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...
40 KB (7,831 words) - 04:28, 20 June 2025
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...
33 KB (6,114 words) - 03:16, 25 May 2025
named elliptic functions because they come from elliptic integrals. Those integrals are in turn named elliptic because they first were encountered for the...
16 KB (2,442 words) - 04:21, 30 March 2025
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, as...
73 KB (13,081 words) - 10:20, 4 July 2025
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 ∞ ( (...
43 KB (7,667 words) - 21:45, 19 June 2025
Fubini's theorem (redirect from An elegant rearrangement of a conditionally convergent iterated integral)
using the Arctangent Integral, also called Inverse Tangent Integral. The same procedure also works for the Complete Elliptic Integral of the second kind...
41 KB (7,862 words) - 10:10, 5 May 2025
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...
54 KB (8,439 words) - 06:57, 19 June 2025
Differential of the first kind (redirect from Hyper-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...
4 KB (530 words) - 17:31, 26 January 2025
Perimeter of an ellipse (section Elliptic integral)
/2}{\sqrt {1-x\sin ^{2}\theta }}\ d\theta ,} known as the complete elliptic integral of the second kind, the perimeter can be expressed in terms of that...
9 KB (1,251 words) - 04:36, 5 July 2025
{1}{b^{2}}}.} The integrals can be expressed in terms of incomplete elliptic integrals. In terms of the Carlson symmetric form elliptic integral R J {\displaystyle...
9 KB (1,299 words) - 03:03, 14 February 2025
{\displaystyle {\sqrt {1-x^{4}}}} (elliptic integral) 1 ln x {\displaystyle {\frac {1}{\ln x}}} (logarithmic integral) e − x 2 {\displaystyle e^{-x^{2}}}...
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Arithmetic–geometric mean (category Elliptic functions)
quickly, it provides an efficient way to compute elliptic integrals, which are used, for example, in elliptic filter design. The arithmetic–geometric mean...
17 KB (3,029 words) - 17:50, 24 March 2025
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...
5 KB (672 words) - 19:34, 11 August 2024
multivalued function of z {\displaystyle z} . Abelian integrals are natural generalizations of elliptic integrals, which arise when F ( x , w ) = w 2 − P ( x )...
6 KB (848 words) - 11:11, 27 May 2025
Elliptic functions: The inverses of elliptic integrals; used to model double-periodic phenomena. Jacobi's elliptic functions Weierstrass's elliptic functions...
10 KB (1,065 words) - 15:31, 16 June 2025
Carlson symmetric form (category Elliptic functions)
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...
14 KB (3,790 words) - 01:01, 11 May 2024
Meridian arc (section Relation to elliptic integrals)
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...
49 KB (6,787 words) - 06:28, 29 June 2025
by an incomplete elliptic integral of the second kind, the arc length of a spherical conic is given by an incomplete elliptic integral of the third kind...
12 KB (1,056 words) - 02:30, 20 January 2025
Legendre's relation (redirect from Legendre relation for elliptic integrals)
forms: as a relation between complete elliptic integrals, or as a relation between periods and quasiperiods of elliptic functions. The two forms are equivalent...
10 KB (2,248 words) - 20:50, 2 March 2023
Gamma function (redirect from Gamma integral)
input x is a non-integer value. Ascending factorial Cahen–Mellin integral Elliptic gamma function Lemniscate constant Pseudogamma function Hadamard's...
90 KB (13,547 words) - 17:59, 24 June 2025
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 /...
9 KB (1,068 words) - 20:47, 11 May 2025
yksikäsitteisyys (The single-valuedness of the inverse function of the elliptic integral of the first kind). His dissertation was the first and still is the...
4 KB (324 words) - 00:14, 19 February 2025
using elliptic integrals and jacobi elliptic functions. Smith uses the third fast Jacobi elliptic function estimation algorithm found in the elliptic functions...
51 KB (8,014 words) - 07:43, 24 June 2025
lemniscate leads to elliptic integrals, as was discovered in the eighteenth century. Around 1800, the elliptic functions inverting those integrals were studied...
9 KB (1,246 words) - 09:50, 5 May 2025
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...
6 KB (891 words) - 09:52, 15 June 2025
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...
64 KB (9,616 words) - 12:39, 28 May 2025
Theta function (category Elliptic functions)
{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 (...
70 KB (14,667 words) - 23:32, 8 June 2025
with coefficients in finite fields, which amounts to counting integral points on an elliptic curve. An unfinished chapter, consisting of work done during...
181 KB (17,941 words) - 14:40, 8 July 2025
Nome (mathematics) (redirect from Elliptic nome)
theta functions and K ( k ) {\displaystyle K(k)} is the complete elliptic integral of the first kind with modulus k {\displaystyle k} shown in the formula...
80 KB (13,966 words) - 04:17, 17 January 2025