JacobiAmplitude

The Jacobi amplitude $\operatorname{am}(u,m)$ — the angle $\varphi$ inverting $u = F(\varphi,m)$ (EllipticF) — in Wolfram/mpmath's $m=k^2$ convention.

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JacobiAmplitude(u, m)the Jacobi amplitude, argument u, parameter m.

Typed signatures

Type	Package	Applies
`(number, number) -> number`	`analytic`	always

Domain: Special functions

Details

$m = k^2$ , the same convention EllipticK / EllipticF use.
$\operatorname{sn}(u,m) = \sin(\operatorname{am}(u,m))$ , $\operatorname{cn}(u,m) = \cos(\operatorname{am}(u,m))$ — the amplitude is what the descending Landen/AGM recursion behind JacobiSN/JacobiCN/JacobiDN computes directly (Abramowitz & Stegun 16.4), continuously (not just modulo $\pi$ , unlike reconstructing it from $\arcsin$ of sn alone).
Real m between 0 and 1 only — unlike the pq family, no parameter transform for m outside [0,1] has been verified for the amplitude itself, so this head declines there.
am(0,m) = 0, for any m in range; am(u,0) = u, for any u.
Numeric only — a symbolic argument (outside the exact table above) stays unevaluated; a floating-point argument (or N()) evaluates directly.
Derivatives of every order in $u$ and in $m$ are closed forms in sn, cn, dn and the Jacobi epsilon $\mathcal{E}(u,m)=Z(u,m)+u\,E(m)/K(m)$ (DLMF 22.13): D, Series in $u$ at $0$ or at a symbolic point, and Series in $m$ at $m=0$ all give concrete coefficients. The $m$ -derivatives evaluate numerically for real $u$ and $m$ between 0 and 1.