Jacobi Elliptic CN Approximation
Jacobicn inversion of real first quarter elliptic integral, PARAMETERm(notmodulusk),0≤m≤.9,|u|≤K,512-panel quadrature and bisection. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Description
Jacobi cn is the cosine of an elliptic amplitude determined by the first-kind integral.
Jacobi cn is the cosine of an elliptic amplitude determined by the first-kind integral.1
This tool approximates it on a restricted real branch using parameter m and fixed-panel quadrature.
When to use Jacobi Elliptic CN Approximation
This tool approximates it on a restricted real branch using parameter m and fixed-panel quadrature.
How to use Jacobi Elliptic CN Approximation and how it works
- Prepare u, parameter m, using the input definitions and matching units below.
- Enter the required fields and run Jacobi Elliptic CN Approximation.
- Read result.cn using the units, signs and ordering explained below.
- Correct any rejected inputs and rerun; interpret the result within the stated model and its assumptions.
| Input field | Meaning | Value type | Requirement |
|---|---|---|---|
| actualU | Real elliptic-integral argument u, restricted to absolute u no greater than the computed complete integral K(m). | number | Required; finite; other constraints below |
| actualParameterM | Dimensionless elliptic parameter m=k squared, from 0 to 0.9 inclusive. | number | Required; finite; other constraints below |
Require real parameter m between 0 and 0.9 inclusive; m is modulus squared, not k. Estimate K(m) by 512-panel Simpson quadrature of 1/sqrt(1-m*sin(theta)^2) on 0 to pi/2. Require absolute u<=that computed K. Use 60 bisection iterations, each with the same quadrature, to invert the integral for absolute u. Set sn=sign(u)*sin(amplitude), cn=cos(amplitude), dn=sqrt(1-m*sn^2), and return the component named by the tool. No periodic extension, complex branch or error certificate occurs.
F(phi,m)=integral(1/sqrt(1-m*sin(theta)^2),theta=0..phi); F(phi,m)=abs(u); sn=sign(u)*sin(phi); cn=cos(phi); dn=sqrt(1-m*sn^2) Apply the input ordering, units and branch conditions stated above.
| Output field | Meaning | Value type |
|---|---|---|
| result | Object containing the calculated or selected fields below. | object |
| result.cn | Dimensionless Jacobi cn on the restricted real branch. | number |
{
"input": {
"actualU": 0,
"actualParameterM": 0.5
},
"output": {
"result": {
"cn": 1
}
}
}Limitations and assumptions
The calculation uses the supplied observations and model inputs. It does not collect measurements, select a reference population or coefficient, or establish that the model applies to the actual case. The stated branch conditions and output units remain part of the result.
All fields require explicit values. Unexpected fields, wrong types, nonfinite numbers and sparse arrays are rejected. Alignment, bounds and supported-token rules are explained above; extremely large inputs or nonfinite results can also fail. Input errors differ from successful false comparisons, unavailable-domain flags and empty match lists.
DLMF relates the elliptic amplitude to the first-kind integral. This tool numerically inverts a restricted real branch using parameter m equal to modulus squared.2
Alternative or Complementary methods
Use Jacobi Elliptic SN Approximation for a complementary operation: Jacobi sn can be obtained from inversion of the elliptic integral and depends on the parameter m equal to modulus squared.
Use Jacobi Elliptic DN Approximation for a complementary operation: Jacobi dn depends on the elliptic amplitude and the parameter m, rather than the modulus k supplied interchangeably.
References
Similar or alternative tools
- Jacobi Elliptic SN Approximation
Jacobi sn can be obtained from inversion of the elliptic integral and depends on the parameter m equal to modulus squared.
- Jacobi Elliptic DN Approximation
Jacobi dn depends on the elliptic amplitude and the parameter m, rather than the modulus k supplied interchangeably.