# amplitude (am) function

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##### 1: 22.16 Related Functions
###### Integral Representation Figure 22.16.1: Jacobi’s amplitude function am ⁡ ( x , k ) for 0 ≤ x ≤ 10 ⁢ π and k = 0.4 , 0.7 , 0.99 , 0.999999 . … Magnify
##### 3: 22.19 Physical Applications
###### §22.19(i) Classical Dynamics: The Pendulum
22.19.3 $\theta(t)=2\operatorname{am}\left(t\sqrt{E/2},\sqrt{2/E}\right),$ Figure 22.19.1: Jacobi’s amplitude function am ⁡ ( x , k ) for 0 ≤ x ≤ 10 ⁢ π and k = 0.5 , 0.9999 , 1.0001 , 2 . … Magnify
##### 4: 22.1 Special Notation
The functions treated in this chapter are the three principal Jacobian elliptic functions $\operatorname{sn}\left(z,k\right)$, $\operatorname{cn}\left(z,k\right)$, $\operatorname{dn}\left(z,k\right)$; the nine subsidiary Jacobian elliptic functions $\operatorname{cd}\left(z,k\right)$, $\operatorname{sd}\left(z,k\right)$, $\operatorname{nd}\left(z,k\right)$, $\operatorname{dc}\left(z,k\right)$, $\operatorname{nc}\left(z,k\right)$, $\operatorname{sc}\left(z,k\right)$, $\operatorname{ns}\left(z,k\right)$, $\operatorname{ds}\left(z,k\right)$, $\operatorname{cs}\left(z,k\right)$; the amplitude function $\operatorname{am}\left(x,k\right)$; Jacobi’s epsilon and zeta functions $\mathcal{E}\left(x,k\right)$ and $\mathrm{Z}\left(x|k\right)$. …
##### 6: 19.25 Relations to Other Functions
19.25.30 $\operatorname{am}\left(u,k\right)=R_{C}\left({\operatorname{cs}}^{2}\left(u,k% \right),{\operatorname{ns}}^{2}\left(u,k\right)\right),$
##### 7: Errata
• Equation (22.19.3)
22.19.3 $\theta(t)=2\operatorname{am}\left(t\sqrt{E/2},\sqrt{2/E}\right)$

Originally the first argument to the function $\operatorname{am}$ was given incorrectly as $t$. The correct argument is $t\,\sqrt{E/2}$.

Reported 2014-03-05 by Svante Janson.

• ##### 8: 22.14 Integrals
22.14.3 $\int\operatorname{dn}\left(x,k\right)\mathrm{d}x=\operatorname{Arcsin}\left(% \operatorname{sn}\left(x,k\right)\right)=\operatorname{am}\left(x,k\right).$
##### 9: 29.2 Differential Equations
29.2.5 $\phi=\tfrac{1}{2}\pi-\operatorname{am}\left(z,k\right).$
##### 10: 29.1 Special Notation
(For other notation see Notation for the Special Functions.) … All derivatives are denoted by differentials, not by primes. … The notation for the eigenvalues and functions is due to Erdélyi et al. (1955, §15.5.1) and that for the polynomials is due to Arscott (1964b, §9.3.2). … where $\psi=\operatorname{am}\left(z,k\right)$; see §22.16(i). The relation to the Lamé functions ${\rm Ec}^{m}_{\nu}$, ${\rm Es}^{m}_{\nu}$ of Ince (1940b) is given by …