Elliptic Integral — Definition, Formula & Examples

An elliptic integral is a type of integral that involves the square root of a polynomial of degree three or four, and cannot be evaluated using elementary functions like polynomials, exponentials, or trigonometric functions.

An elliptic integral is any integral of the form $\int R\bigl(x,\,\sqrt{P(x)}\bigr)\,dx$ , where $R$ is a rational function of its arguments and $P(x)$ is a polynomial of degree 3 or 4 with no repeated roots. Such integrals are not expressible in closed form using elementary functions and instead define new transcendental functions.

Key Formula

F(\phi,\,k) = \int_0^{\phi} \frac{d\theta}{\sqrt{1 - k^2 \sin^2 \theta}}$$ $$E(\phi,\,k) = \int_0^{\phi} \sqrt{1 - k^2 \sin^2 \theta}\;d\theta

Where:

$\phi$ = Amplitude — the upper limit of integration (in radians)
$k$ = Modulus, satisfying 0 ≤ k < 1, controlling the shape of the integrand
$\theta$ = Variable of integration
$F(\phi, k)$ = Incomplete elliptic integral of the first kind
$E(\phi, k)$ = Incomplete elliptic integral of the second kind

How It Works

Elliptic integrals arise when you try to compute quantities like the arc length of an ellipse — which is where the name originates. Legendre showed that every elliptic integral can be reduced to a combination of three standard forms called the elliptic integrals of the first, second, and third kind. The incomplete elliptic integral of the first kind is

F(\phi, k)

, and the second kind is

E(\phi, k)

. When the upper limit is

\phi = \pi/2

, these become the complete elliptic integrals

K(k)

and

E(k)

. In practice, you evaluate them using tables, series expansions, or numerical methods rather than antiderivative techniques.

Worked Example

Problem: Write the integral for the arc length of one quarter of an ellipse with semi-major axis

a = 2

and semi-minor axis

b = 1

, and express it as a complete elliptic integral.

Step 1: The total circumference of an ellipse involves the integral

C = 4a \int_0^{\pi/2} \sqrt{1 - e^2 \sin^2\theta}\;d\theta

, where

e

is the eccentricity. One quarter of the perimeter uses this integral directly.

\text{Quarter arc length} = a\int_0^{\pi/2}\sqrt{1 - e^2 \sin^2\theta}\;d\theta

Step 2: Compute the eccentricity:

e = \sqrt{1 - b^2/a^2} = \sqrt{1 - 1/4} = \sqrt{3}/2 \approx 0.866

e = \frac{\sqrt{3}}{2}

Step 3: Recognize this as the complete elliptic integral of the second kind with modulus

k = e

\text{Quarter arc length} = 2\,E\!\left(\frac{\sqrt{3}}{2}\right) \approx 2 \times 1.2111 \approx 2.422

Answer: The quarter arc length is

2\,E(\sqrt{3}/2) \approx 2.422

units. This value cannot be expressed in terms of elementary functions.

Why It Matters

Elliptic integrals appear in physics when calculating the period of a pendulum with large swings, the magnetic field of a current loop, and orbital mechanics. They also serve as a gateway to elliptic functions and modular forms, which are central topics in number theory and advanced mathematics.

Common Mistakes

Mistake: Assuming every integral with a square root of a polynomial is an elliptic integral.

Correction: The polynomial under the square root must be of degree 3 or 4 with no repeated roots. Integrals involving

\sqrt{ax^2+bx+c}

(degree 2) are evaluated with standard trigonometric or hyperbolic substitution and are not elliptic.