Chapter 24 — Further Reading

Each entry below says what to read it for and, where applicable, maps to the matching section in our two reference textbooks. (Full chapter-by-chapter mappings live in appendices/appendix-h-stewart-chapter-mapping.md and appendices/appendix-i-openstax-chapter-mapping.md.)

Standard Treatments

  • Stewart, Calculus: Early Transcendentals (9th ed.), Section 11.10 and Appendix H. Section 11.10 develops Taylor and Maclaurin series — the foundation for the Euler-formula derivation of our Section 24.3. Appendix H (Complex Numbers) is the direct match for this chapter: it covers the complex plane, polar form $re^{i\theta}$, de Moivre's theorem, and complex exponentials. Read Appendix H alongside our Sections 24.3–24.5.
  • OpenStax Calculus Volume 2 (Strang & Herman), Sections 6.3–6.4. Free and careful coverage of power series, Taylor/Maclaurin series, and the radius/interval of convergence that our Section 24.2 relies on. OpenStax keeps complex numbers light, so pair it with Stewart's Appendix H for the Euler material.
  • OpenStax Calculus Volume 2, Section 5.3–5.4 (the divergence, ratio, and alternating-series tests). Re-read for the alternating-series error bound that underlies every "how many terms?" estimate in our Section 24.6 and exercises C1–C6.

Euler's Formula, Identity, and Complex Analysis

  • Nahin, Dr. Euler's Fabulous Formula (Princeton). A whole accessible book devoted to $e^{i\pi}+1=0$ and its consequences, including a full chapter on the AC-circuit / phasor application of Case Study 1. The most enjoyable next step after Section 24.4.
  • Needham, Visual Complex Analysis (Oxford). The unmatched geometric account of why $e^{i\theta}$ rotates and why "multiplication adds arguments" (our Section 24.5). Picture-first, proof-second — perfectly aligned with this book's style.
  • Stewart (James), Calculus, Appendix H exercises on de Moivre and roots of unity. Drill problems that extend our A8 and B-series exercises.

Fourier Series and Signal Processing

  • Stein & Shakarchi, Fourier Analysis: An Introduction (Princeton). The rigorous sequel to our Section 24.7: convergence in the mean, the Gibbs phenomenon stated and proved, and the complex Fourier series $\sum c_n e^{inx}$ developed properly.
  • Bracewell, The Fourier Transform and Its Applications. The engineering classic; read its Gibbs and windowing chapters to see Case Study 2's overshoot become real filter design.
  • Smith, The Scientist and Engineer's Guide to Digital Signal Processing (free at dspguide.com). Reader-friendly chapters on the FFT, the DCT behind JPEG, and why sharp filters ring — the applied payoff of Section 24.7.

The Basel Problem and the Zeta Function

  • Dunham, Euler: The Master of Us All (MAA). A guided tour of Euler's actual arguments, including the sine-product solution of the Basel problem from our Section 24.8, written for a general calculus audience.
  • Aigner & Ziegler, Proofs from THE BOOK, chapter "$\pi^2/6$ again." Several elegant rigorous proofs of the Basel result by different routes (Fourier series, double integrals), filling the gap our Math Major Sidebar flags in Euler's original.
  • Derbyshire, Prime Obsession. A narrative history of the Riemann zeta function and hypothesis (our Section 24.8's "door it opened"), accessible with only the series background of this chapter.

Computation and Probability

  • Muller, Elementary Functions: Algorithms and Implementation (Birkhäuser). How calculators and math libraries really evaluate $\sin$, $\cos$, $e^x$ — range reduction, minimax polynomials, and CORDIC (our Section 24.6) in full engineering detail.
  • Grimmett & Stirzaker, Probability and Random Processes (Oxford), chapter on generating functions. The standard development of probability generating functions (our Section 24.10), including branching processes and the extinction-probability / $R_0$ result that connects to the Chapter 39 capstone.