Measuring the edge of an ellipse needed a whole new family of functions
Finding the length of a circle's rim is easy, but squash it into an ellipse and ordinary formulas fail. The integrals that arise, first studied by Giulio Fagnano and Leonhard Euler around 1750, cannot generally be written with elementary functions. They became known as elliptic integrals.
In modern terms, an elliptic integral is any integral of a rational expression involving the square root of a polynomial of degree three or four with no repeated roots. Such integrals usually have no answer in terms of familiar functions like powers, logarithms or sines. There are exceptions, for instance when the polynomial does have repeated roots, but most cases stay stubbornly irreducible.
Legendre's classification brought order to the zoo. With the right reduction formulas, every elliptic integral can be rewritten using ordinary rational integrals plus three standard shapes, known simply as the first, second and third kinds. Each comes in an incomplete version, depending on two inputs, an angle-like amplitude and a modulus describing how squashed the ellipse is, and a complete version with a single input. They can be written in Legendre's trigonometric form or, after a substitution, in Jacobi's algebraic form, and an alternative scheme called the Carlson symmetric form also exists.
Turning the problem inside out proved fruitful. Historically, elliptic functions were first found by inverting these integrals, swapping the roles of input and output. The Jacobi elliptic functions, with terse names like sn, cn and dn, belong to this family, and sn is simply the inverse of the first-kind integral. That integral even obeys its own addition theorem, a formula for combining two values into one.
The subject is also a notation minefield. Different reference works place arguments in different orders and use vertical bars, backslashes and semicolons to signal whether an input is the modulus, the parameter or a modular angle. Standard tables by Abramowitz and Stegun and by Gradshteyn and Ryzhik disagree in places, and Mathematica defines the complete first-kind integral using the parameter rather than the modulus, so careless users can get the wrong number. The second-kind integral has a very concrete use: it gives the length of a meridian arc from the equator up to any chosen latitude.
Source: Elliptic integral
