The variable 
(also denoted 
)
used in elliptic functions and elliptic
integrals is called the amplitude (or Jacobi amplitude). It can be defined by
 |  |  |
(1)
|
 |  |  |
(2)
|
where 
is a Jacobi elliptic function with elliptic modulus. As is common with Jacobi
elliptic functions, the modulus 
is often suppressed for conciseness. The Jacobi amplitude
is the inverse function of the elliptic
integral of the first kind. The amplitude function is implemented in the Wolfram
Language as JacobiAmplitude[u,
m], where 
is the parameter.
It is related to the elliptic integral of the first kind 
by
 |
(3)
|
(Abramowitz and Stegun 1972, p. 589).
The derivative of the Jacobi amplitude is given by
 |
(4)
|
or using the notation 
,
 |
(5)
|
The amplitude function has the special values
 |  |  |
(6)
|
 |  |  |
(7)
|
where 
is a complete elliptic integral
of the first kind. In addition, it obeys the identities
 |  |  |
(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
 |  |  |
(11)
|
 |  |  |
(12)
|
 |  |  |
(13)
|
which serve as definitions for the Jacobi elliptic functions.
