Search results
Results from the WOW.Com Content Network
This is also called an elliptical head. The shape of this head is more economical, because the height of the head is just a fraction of the diameter. Its radius varies between the major and minor axis; usually the ratio is 2:1.
Solving for r x yields r x = 1 / 2 (l − a + c); furthermore r 2 y = r 2 x − c 2. From the upper diagram we see that S 1 and S 2 are the foci of the ellipse section of the ellipsoid in the xz-plane and that r 2 z = r 2 x − a 2.
A holomorphic elliptic function is constant. [2] This is the original form of Liouville's theorem and can be derived from it. [3] A holomorphic elliptic function is bounded since it takes on all of its values on the fundamental domain which is compact. So it is constant by Liouville's theorem.
Plot of the Jacobi ellipse (x 2 + y 2 /b 2 = 1, b real) and the twelve Jacobi elliptic functions pq(u,m) for particular values of angle φ and parameter b. The solid curve is the ellipse, with m = 1 − 1/b 2 and u = F(φ,m) where F(⋅,⋅) is the elliptic integral of the first kind (with parameter =). The dotted curve is the unit circle.
The Weierstrass's elliptic function is usually written with a rather special, lower case script letter ℘, which was Weierstrass's own notation introduced in his lectures of 1862–1863. [footnote 1] It should not be confused with the normal mathematical script letters P: 𝒫 and 𝓅. In computing, the letter ℘ is available as \wp in TeX.
In mathematics, specifically the theory of elliptic functions, the nome is a special function that belongs to the non-elementary functions. This function is of great importance in the description of the elliptic functions, especially in the description of the modular identity of the Jacobi theta function, the Hermite elliptic transcendents and the Weber modular functions, that are used for ...
with j-invariant j(τ) and λ(τ) is sometimes called the modular lambda function. For example, let τ = 2i, then λ(2i) = (−1 + √ 2) 4 which implies g ′ 2, g ′ 3, and therefore g ′ 2 3 − 27g ′ 3 2 of the formula above are all algebraic numbers if τ involves an imaginary quadratic field. In fact, it yields the integer j(2i) = 66 ...
The hyperspherical model is the generalization of the spherical model to higher dimensions. The points of n-dimensional elliptic space are the pairs of unit vectors (x, −x) in R n+1, that is, pairs of antipodal points on the surface of the unit ball in (n + 1)-dimensional space (the n-dimensional hypersphere).