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Pathological functions that are continuous but differentiable only on a set of points of measure zero are sometimes known as monsters of real analysis. Examples include the ...

Multivariable Calculus

A Lambert series is a series of the form F(x)=sum_(n=1)^inftya_n(x^n)/(1-x^n) (1) for |x|<1. Then F(x) = sum_(n=1)^(infty)a_nsum_(m=1)^(infty)x^(mn) (2) = ...

The Bessel differential equation is the linear second-order ordinary differential equation given by x^2(d^2y)/(dx^2)+x(dy)/(dx)+(x^2-n^2)y=0. (1) Equivalently, dividing ...

The Fourier transform is a generalization of the complex Fourier series in the limit as L->infty. Replace the discrete A_n with the continuous F(k)dk while letting n/L->k. ...

The sign of a real number, also called sgn or signum, is -1 for a negative number (i.e., one with a minus sign "-"), 0 for the number zero, or +1 for a positive number (i.e., ...

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) ...

An atlas is a collection of consistent coordinate charts on a manifold, where "consistent" most commonly means that the transition functions of the charts are smooth. As the ...

A general type of statistical distribution which is related to the gamma distribution. Beta distributions have two free parameters, which are labeled according to one of two ...

The Hankel transform (of order zero) is an integral transform equivalent to a two-dimensional Fourier transform with a radially symmetric integral kernel and also called the ...