute $e^{i\pi/3}$ in rectangular form $a+bi$.

B3. ⭐ Find the modulus and argument of $z=1+i$, then write $z$ as $re^{i\theta}$.

B4. ⭐ Using polar form, compute $(1+i)(1-i)$ by adding arguments and multiplying moduli. Check against direct multiplication.

B5. ⭐⭐ Derive the Pythagorean identity $\cos^2\theta+\sin^2\theta=1$ from $e^{i\theta}\,e^{-i\theta}=1$ (Section 24.5).

B6. ⭐⭐ Use $e^{i(\alpha+\beta)}=e^{i\alpha}e^{i\beta}$ to derive both angle-addition formulas for $\sin$ and $\cos$ at once (Section 24.5).

B7. ⭐⭐ Express $\cos^2\theta$ in terms of $\cos 2\theta$ by squaring $\cos\theta=\tfrac12(e^{i\theta}+e^{-i\theta})$. (This power-reduction identity is used constantly in integration, Chapter 15.)

Part C — Evaluating Functions by Series (Section 24.6)

C1. ⭐⭐ Use the first four nonzero terms of the Maclaurin series for $\sin x$ to estimate $\sin(0.3)$. State an error bound via the alternating-series test (Chapter 22) before comparing to a calculator.

C2. ⭐⭐ Estimate $\cos(0.5)$ using terms through $x^4$. How many correct decimal places does the alternating-series bound guarantee?

C3. ⭐⭐ Estimate $e^{0.2}$ using terms through $x^3$ of the exponential series. Because this series is not alternating, bound the error by comparing its tail to a geometric series.

C4. ⭐⭐ Reduce $\sin(40°)$ to radians and estimate it with five terms of the sine series, reproducing the worked computation of Section 24.6. Give six decimals.

C5. ⭐⭐ Section 24.6 claims that to guarantee $\sin(0.5)$ to within $10^{-9}$ you need terms through $x^9$. Verify by computing $x^{11}/11!$ at $x=0.5$ and showing it is below $10^{-9}$.

C6. ⭐⭐⭐ You must compute $e^{x}$ for $x\in[-0.5,0.5]$ to within $10^{-6}$. Find the smallest number of series terms that suffices across the whole interval, justifying the worst-case $x$.

C7. ⭐⭐⭐ In your own words and with a short formula, explain how "range reduction first, then a short series" (Section 24.6) lets a calculator evaluate $\sin(1000)$ accurately.

C8. ⭐⭐⭐ The Python below computes $\cos$ from its series. Predict the error bound by hand before reading the comment, then explain why six terms suffice on $[-\pi/4,\pi/4]$.

# Maclaurin cosine vs library value at x = pi/4
from math import factorial, cos, pi
def cos_series(x, n=6):
    return sum((-1)**k * x**(2*k) / factorial(2*k) for k in range(n))
x = pi/4
print(cos_series(x))   # 0.7071067811865...
print(cos(x))          # 0.7071067811865...
# abs error ~ 3e-12

Part D — Fourier Series Basics (Section 24.7)

D1. ⭐⭐ State the Fourier coefficient formulas $a_n$ and $b_n$ for a $2\pi$-periodic function (Section 24.7). Which chapter's integral calculus do they rely on?

D2. ⭐⭐ Explain why the square wave of Section 24.7 has all cosine coefficients $a_n=0$. (Hint: odd/even symmetry.)

D3. ⭐⭐ Write the first three nonzero terms of the square-wave Fourier series and evaluate the partial sum at $x=\pi/2$. Compare to the true value $+1$.

D4. ⭐⭐⭐ At $x=\pi/2$ the square-wave series gives $\tfrac{4}{\pi}(1-\tfrac13+\tfrac15-\cdots)$. Setting this equal to $1$ recovers a famous series for $\pi$. Identify it and connect it to Section 24.9.

D5. ⭐⭐⭐ Describe the Gibbs phenomenon (Section 24.7) in two sentences: what overshoot persists, and what does the series converge to at the jump?

D6. ⭐⭐⭐ Sketch (or describe) the 1-, 3-, and 5-harmonic approximations of the square wave, noting how each added odd harmonic sharpens the corners.

Part E — Basel Sums, Applications, and Challenges (Sections 24.8–24.10)

E1. ⭐⭐⭐ Confirm $\sum 1/n^2$ is a convergent $p$-series (state $p$), and use the integral-test tail estimate (Chapter 22) to find how many terms give two-decimal accuracy toward $\pi^2/6$ (Section 24.8).

E2. ⭐⭐⭐ Section 24.8 lists $\zeta(2)=\pi^2/6$ and $\zeta(4)=\pi^4/90$. Compute both numerically, verify $\zeta(4)<\zeta(2)$, and explain why $\zeta(s)$ decreases in $s$.

E3. ⭐⭐⭐ (Probability — PGF.) A Poisson variable has PGF $G(s)=e^{\lambda(s-1)}$ (Section 24.10). Differentiate to confirm $G'(1)=\lambda=E[X]$, and verify $G(1)=1$.

E4. ⭐⭐⭐⭐ (Electrical engineering — phasors.) A source is $v_1(t)=10\cos(\omega t+\pi/6)$. Write it as $\operatorname{Re}(V_1 e^{i\omega t})$ with phasor $V_1=10e^{i\pi/6}$ (Section 24.5). A second source $v_2(t)=10\cos(\omega t-\pi/6)$ is added in series. Compute the combined phasor $V_1+V_2$ and express the sum as $A\cos(\omega t+\phi)$.

E5. ⭐⭐⭐⭐ (Number theory — Basel argument.) Follow Euler's sketch (Section 24.8): write $\dfrac{\sin x}{x}=\prod_{n\ge1}\left(1-\dfrac{x^2}{n^2\pi^2}\right)$, match the coefficient of $x^2$ against $1-\tfrac{x^2}{6}+\cdots$, and derive $\sum 1/n^2=\pi^2/6$. State explicitly which step Euler did not rigorously justify and which later theorem rescues it.

E6. ⭐⭐⭐⭐ (Physics — small oscillations.) A pendulum's potential is $U(\theta)=mgL(1-\cos\theta)$. Expand $\cos\theta$ and keep terms through $\theta^2$ to show $U\approx\tfrac12 mgL\,\theta^2$ (simple harmonic). Then write $\theta(t)=\operatorname{Re}(Ae^{i\omega t})$ using Euler's formula and identify $\omega$.

E7. ⭐⭐⭐⭐ (Signal processing — synthesis.) Using $\cos\theta=\tfrac12(e^{i\theta}+e^{-i\theta})$, rewrite the real Fourier series $\sum(a_n\cos nx+b_n\sin nx)$ as the complex form $\sum_{n=-\infty}^{\infty}c_n e^{inx}$ (Section 24.7). Express $c_n$ and $c_{-n}$ in terms of $a_n,b_n$, and explain why $c_{-n}=\overline{c_n}$ for a real signal.