By Bruce P. Palka

ISBN-10: 038797427X

ISBN-13: 9780387974279

This publication offers a rigorous but hassle-free advent to the idea of analytic services of a unmarried complicated variable. whereas presupposing in its readership a level of mathematical adulthood, it insists on no formal must haves past a legitimate wisdom of calculus. ranging from uncomplicated definitions, the textual content slowly and thoroughly develops the guidelines of complicated research to the purpose the place such landmarks of the topic as Cauchy's theorem, the Riemann mapping theorem, and the theory of Mittag-Leffler will be handled with no sidestepping any problems with rigor. The emphasis all through is a geometrical one, so much said within the broad bankruptcy facing conformal mapping, which quantities primarily to a "short path" in that vital sector of complicated functionality idea. each one bankruptcy concludes with a big variety of workouts, starting from hassle-free computations to difficulties of a extra conceptual and thought-provoking nature.

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7 Let aj , bj , j = 1, . . , g be a canonical homology basis of R and ωk , k = 1, . . , g the dual basis of H 1 (R, C). The matrix Bij = ωj (69) bi is called the period matrix of R. 14 The period matrix is symmetric and its real part is negative definite Bij = Bji , (70) g Re(Bα, α) < 0, ∀α ∈ R . (71) Proof For the proof of (70) substitute two normalized holomorphic differentials ω = ωi and ω = ωj into the Riemann bilinear identity (55). The vanishing of the left hand side ωi ∧ ωj ≡ 0 implies (70).

6 A homology basis a1 , b1 , . . , ag , bg of a compact Riemann surface of genus g with the intersection numbers (46) is called canonical basis of cycles. Remark Canonical basis of cycles is by no means unique. Let (a, b) be a canonical basis of cycles. We represent it by a 2g-dimensional vector a1 b1 a , a = ... , b = ... . b ag bg 3 TOPOLOGY OF RIEMANN SURFACES 31 Any other basis (˜ a, ˜b) of H1 (R, Z) is then given by the transformation a ˜ ˜b =A a b , A ∈ SL(2g, Z).

0,1 (Pk ) . . α0,g (Pk ) This is a linear map H : Cg → CdegD , and due to (92) dim Image d = dim ker H T = degD − rankH. (93) Near the points Pj the normalized holomorphic differentials ωi have the following asymptotics ωi = (α0,i (Pj ) + o(1))dzj . This shows that the linear spaces ker H and H(D) are isomorphic g (β1 , . . , βg ) ∈ ker H ⇐⇒ βi ωi ∈ H(D). 5. 9 The degree of the canonical class is deg C = 2g − 2. Proof The differential dz on the complex sphere has a double pole at z = ∞ dz = − 1 dτ, τ2 τ= 1 .

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