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This radical first course on complex analysis brings a beautiful and powerful subject to life by consistently using geometry (not calculation) as the means of explanation. Aimed at undergraduate students in mathematics, physics, and engineering, the book's intuitive explanations, lack of advanced prerequisites, and consciously user-friendly prose style will help students to master the subject more readily than was previously possible. The key to this is Visual Complex Analysis's use of new geometric arguments in place of the standard calculational ones. These geometric arguments are communicated with the aid of hundreds of diagrams of a standard seldom encountered in mathematical works. A new approach to a classical topic, Visual Complex Analysis will be of interest to students in mathematics, physics, and engineering, as well as to professionals in these fields.
1. Geometry and Complex Arithmetic
2. Complex functions as Transformations
3. Mobius Transformations and Inversion
4. Differentiation: The Amplitwist Concept
5. Further Geometry of Differentiation
6. Non-Euclidean Geometry
7. Winding Numbers and Topology
8. Complex Integration: Cuachy's Theorem
9. Cauchy's Formula and Its Applications
10. Vector Fields: Physics and Topology
11. Vector Fields and Complex Integration
12. Flows and Harmonic Functions
"[...] a fascinating and refreshing look at a familiar subject [...] essential reading for anybody with any interest at all in this absorbing area of mathematics."
- Times Higher Education Supplement
"Visual Complex Analysis is a delight, and a book after my own heart. By his innovative and exclusive use of the geometrical perspective, Tristan Needham uncovers many surprising and largely unappreciated aspects of the beauty of complex analysis."
- Roger Penrose
"One of the saddest developments in school mathematics has been the downgrading of the visual for the formal. I'm not lamenting the loss of traditional Euclidean geometry, despite its virtues, because it too emphasised stilted formalities. But to replace our rich visual tradition by silly games with 2 × 2 matrices has always seemed to me to be the height of folly. It is therefore a special pleasure to see Tristan Needham's Visual Complex Analysis with its elegantly illustrated visual approach. Yes, he has 2 × 2 matrices – but his are interesting."
- Ian Stewart, New Scientist, 11 October 1997
"[...] an engaging, broad, thorough, and often deep, development of undergraduate complex analysis and related areas from a geometric point of view. The style is lucid, informal, reader-friendly, and rich with helpful images (e.g. the complex derivative as an "amplitwist"). A truly unusual and notably creative look at a classical subject."
- Paul Zorn, American Mathematical Monthly
"I was delighted when I came across [Visual Complex Analysis]. As soon as I thumbed through it, I realized that this was the book I was looking for ten years ago."
- Ed Catmull, founder of Pixar
