# Generalized Analytic Functions on Riemann Surfaces by Yuri L. Rodin By Yuri L. Rodin

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Example text

Be two a r b i t r a r y have poles I the s y m b o l residues ~ I of the surface. de (s) and q0 q at the p o i n t s and periods = 0 j = I, .... 6) 31 Re I d~q0q(S) K 3 j = I, .... 2g . 6) Now several known relations connecting Abelian differentials example, G. Springer [a]) are listed. (see, for k,j = I ..... 8) ql I dtq,z(S) j = 1 ..... 2g , = 2zi dn@j (q) (n-l) ! d[z(q) in , j = 1, . . 9) K2j I dT~ z(S) = , K 2zi dn@ (q) (n-l)' Im J . dLzlqj r . jn ~ 3 q2 [ dt n z(S) = I J q, - ~ ql dn~ (q) qlq2 d[z(q)] n ~s0s(q) - ~s0s(q 0) = ~q0q(S) - eq0q(S0) ~s0s(q) - ~s0s(q 0) = ~q0q(S) - ~q0q(S0) -2zi ~ j=1 j = I .....

Of all, be u s e d [a]). The ferential (1,0) we form famous Abelian list Stein on a R i e m a n n constructed [a]). the A b e l i a n ~ = ~(z(p)) dz(p) is c a l l e d ~(z(p)) denotes (z = z(p) surface we n e e d several authors the later paper and integrals of this relations along the c y c l e s space dw. 3 I dw k = 6jk on M [a], by an which G. ~,g) K2j_I of the to first g dif- kind. linear normalized (j = 1 .... 1) I ..... g . of this space I K2j- I de2k = - Im I K2j d~2k_1 = 6ki use by the p e r i o d of the h o m o l o g y de, ] periods Im of type K2j-I We an singularities.

2. v. 8) 58 (l,v I) ... -- (l,V2g) (i,v I) ... (i,Vg 0) (i,Vg0+ I) --- (i,V2g) (tpl,v I) ... )+Ira Z{(pn ) Pn gO+ 1 The matrix Figure Let some zero c o m b i n a t i o n s to the c o e f f i c i e n t s ... -(tp1'V2g ) +Re Z'2g-g 0 (pl) (itpl,vg0+l) +Re Z~(pl) ... (itpl,Vg 0) ... ) ... Pn ± Pl (itpl,V2g) + Im Z' (pll 2g-g 0 ... -(t ... 9) tpk(s) Then (ty,vj) (ty,Vg0+ j) - Re = 0 , n [ k=1 j = 1 ..... ,2g-g 0 . 10) it follows be a solution. 11) - Re = (u0,S*Vg0+ j) - Re Su = t Y can be t r a n s f o r m e d n ~ k=l n [ k=1 (-e2k + i~2k+1) (-~2k + ie2k+l) is solvable.

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