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Given a simple harmonic oscillator with a quadratic perturbation, write the perturbation term in the form alphaepsilonx^2, x^..+omega_0^2x-alphaepsilonx^2=0, (1) find the ...

Take the Helmholtz differential equation del ^2F+k^2F=0 (1) in spherical coordinates. This is just Laplace's equation in spherical coordinates with an additional term, (2) ...

The ordinary differential equation z^2y^('')+zy^'+(z^2-nu^2)y=(4(1/2z)^(nu+1))/(sqrt(pi)Gamma(nu+1/2)), where Gamma(z) is the gamma function (Abramowitz and Stegun 1972, p. ...

For a second-order ordinary differential equation, y^('')+p(x)y^'+q(x)y=g(x). (1) Assume that linearly independent solutions y_1(x) and y_2(x) are known to the homogeneous ...

(d^2u)/(dz^2)+(du)/(dz)+(k/z+(1/4-m^2)/(z^2))u=0. (1) Let u=e^(-z/2)W_(k,m)(z), where W_(k,m)(z) denotes a Whittaker function. Then (1) becomes ...

The term "closure" has various meanings in mathematics. The topological closure of a subset A of a topological space X is the smallest closed subset of X containing A. If R ...

The colon is the symbol ":". It is used in a number of different ways in mathematics. 1. To denote ratio or odds, as in 2:1 (voiced "two to one"). 2. To mean such that in ...

The tangent space at a point p in an abstract manifold M can be described without the use of embeddings or coordinate charts. The elements of the tangent space are called ...

Given a smooth manifold M with an open cover U_i, a partition of unity subject to the cover U_i is a collection of smooth, nonnegative functions psi_i, such that the support ...

The highest order power in a univariate polynomial is known as its order (or, more properly, its polynomial degree). For example, the polynomial ...