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...

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

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...

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...

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