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...

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...

to proceed to calculate the elliptic integral. Given Eq. 3 and the Legendre polynomial solution for the elliptic integral: K ( k ) = π 2 ∑ n = 0 ∞ ( (...

named elliptic functions because they come from elliptic integrals. Those integrals are in turn named elliptic because they first were encountered for the...

using the Arctangent Integral, also called Inverse Tangent Integral. The same procedure also works for the Complete Elliptic Integral of the second kind...

which has genus zero: see elliptic integral for the origin of the term. However, there is a natural representation of real elliptic curves with shape invariant...

 Complete Elliptic integral of  τ 1 X 1 ′ =  Complete Elliptic integral of  1 − τ 1 2 X 2 =  Complete Elliptic integral of  τ 2 X 2 ′ =  Complete Elliptic integral...

to integrals that generalise the elliptic integrals to all curves over the complex numbers. They include for example the hyperelliptic integrals of type...

quickly, it provides an efficient way to compute elliptic integrals, which are used, for example, in elliptic filter design. The arithmetic–geometric mean...

eccentricity, and the function E {\displaystyle E} is the complete elliptic integral of the second kind, E ( e ) = ∫ 0 π / 2 1 − e 2 sin 2 ⁡ θ   d θ {\displaystyle...

latitude μ, are unrestricted. The above integral is related to a special case of an incomplete elliptic integral of the third kind. In the notation of the...

lemniscate leads to elliptic integrals, as was discovered in the eighteenth century. Around 1800, the elliptic functions inverting those integrals were studied...

{\displaystyle {\sqrt {1-x^{4}}}} (elliptic integral) 1 ln ⁡ x {\displaystyle {\frac {1}{\ln x}}} (logarithmic integral) e − x 2 {\displaystyle e^{-x^{2}}}...

2K(\sin \varphi )}} where K ( k ) {\displaystyle K(k)} is the complete elliptic integral of the first kind K ( k ) = ∫ 0 π / 2 d θ 1 − k 2 sin 2 ⁡ θ . {\displaystyle...

Elliptic functions: The inverses of elliptic integrals; 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 elliptic integrals. In particular, let: u ( z ) = −...

In mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied...

is a mapping of the parameters of an elliptic integral, useful for the efficient numerical evaluation of elliptic functions. It was originally due to John...

forms: as a relation between complete elliptic integrals, or as a relation between periods and quasiperiods of elliptic functions. The two forms are equivalent...

input x is a non-integer value. Ascending factorial Cahen–Mellin integral Elliptic gamma function Gauss's constant Pseudogamma function Hadamard's gamma...

multivalued function of z {\displaystyle z} . Abelian integrals are natural generalizations of elliptic integrals, which arise when F ( x , w ) = w 2 − P ( x )...

mathematics, the Carlson symmetric forms of elliptic integrals are a small canonical set of elliptic integrals to which all others may be reduced. They are...

{x^{n+2}+1}}}\,\mathrm {d} x} In the following some Elliptic Integral Singular Values are derived: The elliptic nome function has these important values: q (...

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 elliptic integral of the second kind. More precisely, C e l l i p s e = 4 a ∫ 0 π /...

one of the Jacobi elliptic functions and K(m) is the complete elliptic integral of the first kind; both are dependent on the elliptic parameter m. The...

k=\cos(\tan ^{-1}(b/a))} is the elliptic modulus and E ( u , k ) {\displaystyle E(u,k)} is the incomplete elliptic integral of the second kind and sn, cn...

of elliptic integrals are a canonical set of three elliptic integrals to which all others may be reduced. Legendre chose the name elliptic integrals because...

are related to Malmsten's integrals. If K(k) is the complete elliptic integral of the first kind, as a function of the elliptic modulus k, then G = 1 2...

{\displaystyle \operatorname {E} (\varphi ,k)} is the incomplete elliptic integral of the second kind with modulus k {\displaystyle k} . It cannot be...