Elliptic Functions

Historically, the starting point of the theory of elliptic functions were the elliptic integrals, named in this way because of their direct connection to computing arc lengths of ellipses. Already in 1718 (G.C. FAGNANO), a very special elliptic integral was extensively investigated,

$$E(x) : = \int^{x}_{0} \frac{dt}{\sqrt{1-t^4}}.$$

It represents in the interval ]0, 1[ a strictly increasing (continuous) function. So we can consider its inverse function f. A result of N.H. ABEL (1827) affirms that f has a meromorphic continuation into the entire C. In addition to an obvious real period, ABEL discovered a hidden complex period. So the function f turned out to be a doubly periodic function. Nowadays, a meromorphic function in the plane with two independent periods is also called elliptic. Many results that were already know for the elliptic integral, as for instance the famous EULER Addition Theorem for elliptic integrals, appeared to be surprisingly simple corollaries of properties of elliptic functions. This motivated K. WEIERSTRASS to turn the tables. In his lectures in the winter term 1862/1863 he gave a purely function theoretical introduction to the theory of elliptic functions. In the center of this new setup, there is a special function, the ℘-function. It satisfies a differential equation which immediately shows the inverse function of ℘ to be an elliptic integral. The theory of elliptic integrals was thus derived as a byproduct of the theory of elliptic functions.