# relation to amplitude (am) function

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## 6 matching pages

##### 1: 22.16 Related Functions
22.16.4 $\operatorname{am}\left(x,0\right)=x,$
22.16.8 $\operatorname{am}\left(x,k\right)=\operatorname{gd}x-\tfrac{1}{4}{k^{\prime}}^% {2}(x-\sinh x\cosh x)\operatorname{sech}x+O\left({k^{\prime}}^{4}\right).$
##### 2: 29.1 Special Notation
(For other notation see Notation for the Special Functions.) … 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). … The relation to the Lamé functions $L^{(m)}_{c\nu}$, $L^{(m)}_{s\nu}$of Jansen (1977) is given by …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 …
##### 3: 29.6 Fourier Series
###### §29.6(i) Function$\mathit{Ec}^{2m}_{\nu}\left(z,k^{2}\right)$
With $\phi=\frac{1}{2}\pi-\operatorname{am}\left(z,k\right)$, as in (29.2.5), we have … In addition, if $H$ satisfies (29.6.2), then (29.6.3) applies. … Consequently, $\mathit{Ec}^{2m}_{\nu}\left(z,k^{2}\right)$ reduces to a Lamé polynomial; compare §§29.12(i) and 29.15(i). …
##### 4: 22.20 Methods of Computation
A powerful way of computing the twelve Jacobian elliptic functions for real or complex values of both the argument $z$ and the modulus $k$ is to use the definitions in terms of theta functions given in §22.2, obtaining the theta functions via methods described in §20.14. …
###### §22.20(vi) RelatedFunctions
$\operatorname{am}\left(x,k\right)$ can be computed from its definition (22.16.1) or from its Fourier series (22.16.9). Alternatively, Sala (1989) shows how to apply the arithmetic-geometric mean to compute $\operatorname{am}\left(x,k\right)$. … For additional information on methods of computation for the Jacobi and related functions, see the introductory sections in the following books: Lawden (1989), Curtis (1964b), Milne-Thomson (1950), and Spenceley and Spenceley (1947). …
##### 6: 29.2 Differential Equations
###### §29.2(ii) Other Forms
29.2.5 $\phi=\tfrac{1}{2}\pi-\operatorname{am}\left(z,k\right).$
For $\operatorname{am}\left(z,k\right)$ see §22.16(i). … we have …For the Weierstrass function $\wp$ see §23.2(ii). …