In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (c. 1750). Their name originates from their originally arising in connection with the problem of finding the arc length of an ellipse.
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. However, with the appropriate reduction formula, every elliptic integral can be brought into a form that involves integrals over rational functions and the three Legendre canonical forms (i.e. the elliptic integrals of the first, second and third kind).
Besides the Legendre form given below, the elliptic integrals may also be expressed in Carlson symmetric form. Additional insight into the theory of the elliptic integral may be gained through the study of the Schwarz–Christoffel mapping. Historically, elliptic functions were discovered as inverse functions of elliptic integrals.
In integral calculus, an ellipticintegral is one of a number of related functions defined as the value of certain integrals, which were first studied...
In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum (see...
named elliptic functions because they come from ellipticintegrals. Those integrals are in turn named elliptic because they first were encountered for the...
to integrals that generalise the ellipticintegrals to all curves over the complex numbers. They include for example the hyperelliptic integrals of type...
which has genus zero: see ellipticintegral for the origin of the term. However, there is a natural representation of real elliptic curves with shape invariant...
eccentricity, and the function E {\displaystyle E} is the complete ellipticintegral of the second kind, E ( e ) = ∫ 0 π / 2 1 − e 2 sin 2 θ d θ {\displaystyle...
multivalued function of z {\displaystyle z} . Abelian integrals are natural generalizations of ellipticintegrals, which arise when F ( x , w ) = w 2 − P ( x )...
latitude μ, are unrestricted. The above integral is related to a special case of an incomplete ellipticintegral of the third kind. In the notation of the...
lemniscate leads to ellipticintegrals, as was discovered in the eighteenth century. Around 1800, the elliptic functions inverting those integrals were studied...
with coefficients in finite fields, which amounts to counting integral points on an elliptic curve. Some 150 years later, Andre Weil remarked that this particular...
circumference of an ellipse can be expressed exactly in terms of the complete ellipticintegral of the second kind. More precisely, C e l l i p s e = 4 a ∫ 0 π /...
input x is a non-integer value. Ascending factorial Cahen–Mellin integralElliptic gamma function Gauss's constant Pseudogamma function Hadamard's gamma...
Elliptic functions: The inverses of ellipticintegrals; used to model double-periodic phenomena. Jacobi's elliptic functions Weierstrass's elliptic functions...
function is an inverse function of an integral function. Elliptic functions are the inverse functions of ellipticintegrals. In particular, let: u ( z ) = −...
mathematics, the Carlson symmetric forms of ellipticintegrals are a small canonical set of ellipticintegrals to which all others may be reduced. They are...
In mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied...
of ellipticintegrals are a canonical set of three ellipticintegrals to which all others may be reduced. Legendre chose the name ellipticintegrals because...
{x^{n+2}+1}}}\,\mathrm {d} x} In the following some EllipticIntegral Singular Values are derived: The elliptic nome function has these important values: q (...
{L_{z}^{2}}{ml^{2}\sin ^{2}\theta }}+mgl\cos \theta \right]} which leads to an ellipticintegral of the first kind for θ {\displaystyle \theta } t ( θ ) = 1 2 m l...
(geometry) Circumference Crofton formula Ellipticintegral Geodesics Intrinsic equation Integral approximations Line integral Meridian arc Multivariable calculus...
one of the Jacobi elliptic functions and K(m) is the complete ellipticintegral of the first kind; both are dependent on the elliptic parameter m. The...
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