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This four-page chart comprises reviews:
Definition of calculus and functions
Types of capabilities and rules
Trigonometric identities
Limits and continuity
Taking derivatives
Using derivatives

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Example text

G. (91) bj The conditions (91) is a system of g linear equations for degD variables fj . This observation immediately implies dim Image d ≥ deg D − g. 6 (Riemann’s inequality) For any positive divisor D l(−D) ≥ deg D + 1 − g. 5 for two simple corollaries of Riemann’s inequality. 7 For any positive divisor D with deg D = g + 1 there exists a non-trivial meromorphic function in L(−D). 8 Any Riemann surface of genus 0 is conformally equivalent to the complex ¯ sphere C. 5 MEROMORPHIC FUNCTIONS ON COMPACT RIEMANN SURFACES 56 Proof Let us consider a divisor which consists of one point D = P .

PM PM − q1 Q1 − . . − qN QN = (f ) is called the divisor of f and is denoted by (f ) . A divisor D is called principal if there exists a function with (f ) = D. Obviously we have (f g) = (f ) + (g), (const = 0) = 0, where f and g are two meromorphic functions on R. 3 Two divisors D and D are called linearly equivalent if the divisor D−D is principal. The corresponding equivalence class is called the divisor class. 5 MEROMORPHIC FUNCTIONS ON COMPACT RIEMANN SURFACES 51 We denote linearly equivalent divisors by D ≡ D .

Mg , mn ∈ Z. and mn ≥ 2(n − 1) (106) because of (105). This observation allows us to estimate the weight of the Weierstrass point Pk . Using (99) we get = = = ∆[h1 , . . , hg ] = ∆[z m1 g1 , . . , z mg gg ] = g (z m1 g1 )g ∆[1, z m2 −m1 gg21 , . . , z mg −m1 gg1 ] g (z m1 g1 )g ∆g−1 [(z m2 −m1 gg21 ) , . . , (z mg −m1 gg1 ) ] (z m1 g1 )g ∆g−1 [z m2 −m1 −1 g˜1 , . . , z mg −m1 −1 g˜g−1 ], = = where g˜k (z) defined by z mk −m1 −1 g˜k = z mk+1 −m1 gk+1 g1 are holomorphic near z = 0 and g˜k (z) = 0.

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