By C. G. Lekkerkerker, N. G. De Bruijn, J. De Groot, A. C. Zaanen

This quantity features a quite whole photograph of the geometry of numbers, together with kinfolk to different branches of arithmetic equivalent to analytic quantity idea, diophantine approximation, coding and numerical research. It offers with convex or non-convex our bodies and lattices in euclidean area, and so forth. This moment variation used to be ready together by means of P.M. Gruber and the writer of the 1st version. The authors have retained the present textual content (with minor corrections) whereas including to every bankruptcy supplementary sections at the more moderen advancements. whereas this technique can have drawbacks, it has the sure benefit of exhibiting basically the place fresh growth has taken position and in what parts attention-grabbing effects might be anticipated sooner or later.

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1) Find the domains of analyticity A of f , and B of g, and sketch B. Find the derivative g (z) for z ∈ B. 2) Denote by Γ any oriented closed curve in B, and ﬁnd the value of the line integral g (z) dz. Γ Let γ denote any oriented curve in B of initial point z = −i and end point z = i. Prove that √ π 4 g (z) dz = i 2 2 sin . 8 γ 1) Clearly, A = C \ {z ∈ C | Re(z) ≤ 0, Im(z) = 0}. The exceptional set of g is given by 1 − z 3 ∈ R− ∪ {0}, hence z 3 ∈ [1, +∞[, and thus B =C\ z ∈C z = r · eiθ , r ≥ 1, θ ∈ − 2π 2π , 0, 3 3 .

2iz i w2 + 1 cos z i e +1 Since we require that tan z is deﬁned, we must have w 2 = −1. Hence, w ∈ C \ {−i , 0 , i} = Ω. com 55 Complex Functions Examples c-3 Trigonometric and hyperbolic functions Then we put w = eiz ∈ Ω into the given equation, and obtain after a rearrangement, w2 − 1 1 w2 − 1 = i w + 1 + i w2 + 1 w2 + 1 2 2 (w + 1) w w +1+w−1 = i(w + 1) · , =i w2 + 1 wr + 1 0 = i 1 + eiz − tan z = i(1 + w) − = i(w + 1) 1 + w−1 w2 + 1 where we shall solve the equation for w ∈ Ω = C \ {−i , 0 , i}.

A) It follows from |F (z)| = exp z 2 = exp x2 − y 2 + 2i xy = exp x2 − y 2 = R > 0, that x2 − y 2 = ln R ∈ R, which is the equation of a system of hyperbolas, supplied with the straight lines y = x and y = −x, both corresponding to R = 1. com 36 Complex Functions Examples c-3 The exponential function and the logarithm function 2 1 –2 –1 0 1 2 –1 –2 2 Figure 6: Some level curves F (z)| = ex −y 2 = R > 0. (b) By using polar coordinates we get the description F r eiθ = exp r2 cos 2θ · exp i r2 sin 2θ .