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An ellipse is a curve that is the locus of all points in the plane the sum of whose distances r_1 and r_2 from two fixed points F_1 and F_2 (the foci) separated by a distance ...

A doubly periodic function with periods 2omega_1 and 2omega_2 such that f(z+2omega_1)=f(z+2omega_2)=f(z), (1) which is analytic and has no singularities except for poles in ...

Let 0<k^2<1. The incomplete elliptic integral of the third kind is then defined as Pi(n;phi,k) = int_0^phi(dtheta)/((1-nsin^2theta)sqrt(1-k^2sin^2theta)) (1) = ...

A second-order partial differential equation, i.e., one of the form Au_(xx)+2Bu_(xy)+Cu_(yy)+Du_x+Eu_y+F=0, (1) is called elliptic if the matrix Z=[A B; B C] (2) is positive ...

A map projection in which areas on a sphere, and the areas of any features contained on it, are mapped to the plane in such a way that two are related by a constant scaling ...

The general nonhomogeneous differential equation is given by x^2(d^2y)/(dx^2)+alphax(dy)/(dx)+betay=S(x), (1) and the homogeneous equation is x^2y^('')+alphaxy^'+betay=0 (2) ...

Let four lines in a plane represent four roads in general position, and let one traveler T_i be walking along each road at a constant (but not necessarily equal to any other ...

In 1750, Benjamin Franklin constructed the above 8×8 semimagic square having magic constant 260. Any half-row or half-column in this square totals 130, and the four corners ...

Let Omega be an open, bounded, and connected subset of R^d for some d and let dx denote d-dimensional Lebesgue measure on R^d. In functional analysis, the Friedrichs ...

A transformation from one reference frame to another moving with a constant velocity v with respect to the first for classical motion. However, special relativity shows that ...