The elliptic functions were expressed by Abel as quotients of doubly infinite products. He did not, however, inquire
rigorously into the convergency of the products. In 1845 Cayley studied these products, and found for them a complete
theory, based in part upon geometrical interpretation, which he made the basis of the whole theory of elliptic functions. Eisenstein discussed by purely analytical methods the general
doubly infinite product, and arrived at results which have been greatly simplified in form by the theory of primary factors, due to Weierstrass. A certain function involving a
doubly infinite product has been called by Weierstrass the sigma-function, and is the basis of his beautiful theory of
elliptic functions. The first systematic presentation of Weierstrass' theory of elliptic functions was published in 1886 by G. H. Halphen in his Théorie des fonctions elliptiques et des
leurs applications. Applications of these functions have been given also by A. G. Greenhill. Generalisations analogous to
those of Weierstrass on elliptic functions have been made by Felix Klein on hyperelliptic functions.
Standard works on elliptic functions have been published by
Briot and Bouquet (1859), by Königsberger, Cayley, Heinrich
Durège of Prague (1821–1893), and others.
Jacobi's work on Abelian and theta-functions was greatly
extended by Adolph Göpel (1812–1847), professor in a gymnasium
near Potsdam, and Johann Georg Rosenhain of Königsberg
(1816–1887). Göpel in his latinTheoriæ transcendentium primi ordinis adumbratio levis (Crelle, 35, 1847) and Rosenhain in several memoirs established each independently, on the analogy of the single theta-functions, the functions of two variables, called double theta-functions, and worked out in connection with them the theory of the Abelian functions of two variables.
The theta-relations established by Göpel and Rosenhain received for thirty years no further development, notwithstanding the fact that the double theta series came to be of increasing importance in analytical, geometrical, and mechanical problems, and that Hermite and Königsberger had considered the
subject of transformation. Finally, the investigations of C. W. Borchardt of Berlin (1817–1880), treating of the representation
of Kummer's surface by Göpel's biquadratic relation between