Christian Houzel

Riemann’s Memoir Über das Verschwinden der \(\vartheta \)-Functionen

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 Gopel 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 Uber das Verschwinden der \(\vartheta \) -Functionen (1865) completes it.