Needham, Chapter 2, Visual Complex Analysis—PDF files with notes, figures, and some answers for exercises

The figures in this list pertain to Visual Complex Analysis (1997) by Tristan Needham. The illustrations pertain to suggested exercises in the text, to exercises at ends of chapters, and to ideas suggested by the text. In a some cases, they are attempts to reproduce figures found in the text. The figures are plotted with Mathematica 10.

Some answers to exercises are provided without figures. The answers are my own and they are not endorsed by the author of Visual Complex Analysis. The answers in the archives of Vasco are in general more consistent and reliable. I have consulted them frequently in my own studies. Current discussions of the book and the exercises appear on Vasco's forum.

Exercises
  1. Cardioid (PDF)
  2. Complex division and unit circle (PDF)
  3. Mapping unit circle and disc to itself (PDF)
  4. Cardano's formula, q^2 > p^3 (PDF)
  5. Lemniscate (PDF)
  6. Orthogonal curves of lemniscate (PDF)
  7. Modular surface of Cassinian curves (PDF)
  8. Standard lemniscate (PDF)
  9. Using real methods to derive power series for 1/(1+x^2) (PDF)
  10. Expansions of 1/(1+x^2) to power series (PDF)
  11. Radius of convergence of power series (PDF)
  12. Error in approximation of geometric series (PDF)
  13. Diverse polynomial forms of e^z (PDF)
  14. Precision in polynomial approximations (PDF)
  15. Disc of convergence of PQ larger than that of P or Q (PDF)
  16. Binomial Theorem w. negative powers (PDF)
  17. Binomial Theorem w. negative powers II (PDF)
  18. Binomial Theorem w. real, negative and rational powers
    • (i, ii) Convergence of B(z,n) = (1+z)^n in unit disc; multiplying series (PDF)
    • (iii, iv) Multiplying binomial series; series for negative n (PDF)
    • (v, vi) Series B[z,1/q] is principle branch of (1+z)^(1/q); B[z,p/q] is principle branch of (1+z)^(p/q)(PDF)
  19. Radius of convergence for (18) (PDF)
  20. Fourier series (PDF)
  21. Trig and Exp to Power Series, (i) and (ii) only (PDF)
  22. Power series e^z (PDF)
  23. Argument of asymptote of hyperbola on x = k (PDF)
  24. Path of Sqrt(z-1)CubeRt(z-i) as z traverses loops on branch points 1 and i. (PDF)
  25. Branch cuts for Sqrt[(z+i)(z-i)] (PDF)
  26. Branch points, singularities, and radius of convergence(PDF)
  27. log(z) and log(-z) have no common values (PDF)
  28. z^i after revolution of e^{-i3Pi} (PDF)
  29. k^z after revolution of e^{-i2Pi} (PDF)
  30. All branches of i^i are real (PDF)
  31. ln(1+x) (PDF)
  32. ln(1+z) (PDF)
  33. cos^{-1}(w) as multifunction (PDF)
  34. Power series (cos z)^(1/2) (PDF)
  35. Power series z/sin z (PDF)
  36. Power series tan^{-1}(x) (PDF)
  37. Fourier series from power series: jagged wave (PDF)
  38. Fourier series from power series: square wave (PDF)
  39. Centroid always found within convex hull (PDF)
  40. Centre of regular n-gon is centroid of vertices (PDF)
  41. Image of regular n-gon is regular n-gon (PDF)
  42. Use Viete's sum of roots to find centroid of multifunction (PDF)
  43. Use Gauss's mean value theorem to find < cos z >_C (PDF)
Notes

  1. Notes and [Exercises] in text (PDF)

Last modified on March 5, 2021.

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