Needham, Chapter 12, 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. See also the animated demos. Both the figures and demos 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.

1. Dual of dipole (PDF)
2. Product of real Phi and dual Phi (PDF)
3. Gauss' Mean Value Theorem rules out local maxima (PDF)
4. The meaning of the Laplacian (3D) (PDF)
5. Gradient of |f| and Laplacian of |f|^2 (PDF)
6. Show f constant with nabla(f + conj(f)) and nabla(f conj(f)) (PDF)
7. Partials on z and conjugate(z) (PDF)
8. Determinant of Jacobian of z --> w (PDF)
9. General harmonic function (PDF)
10. Kasner's Invariant (PDF)
11. Curvature from stream function (PDF)
12. Analytic f applied to source (PDF)
13. Analytic f applied to dipole (PDF)
14. Transposition of vortices at p, 0, and 1/p* projected onto Riemann sphere (PDF)
15. Green's function where X = (u nabla v - v nabla u) (PDF)
16. T(z) on disc with boundaries UH T = Pi/2 and LH T = -Pi/2. (PDF)
17. Harnack's inequality. (PDF)
18. Analogue of Liouville's Theorem derived from Harnack's inequality. (PDF)
19. Substitute Green's function to obtain heat flux at boundary. (PDF)
20. Green's function for upper half plane and Green to Poisson (PDF)
21. Green's function for east half-plane (PDF)

General notes (PDF)

1. Harmonic Duals (PDF)
2. Conformal Invariance (PDF)
3. A Powerful Computational Tool (PDF)
4. The Complex Curvature Revisited (PDF)
5. Flow Around an Obstacle
   1. Flow Around an Obstacle (PDF)
   2. Disturbed and Undisturbed (PDF)
   3. Schwartzian symmetry (PDF)
   4. Flow around ellipse (PDF)
   5. Plotting streamlines, V. pp. 527-536 (PDF)
   6. Flow around ellipse, V.4, pp. 538-540 (PDF)
6. The Physics of Riemann's Mapping
   1. Framework for V and VI, Figures [24], [25] (PDF)
   2. The Physics of Riemann's Mapping, pp. 540-544 (PDF)
   3. The Physics of Riemann's Mapping, Expansion, pp. 540-544 (PDF)
   4. Example, Automorphisms of the Disc, [exercise], p. 549; Green's Function [exercise], p. 554 (PDF)
   5. Harmonic functions (PDF)
7. Dirichlet's Problem (PDF)

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