Amplitude of an elliptic integral

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The variable , considered as a function of , in an elliptic integral of the first kind

in the normal Legendre form. The concept of the amplitude of an elliptic integral and the notation were introduced by C.G.J. Jacobi in 1829. The amplitude of an elliptic integral is an infinite-valued periodic function of . The basic elliptic Jacobi functions , , are all single-valued. It is convenient, however (e.g. for purposes of tabulation), to consider an elliptic integral as a function of the amplitude and the modulus . See also Jacobi elliptic functions.

How to Cite This Entry:
Amplitude of an elliptic integral. E.D. Solomentsev (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098