Calculator
How to use this calculator
- Select the integral kind: first, second, or third.
- Pick incomplete (uses φ) or complete (φ = π/2).
- Enter modulus k or parameter m = k².
- For incomplete cases, enter φ and choose units.
- For the third kind, enter the characteristic n.
- Click Compute, then download CSV or PDF if needed.
Formula used
Legendre incomplete elliptic integrals (real domain):
- First kind: F(φ, k) = ∫0φ dθ / √(1 − k² sin²θ)
- Second kind: E(φ, k) = ∫0φ √(1 − k² sin²θ) dθ
- Third kind: Π(n; φ, k) = ∫0φ dθ / ((1 − n sin²θ) √(1 − k² sin²θ))
This calculator evaluates these using Carlson symmetric integrals RF, RD, and RJ, which improve numerical stability across wide parameter ranges.
Example data table
| Case | φ | k | n | Expected value |
|---|---|---|---|---|
| Complete K(k) | N/A | 0.5 | N/A | 1.685750354813 |
| Complete E(k) | N/A | 0.5 | N/A | 1.467462209339 |
| Incomplete F(φ,k) | 45° | 0.8 | N/A | 0.839622346804 |
| Incomplete E(φ,k) | 45° | 0.8 | N/A | 0.737136287090 |
| Incomplete Π(n;φ,k) | 30° | 0.7 | 0.2 | 0.545089130334 |
| Complete Π(n,k) | N/A | 0.6 | 0.5 | 2.523900708449 |
Values are shown to 12 decimal places for reference.
Physics notes
Elliptic integrals appear in pendulum motion, magnetic fields from current loops, waveguides, and orbital mechanics. Using m = k² can match many physics references, while k is common in geometry and optics.
Accurate elliptic integrals help model complex physical systems today.
Professional article
1) What elliptic integrals measure
Elliptic integrals arise when a physical quantity depends on a square‑root of a quadratic or quartic expression. In Legendre form, the modulus k (or parameter m = k²) controls how strongly the integrand varies with angle. The first kind accumulates phase stretching, the second kind tracks energy‑weighted arc behavior, and the third kind adds a pole term controlled by n.
2) Complete vs incomplete in real systems
Incomplete integrals use an upper limit φ and model partial motion or finite apertures, such as a field sampled over a limited angular span. Complete integrals fix φ = π/2 and often represent full periods, quarter‑periods, or total geometric contributions. For example, K(0.5) ≈ 1.685750 and E(0.5) ≈ 1.467462, matching the reference table above.
3) Nonlinear pendulum timing data
The exact period of a simple pendulum with amplitude θ0 is T = 4 √(L/g) K(k) with k = sin(θ0/2). At small angles, k is small and K(k) ≈ π/2, reproducing the familiar linear period. As θ0 approaches 180°, k → 1 and K(k) grows large, predicting slower swings consistent with experiments.
4) Magnetic field and inductance modeling
Circular current loops and coaxial coil inductance commonly reduce to combinations of K(k) and E(k). Engineers often tabulate these to compute on‑axis or off‑axis magnetic fields and mutual inductance without brute‑force numerical quadrature. Using stable symmetric‑integral evaluation helps keep accuracy when k is moderate to high, where naive series can lose precision.
5) Geometry and arc‑length datasets
Ellipse perimeter and ellipse arc length are classic applications of the second kind. A useful approximation is P ≈ 4a E(e)
6) Why the third kind matters
The third kind Π introduces the factor (1 − n sin²θ)−1, which models resonant denominators and constraints. It appears in rigid‑body rotation with conserved quantities, in certain gravitational potentials, and in impedance integrals. Because Π can approach large values as the denominator nears zero, the calculator checks the real‑domain condition 1 − n sin²φ > 0.
7) Numerical stability and tolerance control
Direct numerical integration can struggle when the integrand becomes steep near k → 1 or when parameters create near‑singular behavior. Carlson symmetric forms (RF, RD, RJ) transform the problem into rapidly convergent duplication iterations with controlled error. The tolerance input lets you balance speed and precision; 1e‑12 is a strong default for most physics work.
8) Reporting results for reproducible workflows
Scientific reporting benefits from recording the exact kind, parameters (k or m), φ units, and n when applicable. The built‑in CSV export is convenient for lab notebooks and spreadsheets, while the PDF output preserves a clean snapshot for sharing. Always include enough digits for your downstream sensitivity; many problems remain stable with 10–12 significant digits.
FAQs
1) What is the difference between k and m?
k is the modulus, while m is the parameter defined as m = k². Many physics tables use m, but both describe the same shape of the integrand.
2) When should I use the complete form?
Use the complete form when the upper limit is effectively π/2, such as full-period or quarter-period problems. Otherwise choose the incomplete form with your φ limit.
3) Why does K(k) get large near k = 1?
As k approaches 1, the integrand 1/√(1 − k² sin²θ) becomes sharply peaked near θ = π/2, and the integral grows without bound in the real domain.
4) What φ range is acceptable?
Any real φ is allowed, but real-valued results require 1 − k² sin²φ > 0. If you hit the boundary, reduce k or φ.
5) What does the characteristic n control in Π?
n scales an extra denominator term (1 − n sin²θ). Larger n can amplify the result and may create near-singular behavior, so the calculator enforces 1 − n sin²φ > 0.
6) Which tolerance should I pick?
For most physics calculations, 1e‑10 to 1e‑12 works well. If you need faster output, loosen tolerance slightly; if results are sensitive, tighten it.
7) Are the example values exact?
The example values are high-precision references rounded to 12 decimals. Your computed output should match closely, depending on chosen tolerance and the parameters.