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.