George Kempf's Abelian integrals PDF

By George Kempf

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2), we have (1 ;=1 and - , = m S , u= 1 for J Z < ~ 1 SECTION 3. LO. 4 where C" = cu(zl,. . 3) exp G ( z ) = m JJ (1 - Z,Z) /=I with some c: = c:(z,, . . ,z,,). This last identity obviously holds for all complex z. 4) Passage to the limit r + 1 gives, with the notation 26 PURE POWER SUMS the important identity Cauchy's inequality then gives First consider the middle term. 5) and Parseval's formula. 1). First we remark that, due to the inequality the inequality also holds in our range. 4- - a 9 1 ; < a R n + 2Rn + 2R, 11/6

Wk as roots. The a , are obviously real. If max a, 2 0 v=l. . k we have nothing to prove. If max a v < O v=l. k . . 4) give successively But because we have a, +,> 0. 2. Suppose, without loss of generality, that (g,. Adding to (z,, z,,. . ,z,) the points . ,<), the new system forms a real system with k = 2n - 2. 2 made no claim to account for all extremal systems. The matter was cleared up by S. Dancs (Dancs [I]). , 3 n - 4 = 1, n23 is that of the continuum. 4). For details we refer to his paper.

The coeficcients of F(z)cp(z) = 1 + .. are nonnegative. Proof. 3) ~ ( z = ) 1 + Az, with A real. Then A > 0, that is, cp(z) = 1 is suitable. 5) = 1 - 2cosa. z + z2 ~ 5 o 1 a . For a / 2 5 a I a we can take ~ ( z=) 1 again, so assume For arbitrary nonnegative integers k one has the identity + - I - - - sin(k 2 ) -a . sin a - + sin(k + l ) oz . 7) are nonnegative. 4) one can turn to the general case. We then write where n , + 2n2 + 2n, with a / 2 5 a, a are = N , the negative zeros are -A,, P, e la/ ( .

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Abelian integrals by George Kempf


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