Inverse Jacobi Elliptic Functions Calculator (arcsn, arccn, arcdn)

Compute the inverse Jacobi elliptic functions arcsn(x,k), arccn(x,k), and arcdn(x,k). Enter the function type, argument x, and modulus k to get the u value expressed via the incomplete elliptic integral of the first kind.

Input

Compute the inverse Jacobi elliptic functions arcsn, arccn, and arcdn. Enter the type, argument x, and modulus k (0 to 1). The modulus is k itself, not m=k^2.

Function type

Choose which Jacobi elliptic function to invert.

Argument x

For sn and cn use -1 to 1; for dn use the complementary modulus k to 1.

Modulus k

A value from 0 to 1. Enter the modulus k, not the parameter m=k^2.

Result

u value of arcsn (x=0.5, k=0.5)

0.5294286271

Modulus k

0.5

Parameter m = k^2

0.25

Check (original arcsn value recomputed from u)

0.5

Graph of the base function for arcsn and the position of u

How it works

The inverse Jacobi elliptic functions are the inverses of sn(u,k), cn(u,k), and dn(u,k). Given a value x and modulus k they return u.
The modulus convention uses the modulus k itself as input, not the parameter m=k^2. The valid input range for k is 0 to 1. The displayed m means k^2.
arcsn uses u=F(arcsin(x),k), arccn uses u=F(arccos(x),k), and arcdn uses sin(phi)=sqrt((1-x^2)/k^2) with u=F(phi,k). Here F is the incomplete elliptic integral of the first kind.
The argument x must lie in -1 to 1 for sn and cn, and in k to 1 for dn (where k is the complementary modulus sqrt(1-k^2)); otherwise the value is undefined.
Computation uses the Carlson symmetric form RF for the incomplete elliptic integral and the AGM method for the amplitude function am. The verification value recomputes the original function from the found u and should closely match the input x.

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