GaussianHeavisideLorentz

Gaussian Heaviside, commonly referred to as the Gaussian-smoothed Heaviside, is a smooth approximation to the Heaviside step function. It replaces the discontinuity at zero with a rapid but continuous transition, enabling differentiation and improved numerical stability.

Definition: For a positive scale parameter ε, the Gaussian Heaviside is Hε(x) = Φ(x/ε) = (1/2)[1 + erf(x/(√2 ε))], where Φ is the standard normal cumulative distribution function and erf is the error function. Its derivative is the Gaussian function dHε/dx = (1/(√(2π) ε)) exp(-x^2/(2 ε^2)). The function tends to 0 as x → −∞ and to 1 as x → ∞; as ε → 0, Hε converges to the standard Heaviside function H.

Properties: Hε is monotone increasing and infinitely differentiable for ε > 0. The transition width is proportional to ε; the area under the derivative equals 1, reflecting a unit jump in the limit. The parameter ε controls the sharpness of the transition.

Applications: It is used in numerical analysis and partial differential equations to regularize nonsmooth terms, in level-set methods for smooth evolution, in image processing for edge-preserving operations, and in optimization where nondifferentiable penalties are approximated by smooth surrogates.

Alternatives and remarks: Other smooth approximations include the logistic sigmoid 1/(1+e−kx) and arctan-based functions. Gaussian smoothing is related to convolving a target function with a Gaussian kernel, yielding a mollified representation of a step.

Implementation tips: Choose ε relative to the desired smoothness and grid resolution. Functions like erf and Φ are available in scientific libraries, and the derivative provides a natural regularization term.