Riemann's memoir Über das verschwinden der $$vartheta $$-functionen
pp. 125-134
in: Lizhen Ji, Athanase Papadopoulos, Sumio Yamada (eds), From Riemann to differential geometry and relativity, Berlin, Springer, 2017Abstract
In the second part of his great memoir Theorie der Abel'schen Functionen (1857), Riemann gives a solution to Jacobi's problem of inversion for the general case of integrals of algebraic functions , now called Abelian integrals. The case of hyperelliptic integrals had been treated, for the genus 2, by Göpel and Rosenhain and, for any genus, by Weierstrass in a series of memoirs between 1848 and 1856. The proof developed by Riemann in his 1857 paper is not complete and the memoir Über das Verschwinden der (vartheta ) -Functionen (1865) completes it.