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The pathological function f_a(x)=sum_(k=1)^infty(sin(pik^ax))/(pik^a) (originally defined for a=2) that is continuous but differentiable only on a set of points of measure ...

The function frac(x) giving the fractional (noninteger) part of a real number x. The symbol {x} is sometimes used instead of frac(x) (Graham et al. 1994, p. 70; Havil 2003, ...

There are two sorts of transforms known as the fractional Fourier transform. The linear fractional Fourier transform is a discrete Fourier transform in which the exponent is ...

The so-called generalized Fourier integral is a pair of integrals--a "lower Fourier integral" and an "upper Fourier integral"--which allow certain complex-valued functions f ...

The rectangle function Pi(x) is a function that is 0 outside the interval [-1/2,1/2] and unity inside it. It is also called the gate function, pulse function, or window ...

The confluent hypergeometric function of the second kind gives the second linearly independent solution to the confluent hypergeometric differential equation. It is also ...

The continuous Fourier transform is defined as f(nu) = F_t[f(t)](nu) (1) = int_(-infty)^inftyf(t)e^(-2piinut)dt. (2) Now consider generalization to the case of a discrete ...

The Heaviside step function is a mathematical function denoted H(x), or sometimes theta(x) or u(x) (Abramowitz and Stegun 1972, p. 1020), and also known as the "unit step ...

The Legendre differential equation is the second-order ordinary differential equation (1-x^2)(d^2y)/(dx^2)-2x(dy)/(dx)+l(l+1)y=0, (1) which can be rewritten ...

The spherical harmonics can be generalized to vector spherical harmonics by looking for a scalar function psi and a constant vector c such that M = del x(cpsi) (1) = psi(del ...