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As Lagrange showed, any irrational number alpha has an infinity of rational approximations p/q which satisfy |alpha-p/q|<1/(sqrt(5)q^2). (1) Furthermore, if there are no ...

Let {f_n(x)} be a sequence of analytic functions regular in a region G, and let this sequence be uniformly convergent in every closed subset of G. If the analytic function ...

For a rectangular hyperbola x = asect (1) y = atant (2) with inversion center at the origin, the inverse curve is x_i = (2kcost)/(a[3-cos(2t)]) (3) y_i = ...

The pedal curve of a rectangular hyperbola with the pedal point at the focus is a circle (left figure; Hilbert and Cohn-Vossen 1999, p. 26). The pedal curve of a rectangular ...

The hyperbolic cylinder is a quadratic surface given by the equation (x^2)/(a^2)-(y^2)/(b^2)=-1. (1) It is a ruled surface. It can be given parametrically by x = asinhu (2) y ...

A hyperbolic version of the Euclidean dodecahedron. Hyperbolic three-space can be tessellated with hyperbolic dodecahedra whose intermediate dihedral angles are 60, 72, or 90 ...

A hyperbolic fixed point of a differential equation is a fixed point for which the stability matrix has eigenvalues lambda_1<0<lambda_2, also called a saddle point. A ...

By analogy with the lemniscate functions, hyperbolic lemniscate functions can also be defined arcsinhlemnx = int_0^x(1+t^4)^(1/2)dt (1) = x_2F_1(-1/2,1/4;5/4;-x^4) (2) ...

In the hyperbolic plane H^2, a pair of lines can be parallel (diverging from one another in one direction and intersecting at an ideal point at infinity in the other), can ...

The hyperbolic polar sine is a function of an n-dimensional simplex in hyperbolic space. It is analogous to the polar sine of an n-dimensional simplex in elliptic or ...