Elliptic integral

1. What an Elliptic Integral Is

An elliptic integral arises when the integrand contains a square root of a cubic or quartic polynomial, e.g. arc length of ellipse, pendulum period — it cannot be expressed in elementary functions, so it defines a new special function.

Normal (Legendre) forms for $0≤k²<1$ (modulus $k$ , parameter $m=k²$ ), angle $φ$ :

Incomplete elliptic integral of the first kind:

F(φ,k)= ∫_{0}^{φ} dθ / √[1 - k² sin²θ]

Second kind:

E(φ,k)= ∫_{0}^{φ} √[1 - k² sin²θ] dθ

Third kind:

Π(φ,n,k)= ∫_{0}^{φ} dθ / [(1 - n sin²θ)√[1 - k² sin²θ]]

Complete versions are $φ=π/2$ :

K(k)=F(π/2,k), E(k)=E(π/2,k)

etc. These occur pervasively.

2. Where They Come From (for intuition)

Arc length of ellipse $x²/a²+ y²/b²=1$ :

L = a ∫_{0}^{π/2} √[1 - e² sin²θ] dθ = a E(e)

where $e$ = eccentricity.

Period of simple pendulum amplitude $φ_0$ :

T = 4√(l/g)· K(sin(φ_0/2))

So any "ellipse perimeter" or "large pendulum" problem is elliptic.

3. Reduction and Evaluation (what exams expect)

General $∫ dx/√[(1-x²)(1-k²x²)]$ via $x= sinθ$ becomes $F(φ,k)$ . Standard trick: put $x= sinθ$ to reach Legendre form.

(a) I = ∫_{0}^{1} dx/√[(1-x²)(1 -k² x²)] : set x= sinθ ⇒ dx= cosθ dθ, √(1-x²)=cosθ ⇒ I=∫_{0}^{π/2} dθ/√(1 -k² sin²θ)=K(k) ✓

(b) Ellipse perimeter for a=5,b=3: e=√(1-b²/a²)=0.8 ⇒ L=4a E(e)=20·E(0.8) ; table gives E(0.8)≈1.276 ⇒ L≈25.5 vs 2π√((a²+b²)/2) approx 25.5 close ✓

(c) ∫_{0}^{π/2} dθ/√(1 -½ sin²θ)= K(1/√2) — leave as K, numerical via series K= π/2[1+ (1/4)k²+9k⁴/64+...] ✓

4. Relation to Beta/Gamma?

For $k=0$ , $F(φ,0)=φ, E(φ,0)=φ$ — elementary. For general $k$ , no reduction to Beta/Gamma; series expansion uses Beta coefficients. Complete integrals have series: $K(k)= π/2·[1+ (1·1/2·2)² k²+...]$ .

5. Exam Points

Define incomplete/complete elliptic integrals of first/second kind — 3-5 marks, frequent.
Show arc length of ellipse / pendulum period reduces to $E$ or $K$ — 5 marks, occasional (often as example of non-elementary integral).
Evaluate $∫_{0}^{1} dx/√[(1-x²)(1-k² x²)]$ as $K(k)$ — 5 marks, frequent.
Mention that $∫ dx/√(quartic)$ is elliptic — 3 marks.

Elliptic integral