Elliptic integral

Modern mathematics defines an "elliptic integral" as any function f which can be expressed in the form

where R is a rational function of its two arguments, P is a polynomial of degree 3 or 4 with no repeated roots, and c is a constant.

In general, integrals in this form cannot be expressed in terms of elementary functions.

Exceptions to this general rule are when P has repeated roots, or when R(x, y) contains no odd powers of y or if the integral is pseudo-elliptic.

Additional insight into the theory of the elliptic integral may be gained through the study of the Schwarz–Christoffel mapping.

These arguments are expressed in a variety of different but equivalent ways as they give the same elliptic integral.

The latter is sometimes called the delta amplitude and written as Δ(φ) = dn u.

In this notation, the use of a vertical bar as delimiter indicates that the argument following it is the "parameter" (as defined above), while the backslash indicates that it is the modular angle.

This potentially confusing use of different argument delimiters is traditional in elliptic integrals and much of the notation is compatible with that used in the reference book by Abramowitz and Stegun and that used in the integral tables by Gradshteyn and Ryzhik.

Moreover, their complete integrals employ the parameter k2 as argument in place of the modulus k, i.e. K(k2) rather than K(k).

And the integral of the third kind defined by Gradshteyn and Ryzhik, Π(φ, n, k), puts the amplitude φ first and not the "characteristic" n. Thus one must be careful with the notation when using these functions, because various reputable references and software packages use different conventions in the definitions of the elliptic functions.

The incomplete elliptic integral of the first kind has following addition theorem[citation needed]:

The incomplete elliptic integral of the second kind E in Legendre's trigonometric form is

The meridian arc length from the equator to latitude φ is written in terms of E:

The incomplete elliptic integral of the second kind has following addition theorem[citation needed]:

The number n is called the characteristic and can take on any value, independently of the other arguments.

Note though that the value Π(1; ⁠π/2⁠ | m) is infinite, for any m. A relation with the Jacobian elliptic functions is

The meridian arc length from the equator to latitude φ is also related to a special case of Π:

In terms of the Gauss hypergeometric function, the complete elliptic integral of the first kind can be expressed as

The complete elliptic integral of the first kind is sometimes called the quarter period.

[citation needed] The differential equation for the elliptic integral of the first kind is

instead, because the squaring function introduces problems when inverting in the complex plane.

The complete elliptic integral of the second kind can be expressed as a power series[9]

In terms of the Gauss hypergeometric function, the complete elliptic integral of the second kind can be expressed as

(where λ is the modular lambda function), then E(k) is expressible in closed form in terms of

Note that sometimes the elliptic integral of the third kind is defined with an inverse sign for the characteristic n,

For example: And for two modules that are tangential counterparts to each other, the following relationship is valid: For example: The Legendre's relation for tangential modular counterparts results directly from the Legendre's identity for Pythagorean modular counterparts by using the Landen modular transformation on the Pythagorean counter modulus.

For the lemniscatic case, the elliptic modulus or specific eccentricity ε is equal to half the square root of two.

Legendre's identity for the lemniscatic case can be proved as follows: According to the Chain rule these derivatives hold: By using the Fundamental theorem of calculus these formulas can be generated: The Linear combination of the two now mentioned integrals leads to the following formula: By forming the original antiderivative related to x from the function now shown using the Product rule this formula results: If the value

The previously determined result shall be combined with the Legendre equation to the modulus

Plot of the complete elliptic integral of the first kind K ( k )
Plot of the complete elliptic integral of the second kind E ( k )
Plot of the complete elliptic integral of the third kind Π( n , k ) with several fixed values of n