Generalized Functions and Related Objects
In many practical situations it is convenient to consider limits in which a fixed amount of something is concentrated into an infinitesimal region. Ordinary mathematical functions of the kind normally encountered in calculus cannot readily represent such limits. However, it is possible to introduce generalized functions or distributions which can represent these limits in integrals and other types of calculations.
|DiracDelta[x]||Dirac delta function|
|HeavisideTheta[x]||Heaviside theta function , equal to 0 for and 1 for|
The Heaviside function HeavisideTheta[x] is the indefinite integral of the delta function. It is variously denoted , , , and . As a generalized function, the Heaviside function is defined only inside an integral. This distinguishes it from the unit step function UnitStep[x], which is a piecewise function.
Dirac delta functions can be used in DSolve to find the impulse response or Green's function of systems represented by linear and certain other differential equations.
|DiracDelta[x1,x2,…]||multidimensional Dirac delta function|
|HeavisideTheta[x1,x2,…]||multidimensional Heaviside theta function|
Related to the multidimensional Dirac delta function are two integer functions: discrete delta and Kronecker delta. Discrete delta is 1 if all the , and is zero otherwise. Kronecker delta is 1 if all the are equal, and is zero otherwise.