and imaginary arguments have been made by Meissel of Kiel,
J. Thomae of Jena, Alfred Enneper of Göttingen (1830–1885).
A general formula for the product of two theta-functions was given in 1854 by H. Schröter of Breslau (1829–1892). These
functions have been studied also by Cauchy, Königsberger of
Heidelberg (born 1837), F. S. Richelot of Königsberg (1808–1875),
Johann Georg Rosenhain of Königsberg (1816–1887),
L. Schläfli of Bern (born 1818).85
Legendre's method of reducing an elliptic differential to its
normal form has called forth many investigations, most important of which are those of Richelot and of Weierstrass of
Berlin.
The algebraic transformations of elliptic functions involve a relation between the old modulus and the new one which Jacobi expressed by a differential equation of the third order, and also by an algebraic equation, called by him "modular equation." The notion of modular equations was familiar to
Abel, but the development of this subject devolved upon later
investigators. These equations have become of importance in the theory of algebraic equations, and have been studied by Sohnke, E. Mathieu, L. Königsberger, E. Betti of Pisa (died
1892), C. Hermite of Paris, Joubert of Angers, Francesco
Brioschi of Milan, Schläfli, H. Schröter, M. Gudermann of
Cleve, Gützlaff.
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.