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Given a function f(x)=f_0(x), write f_1=f^'(x) and define the Sturm functions by f_n(x)=-{f_(n-2)(x)-f_(n-1)(x)[(f_(n-2)(x))/(f_(n-1)(x))]}, (1) where [P(x)/Q(x)] is a ...

The tangent plane to a surface at a point p is the tangent space at p (after translating to the origin). The elements of the tangent space are called tangent vectors, and ...

Any four mutually tangent spheres determine six points of tangency. A pair of tangencies (t_i,t_j) is said to be opposite if the two spheres determining t_i are distinct from ...

The problem of finding the curve down which a bead placed anywhere will fall to the bottom in the same amount of time. The solution is a cycloid, a fact first discovered and ...

A figurate number Te_n of the form Te_n = sum_(k=1)^(n)T_k (1) = 1/6n(n+1)(n+2) (2) = (n+2; 3), (3) where T_k is the kth triangular number and (n; m) is a binomial ...

The tractrix arises in the following problem posed to Leibniz: What is the path of an object starting off with a vertical offset when it is dragged along by a string of ...

Angle addition formulas express trigonometric functions of sums of angles alpha+/-beta in terms of functions of alpha and beta. The fundamental formulas of angle addition in ...

The Wiener-Hopf method is a powerful technique which enables certain linear partial differential equations subject to boundary conditions on semi-infinite domains to be ...

The associated Legendre polynomials P_l^m(x) and P_l^(-m)(x) generalize the Legendre polynomials P_l(x) and are solutions to the associated Legendre differential equation, ...

The Chebyshev polynomials of the first kind are a set of orthogonal polynomials defined as the solutions to the Chebyshev differential equation and denoted T_n(x). They are ...