Felix Klein of Göttingen has made an extensive study of modular functions, dealing with a type of operations lying between the two extreme types, known as the theory of substitutions and the theory of invariants and covariants. Klein's theory has been presented in book-form by his pupil, Robert Fricke. The bolder features of it were first published in his Ikosaeder, 1884. His researches embrace the theory of modular functions as a specific class of elliptic functions, the statement of a more general problem as based on the doctrine of groups of operations, and the further development of the subject in connection with a class of Riemann's surfaces.
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.
