Inversion of an elliptic integral

The problem of constructing the function $u$ as a function of $z$ or of constructing single-valued composite functions of the form $f(u(z))$ in the case of an elliptic integral

$$z=\int\limits^uR(z,w)dz,$$

where $R$ is a rational function of variables $z$ and $w$ which are related by the equation $w^2=F(z)$, where $F(z)$ is a polynomial of degree 3 or 4 without multiple roots. The complete solution of this problem was given almost simultaneously in 1827–1829 by N.H. Abel and C.G.J. Jacobi, who showed that its solution led to new transcendental elliptic functions (cf. Elliptic function).

An essentially different approach to the theory of elliptic functions is due to K. Weierstrass. For the elliptic integral of the first kind in Weierstrass normal form,

$$z=\int\limits^u\frac{dz}{w},\quad w^2=4z^3-g_2z-g_3,$$

$u=\mathrm p(z)$ turns out to be the Weierstrass $\mathrm p$-function with invariants $g_2,g_3$ (see Weierstrass elliptic functions). For the elliptic integral of the first kind in Legendre normal form,

$$z=\int\limits_0^\phi\frac{dt}{\sqrt{1-k^2\sin^2t}},$$

inversion leads to the Jacobi elliptic functions.

References

[1]	N.I. Akhiezer, "Elements of the theory of elliptic functions" , Amer. Math. Soc. (1990) (Translated from Russian)
[2]	A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , Springer (1964) pp. Chapt. 3, Abschnitt 2