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Denote H : R := [a, b] × I −→ G a homotopy between the loops λ 2 ˜ Let R = and λ. 1≤i,j≤n Rij be the decomposition of R into n congruent rectangles of size n1 the size of R. For sufficiently big n ∈ N we have ω− λ ω= ω= ˜ λ H(∂R) ω = 0, i,j H(∂Rij ) since the ”vertical edges” H(b × [0, 1]) and H(a × [1, 0]) of the ”rectangle” H(∂R) are inverse one to the other and for n 0 any ”rectangle” H(∂Rij ) is contained in an open set, where ω admits a primitive function. 4. , λs in G. It is called nullhomologous in G if and only if s ω=0 ω := λ j=1 λj holds for all locally integrable differential forms ω ∈ D(G).

Z = K(z, ζ)dζ = γ Taking γ = ∂D and K(z, ζ) := with f (ζ) (ζ − z)n+1 ∂K f (ζ) (z, ζ) = n ∂z (ζ − z)(n+2) we obtain with Rem. 5 the case n + 1. 6. Let f ∈ O(G) and Dr (z0 ) ⊂ G be an open disc. , f (z) = n=0 for all z ∈ Dr (z0 ), the right hand side converging uniformly on every closed disc D (z0 ), < r. 38 A function (of one or several real or complex) variables is called (real or complex) analytic if near any point of its domain of definition it can be written as a power series. So Th. 6 tells us that holomorphic functions are complex analytic functions.

First of all, the integral does not change, if we replace λ with a loop λ homotopic to λ in G. 1. Two continuous loops λ, λ topic in G, if there is a continuous map H : [a, b] × I −→ G ˜ with H(t, 0) = λ(t), H(t, 1) = λ(t) and H(a, s) = H(b, s) for all s ∈ I := [0, 1]. 2. , n. , n. Then we set n Fν (γ(tν )) − Fν (γ(tν−1 )). ω := γ ν=1 49 We leave it to the reader to check that the given definition does not depend on n ∈ N. ˜ be loops in G. 3. Let λ, λ ω= ˜ λ ω λ for any locally integrable differential form ω ∈ D(G).

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Complex Analysis [Lecture notes] by Karl-Heinz Fieseler


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