Chapter 24 — Key Takeaways

This chapter cashed in five chapters of machinery. The single thread tying it together: the Taylor series of $e^x$, $\sin x$, and $\cos x$ (Chapter 23), combined with the imaginary unit $i$, are three faces of one function — the complex exponential.

The Climax: Euler's Formula and Identity

  • Euler's formula (Section 24.3): substituting $x=i\theta$ into the exponential series and regrouping real and imaginary terms gives $$e^{i\theta}=\cos\theta+i\sin\theta.$$ The regrouping is legal because the exponential series converges absolutely for all complex arguments (Chapter 22).
  • Euler's identity (Section 24.4): setting $\theta=\pi$ gives $e^{i\pi}=-1$, hence $$e^{i\pi}+1=0,$$ welding five constants — $e,\ i,\ \pi,\ 1,\ 0$ — with addition, multiplication, and exponentiation. It is provable, not poetic: three Taylor series, one substitution, one regrouping.
  • Geometry: $e^{i\theta}$ is the point on the unit circle at angle $\theta$. An imaginary exponent rotates rather than grows. Multiplying by $i=e^{i\pi/2}$ is a quarter-turn.

Complex Exponentials as a Toolkit (Section 24.5)

  • Polar form: $z=re^{i\theta}$; multiplication gives $z_1z_2=r_1r_2e^{i(\theta_1+\theta_2)}$ — moduli multiply, arguments add.
  • de Moivre: $(\cos\theta+i\sin\theta)^n=\cos(n\theta)+i\sin(n\theta)$.
  • Trig from exponentials: $\cos\theta=\dfrac{e^{i\theta}+e^{-i\theta}}{2}$, $\sin\theta=\dfrac{e^{i\theta}-e^{-i\theta}}{2i}$. Angle-addition and Pythagorean identities fall out as two-line algebra.

Evaluating Functions by Series (Section 24.6)

  • Calculators compute $\sin$, $\cos$, $e^x$ by range reduction then a short series, stopping when the alternating-series error bound (Chapter 22) meets the tolerance.
  • Production libraries refine this with minimax polynomials (Remez algorithm) and, on multiplier-free hardware, CORDIC — Euler's unit-circle rotation turned into shifts and adds.

The Fourier Idea (Section 24.7)

  • Fourier series represent a periodic function over its whole period as a sum of sines and cosines (where Taylor is local, Fourier is global). Coefficients come from integration (Chapter 14).
  • The square wave $=\dfrac{4}{\pi}\sum_{k\ge0}\dfrac{\sin((2k+1)x)}{2k+1}$ — odd harmonics only, since the wave is odd.
  • Gibbs phenomenon: partial sums overshoot ~9% beside each jump and never stop; at the jump the series converges to the average of the two sides. Convergence is mean-square ($L^2$), not uniform.

The Basel Result (Section 24.8)

  • $$\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}\approx 1.644934.$$
  • Euler's route: factor $\dfrac{\sin x}{x}=\prod\left(1-\dfrac{x^2}{n^2\pi^2}\right)$ from its roots, match the $x^2$ coefficient against the Taylor series. The $\pi$ enters through the roots of $\sin$.
  • This is $\zeta(2)$, the first value of the Riemann zeta function, which opens onto the Euler product, $\zeta(4)=\pi^4/90$, and the Riemann hypothesis.

More Series at Work

  • Computing $\pi$ (Section 24.9): the Leibniz series converges uselessly slowly; Machin's formula uses small arctan arguments to converge fast.
  • Probability generating functions (Section 24.10): $G(s)=\sum p_n s^n=E[s^X]$, with $G(1)=1$, $G'(1)=E[X]$, and $G_{X+Y}=G_XG_Y$ for independent variables. The Poisson PGF $e^{\lambda(s-1)}$ comes straight from the exponential series.

Common Errors to Avoid

  • Treating $e^{i\pi}=-1$ as a real-logarithm fact. The complex exponential is not one-to-one ($e^{i\pi}=e^{3i\pi}=\cdots$), so $\ln(-1)=i\pi+2\pi ik$ is multi-valued; ordinary log laws are unsafe (Section 24.4).
  • Expecting Gibbs overshoot to vanish with more terms. It narrows but keeps its ~9% height (Section 24.7).
  • Forgetting range reduction when evaluating a series far from $0$: the raw series suffers catastrophic cancellation (Section 24.6).
  • Thinking Euler's Basel factorization is rigorous as written. It needs the Weierstrass factorization theorem; Euler verified it numerically (Section 24.8).

Connections

  • Backward: every result rests on Chapter 23's Taylor series and Chapter 22's convergence/error tools; Fourier coefficients use Chapter 14's integrals; Euler's formula was first hinted at in Chapter 11.
  • Forward: polar form $re^{i\theta}$ becomes a full coordinate geometry in Chapter 26; PGFs and the SIR connection feed the Chapter 39 capstone.
  • Themes: approximation is the soul of calculus (every transcendental value here is a truncated series), geometry and algebra are inseparable (the complex exponential turns rotation into arithmetic), and calculus appears in every quantitative field (EE, signal processing, number theory, probability, physics).