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By Karl-Heinz Fieseler

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Extra resources for Complex Analysis [Lecture notes]

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Let f ∈ C(G) and γ be a piecewise smooth path in G. Then we have f (z)dz ≤ ||f ||γ · L(γ). γ Here ||f ||γ := max{|f (z)|; z ∈ |γ|} denotes the maximum of |f | along the trace of γ. Proof. We may assume that γ is smooth, say γ : [a, b] −→ G. For a complex valued, continuous function g : [a, b] −→ C we have b b |g(t)|dt, g(t)dt ≤ a a since s s g(ti )(ti − ti−1 ) ≤ i=1 |g(ti )| · (ti − ti−1 ), i=1 where t0 = a < t1 < ... < ts = b, and, with shrinking interval lengths ti −ti−1 , the sum on the right hand side converges to the corresponding integral, and the same is true for the left hand side.

1 provides a polynomial hν : C −→ C satisfying ||qν − hν ||Kν < 2−ν . 63 Finally set ∞ (qν (z) − hν (z)). f (z) = ν=0 1 . 4. 1. Take G = C, an = −n (where n ∈ N) and pn := z+n 1 Then we can choose g0 = 0 and gn (z) = − n for n ≥ 1, the resulting function is ∞ 1 1 1 − . f (z) = + z n=1 z + n n 2. Take the above example, but replace the indexing set N with Z. We obtain the function π cot(πz) = 1 + z n∈Z∗ 1 1 − z+n n , where Z∗ := Z \ {0}. 3. Consider a lattice Λ = Zω1 + Zω2 , where the complex numbers ω1 , ω2 ∈ C are linearly independent over R.

Be loops in G. 3. Let λ, λ ω= ˜ λ ω λ for any locally integrable differential form ω ∈ D(G). Proof. 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.

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