Theory of Functions

coefficients, and cubic residues. He read Legendre's Exercises, which give an account of elliptic integrals. When he returned the book to the library, he was depressed in spirits and said that important books generally excited in him new ideas, but that this time he had not been led to a single original thought. Though slow at first, his ideas flowed all the richer afterwards. Many of his discoveries in elliptic functions were made independently by Abel. Jacobi communicated his first researches to Crelle's Journal. In 1829, at the age

of twenty-five, he published his Fundamenta Nova Theoriæ Functionum Ellipticarum, which contains in condensed form the main results in elliptic functions. This work at once secured for him a wide reputation. He then made a closer study of theta-functions and lectured to his pupils on a new

theory of elliptic functions based on the theta-functions. He developed a theory of transformation which led him to a multitude of formulæ containing $q$, a transcendental function of the modulus, defined by the equation $q=e^{-\pi k^{\prime }/k}$. He was also led by it to consider the two new functions $H$ and $\Theta$, which taken each separately with two different arguments are the four (single) theta-functions designated by the ${\Theta }_1$, ${\Theta }_2$, ${\Theta }_3$, ${\Theta }_4$.56 In a short but very important memoir of 1832, he shows that for the hyperelliptic integral of any class the direct functions

to which Abel's theorem has reference are not functions of a

single variable, such as the elliptic $\mathrm{\backslash sn}$, $\mathrm{\backslash cn}$, $\mathrm{\backslash dn}$, but functions of $p$ variables.56 Thus in the case $p=2$, which Jacobi especially considers, it is shown that Abel's theorem has reference to two functions $\lambda (u,v)$, ${\lambda }_1(u,v)$, each of two variables, and gives in effect an addition-theorem for the expression of the functions $\lambda (u+u^{\prime },v+v^{\prime })$, ${\lambda }_1(u+u^{\prime },v+v^{\prime })$ algebraically in terms of the functions $\lambda (u,v)$, ${\lambda }_1(u,v)$, $\lambda (u^{\prime },v^{\prime })$, ${\lambda }_1(u^{\prime },v^{\prime })$. By the memoirs of Abel and Jacobi it may be considered that the notion of the Abelian function of $p$ variables was established and the addition-theorem for these functions given. Recent studies touching Abelian functions have been made by Weierstrass,

E. Picard, Madame Kowalevski, and Poincaré. Jacobi's

work on differential equations, determinants, dynamics, and the theory of numbers is mentioned elsewhere.

In 1842 Jacobi visited Italy for a few months to recuperate

his health. At this time the Prussian government gave him a pension, and he moved to Berlin, where the last years of his life were spent.

The researches on functions mentioned thus far have been greatly extended. In 1858 Charles Hermite of Paris (born 1822),

introduced in place of the variable $q$ of Jacobi a new variable $\omega$

connected with it by the equation $q=e^{i\pi \omega }$, so that $\omega =i{k}^{\prime }/k$, and was led to consider the functions $\varphi (\omega )$, $\psi (\omega )$, $\chi (\omega )$.56 Henry Smith regarded a theta-function with the argument equal to

zero, as a function of $\omega$. This he called an omega-function,

while the three functions $\varphi (\omega )$, $\psi (\omega )$, $\chi (\omega )$, are his modular functions. Researches on theta-functions with respect to real