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Like the integral of the first kind, the complete elliptic integral of the second kind can be computed very efficiently using the arithmetic–geometric mean. [1] Define sequences a n and g n, where a 0 = 1, g 0 = √ 1 − k 2 = k ′ and the recurrence relations a n + 1 = a n + g n / 2 , g n + 1 = √ a n g n hold.
The relation to elliptic integrals has mainly a historical background. Elliptic integrals had been studied by Legendre, whose work was taken on by Niels Henrik Abel and Carl Gustav Jacobi. Abel discovered elliptic functions by taking the inverse function of the elliptic integral function
The integral may also be recognized as a multiple of Legendre's complete elliptic integral of the first kind. Putting = ) = () ...
The incomplete elliptic integral of the first kind is defined as, (,) = (),the second kind as (,) = (),and the third kind as (,,) = ( ()) ().The argument n of the third kind of integral is known as the characteristic, which in different notational conventions can appear as either the first, second or third argument of Π and furthermore is sometimes defined with the opposite sign.
where K and K′ are the complete elliptic integrals of the first kind for values satisfying k 2 + k′ 2 = 1, and E and E′ are the complete elliptic integrals of the second kind. This form of Legendre's relation expresses the fact that the Wronskian of the complete elliptic integrals (considered as solutions of a differential equation) is a ...
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This fact can be understood and proven with the help of some general theory; see local zeta function and étale cohomology for example. Set of affine points of elliptic curve y 2 = x 3 − x over finite field F 89. The set of points E(F q) is a finite abelian group. It is always cyclic or the product of two cyclic groups. For example, [17] the ...
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