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Special functions which arise as solutions to second order ordinary differential equations are commonly said to be "of the first kind" if they are nonsingular at the origin, ...

The sigmoid function, also called the sigmoidal curve (von Seggern 2007, p. 148) or logistic function, is the function y=1/(1+e^(-x)). (1) It has derivative (dy)/(dx) = ...

The expansion of the two sides of a sum equality in terms of polynomials in x^m and y^k, followed by closed form summation in terms of x and y. For an example of the ...

The spherical Bessel function of the second kind, denoted y_nu(z) or n_nu(z), is defined by y_nu(z)=sqrt(pi/(2z))Y_(nu+1/2)(z), (1) where Y_nu(z) is a Bessel function of the ...

Toroidal functions are a class of functions also called ring functions that appear in systems having toroidal symmetry. Toroidal functions can be expressed in terms of the ...

Cylindrical coordinates are a generalization of two-dimensional polar coordinates to three dimensions by superposing a height (z) axis. Unfortunately, there are a number of ...

An orthogonal coordinate system is a system of curvilinear coordinates in which each family of surfaces intersects the others at right angles. Orthogonal coordinates ...

Algebraic topology is the study of intrinsic qualitative aspects of spatial objects (e.g., surfaces, spheres, tori, circles, knots, links, configuration spaces, etc.) that ...

Cartesian coordinates are rectilinear two- or three-dimensional coordinates (and therefore a special case of curvilinear coordinates) which are also called rectangular ...

A cyclide is a pair of focal conics which are the envelopes of two one-parameter families of spheres, sometimes also called a cyclid. The cyclide is a quartic surface, and ...