Carathéodory theorem

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Carathéodory theorem

[‚‚kär·ə‚tā·ə′dȯr·ē ‚thir·əm]
(mathematics)
The theorem that each point of the convex span of a set in an n-dimensional Euclidean space is a convex linear combination of points in that set.
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He starts with Cauchy-Riemann equations in the introduction, then proceeds to power series, results on holomorphic functions, logarithms, winding numbers, Couchy's theorem, counting zeros and the open mapping theorem, Eulers formula for sin(z), inverses of holomorphic maps, conformal mappings, normal families and the Riemann mapping theorem, harmonic functions, simply connected open sets, Runge's theorem and the Mittag-Leffler theorem, the Weierstrass factorization theorem, Caratheodory's theorem, analytic continuation, orientation, the modular function, and the promised Picard theorems.