Elliptic Integral Third Kind Approximation
Approximate complete real elliptic integral of the third kind on [0,π/2] with caller even-panel Simpson rule; parameter m=k² explicitly differs from modulus k, no hidden error guarantee near singular m=1 or characteristic n=1.
Description
The complete elliptic integral of the third kind depends on a characteristic and a parameter.
The complete elliptic integral of the third kind depends on a characteristic and a parameter. The parameter m is modulus squared, and using k directly would change the integral.1
Approximate complete real elliptic integral of the third kind on [0,π/2] with caller even-panel Simpson rule; parameter m=k² explicitly differs from modulus k, no hidden error guarantee near singular m=1 or characteristic n=1.
When to use Elliptic Integral Third Kind Approximation
Use this tool to approximate complete real elliptic integral of the third kind on [0,π/2] with caller even-panel Simpson rule; parameter m=k² explicitly differs from modulus k, no hidden error guarantee near singular m=1 or characteristic n=1.
Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
How to use Elliptic Integral Third Kind Approximation and understand its results
- Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
- Enter ellipticParameterM, simpsonPanels, characteristicN using the field table and calculation rules, then run the tool.
- Read completeEllipticIntegral. The explanation below defines the units, signs, ordering and any flags or placeholder values.
- If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.
| Input field | Meaning | Value type; see rules below | Requirement |
|---|---|---|---|
| ellipticParameterM | Elliptic parameter m | number | Required |
| simpsonPanels | Simpson panels | number | Required |
| characteristicN | Characteristic n | number | Required |
Use m in [0,1), characteristic n<1 with |n|≤100 and even Simpson panels 2–10,000. Apply endpoint and alternating 4/2 weights over [0,π/2].
Π(n|m)=∫[0..π/2]1/[(1−n sin²t)sqrt(1−m sin²t)]dt Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.
| Output field | Meaning | Value type |
|---|---|---|
| completeEllipticIntegral | Complete elliptic integral | number |
{
"input": {
"ellipticParameterM": 0,
"simpsonPanels": 100,
"characteristicN": 0.5
},
"output": {
"completeEllipticIntegral": 2.221441469079183
}
}Limitations and assumptions
No automatic quadrature-error guarantee is supplied, particularly near m=1 or n=1. Complex and incomplete integrals are excluded.
Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.
Alternative or Complementary methods
Use Elliptic Integral First Kind Approximation: Approximate complete real elliptic integral of the first kind on [0,π/2] with caller even-panel Simpson rule; parameter m=k² explicitly differs from modulus k, no hidden error guarantee near singular m=1. For a complementary task, use Elliptic Integral Second Kind Approximation: Approximate complete real elliptic integral of the second kind on [0,π/2] with caller even-panel Simpson rule; parameter m=k² explicitly differs from modulus k, no hidden error guarantee near singular m=1.
References
Similar or alternative tools
- Elliptic Integral First Kind Approximation
Approximate complete real elliptic integral of the first kind on [0,π/2] with caller even-panel Simpson rule; parameter m=k² explicitly differs from modulus k, no hidden error guarantee near singular m=1.
- Elliptic Integral Second Kind Approximation
Approximate complete real elliptic integral of the second kind on [0,π/2] with caller even-panel Simpson rule; parameter m=k² explicitly differs from modulus k, no hidden error guarantee near singular m=1.