Jacobi Amplitude am(u,k) Calculator

Enter u and the modulus k to compute the Jacobi amplitude am(u,k) in degrees and radians, along with the elliptic functions sn, cn, and dn.

Input

Enter u and the modulus k to compute the Jacobi amplitude am(u,k) in degrees and radians, along with the elliptic functions sn, cn, and dn.

u (argument)

Enter the argument u of the Jacobi amplitude function.

Modulus k

Enter the modulus k. Its relation to the parameter m is m=k^2.

Amplitude am(u=1, k=0.5)

55.3495021947 deg

0.9660310526 rad

sn(u,k)

0.8226355781

cn(u,k)

0.5685689981

dn(u,k)

0.9114920057

Curve of am(u,k) at fixed modulus k

The Jacobi amplitude am(u,k) is defined as the angle satisfying sn(u,k)=sin(am). Thus sn(u,k)=sin(am(u,k)) and cn(u,k)=cos(am(u,k)).
Modulus convention: this tool takes the modulus k directly. Its relation to the parameter m is m=k^2. For k between 0 and 1 the function shows periodic oscillation.
The inverse relation is the incomplete elliptic integral of the first kind F(am,k)=u. That is, am is the upper-limit angle that makes the integral F equal to u.
The computation uses an original implementation of the arithmetic-geometric mean (AGM) descending sequence: start from a0=1, b0=sqrt(1-k^2), c0=k, iterate, then refine the phase along the descending path.
Degenerate cases: for k=0, am=u with sn=sin(u), cn=cos(u), dn=1. For k=1, sn=tanh(u), cn=dn=sech(u), and am=arcsin(tanh(u)).
The identities sn^2+cn^2=1 and dn^2+k^2*sn^2=1 always hold. The amplitude am is shown in both degrees and radians.

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