# Amplitude of an elliptic integral

The variable $\phi$, considered as a function of $z$, in an elliptic integral of the first kind
$$z = F ( \phi , k ) = \int\limits _ { 0 } ^ \phi \frac{dt} {\sqrt {1 - k ^ {2} \sin ^ {2} t }}$$
in the normal Legendre form. The concept of the amplitude of an elliptic integral and the notation $\phi = \mathop{\rm am} z$ were introduced by C.G.J. Jacobi in 1829. The amplitude of an elliptic integral is an infinite-valued periodic function of $z$. The basic elliptic Jacobi functions $\sin \mathop{\rm am} z = \mathop{\rm sn} z$, $\cos \mathop{\rm am} z = \mathop{\rm cn} z$, $\Delta \mathop{\rm am} z = \mathop{\rm dn} z$ are all single-valued. It is convenient, however (e.g. for purposes of tabulation), to consider an elliptic integral as a function $F ( \phi , k)$ of the amplitude $\phi$ and the modulus $k$. See also Jacobi elliptic functions.