This concept defines conformal mappings as functions that preserve the angle between intersecting curves at a point, expresses that angle via a complex-number formula analogous to the vector dot-product cosine formula, and proves that any function analytic at a point with nonzero derivative there is conformal at that point. It then introduces Möbius transformations—invertible maps of the extended complex plane of the form (az+b)/(cz+d) with ad−bc ≠ 0—showing they are uniquely determined by images of three points and that they map circles (including lines, viewed as circles through infinity) to circles. It belongs to complex analysis, extending the theory of analytic functions and the extended complex plane into geometric function theory.
This concept defines conformal mappings as functions that preserve the angle between intersecting curves at a point, expresses that angle via a complex-number formula analogous to the vector dot-prod…