Elliptic integral

An integral involving a rational function and the square root of a cubic or quartic polynomial with distinct roots. Its values generally cannot be expressed using elementary functions.

Elliptic curve: A nonsingular curve of genus one equipped with a specified point, often represented by a cubic equation. The square root of a cubic with distinct roots defines a curve of genus one.
Legendre normal form: A standard representation of elliptic integrals using a parameter and an upper integration limit. A change of variables often converts a general elliptic integral into this form.
Giulio Fagnano: An Italian mathematician who studied lemniscate arc lengths and discovered addition properties for them. His lemniscate calculations supplied early examples of elliptic-integral addition formulas.
Pendulum: A suspended body that swings under gravity about a pivot or axis. The exact period of a pendulum at finite amplitude is expressed using a first-kind elliptic integral.
Algebraic function: A function satisfying a polynomial equation whose coefficients are rational functions of its input. The polynomial square root in an elliptic integral is an algebraic function.
Elliptic integral of the first kind: An elliptic integral with integrand proportional to the reciprocal square root of a quadratic expression in sine. It captures the basic inversion of an elliptic integral without a factor of sine squared.
Leonhard Euler: An eighteenth-century mathematician whose work shaped analysis, mechanics, and number theory. Euler developed addition relations that advanced the theory of elliptic integrals.
Elliptic integral of the lemniscatic type: An elliptic integral associated with the lemniscate and the curve defined by a quartic with symmetric roots. The lemniscate’s arc length is a classic geometric source of elliptic integrals.
Genus (mathematics): A topological invariant that, for a compact surface, counts its number of handles. Genus one distinguishes elliptic integrals from the simpler genus-zero case.
Elliptic integral of the second kind: An elliptic integral whose Legendre form includes the square root of a quadratic expression in sine. Its extra square-root factor produces a distinct canonical integral.

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