Heaviside step function

The Heaviside function, also known as Theta, stairs, thresholds, stairs, jump or unit step function, is a function often used in mathematics and physics. It is named after the British mathematician and physicist Oliver Heaviside ( 1850-1925 ).

General

The Heaviside function has for any negative number is null, otherwise the value one. The Heaviside function is continuous everywhere, with the exception of the location. Written in formulas, this means:

So it is the characteristic function of the interval of non-negative real numbers.

In the literature a deviating nomenclature is also commonly held:

  • Which is based on the name of Oliver Heaviside.
  • And according to the label skip function.
  • After the Designation English unit step function.
  • It is also often used.
  • In system theory, one also uses the symbol.

The function has numerous applications, such as in communication engineering or mathematical filter: Multiplying pointwise each value of any continuous function with the corresponding value of the Heaviside function, results in a function that left has the value zero ( deterministic function ), right thereof but corresponds to the original function.

Alternative representations

The value of the Heaviside function on the site you can also define as follows. To identify the definition to write

With. Thus it can represent any size, as long as it contains 0 to 1. Usually, however, be used.

This definition is characterized by the property that is then.

By the selection and hence can be reached that the equations

Are valid for all real.

An integral representation of the Heaviside step function is as follows:

A further representation is given by:

Properties

Differentiability

The Heaviside function is not differentiable in the classical sense, nor is it weakly differentiable. Nevertheless, one can define a derivation of the theory of distributions. The derivative of the Heaviside function in this sense is the Dirac delta function, which is used in physics to describe point-like sources of fields.

A heuristic reasoning for this formula is obtained if one approximates and is suitable, for example by

As well as

And

Integration

The antiderivative of the Heaviside step function is obtained by integration by parts and application of the convolution property of the delta function:

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