step function in nLab

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
=--
=--


#Contents#
* table of contents
{:toc}

## Idea

Generally, a _step function_ is a  [[function]] from the [[real numbers]] to themselves which is [[constant function|constant]] everywhere except at one single point (or a [[finite number]] of points).

Specifically the function

$$
  \Theta \colon x
  \mapsto
  \left\{
    \array{
      0 &\vert& x \lt 0
      \\
      1 &\vert& x \geq 0
    }
  \right.
$$

is sometimes called the _Heaviside step function_. 

This may be regarded as the [[generalized function]]-expression for the [[distributional density]] $\Theta \in \mathcal{D}'(\mathbb{R})$ which sends a [[bump function]] $b \in C^\infty_c(\mathbb{R})$ to its [[integral]] over the positive half-axis:

$$
  \begin{aligned}
    \int_{\mathbb{R}} b(x) \Theta(x) d x
    & \coloneqq
    \langle \Theta, b\rangle
    \\
    & \coloneqq
    \int_0^\infty b(x) d x
  \end{aligned}
  \,.
$$

As such $\Theta$ is also called the _Heaviside distribution_.

The [[derivative of a distribution|distributional derivative]] of the Heaviside distribution is the [[Dirac delta distribution]] (prop. \ref{DerivativeOfHeavisideDistribution} below).


## Properties

+-- {: .num_prop #DerivativeOfHeavisideDistribution}
###### Proposition

The [[derivative of distributions|distributional derivative]] of the Heaviside distribution $\Theta \in \mathcal{D}'(\mathbb{R})$ is the [[delta distribution]] $\delta \in \mathcal{D}'(\mathbb{R})$:

$$
  \partial \Theta = \delta
  \,.
$$

=--

+-- {: .proof}
###### Proof

For $b \in C^\infty_c(\mathbb{R})$ any [[bump function]] we compute:

$$
  \begin{aligned}
    \int \partial\Theta(x) b(x) \, d x
    & =
    - \int \Theta(x) \partial b(x)\, dx
    \\
    & =
    - \int_0^\infty \partial b(x) d x
    \\
    & =
    - \left( b(x)\vert_{x \to \infty} - b(0) \right)
    \\
    & =
    b(0)
    \\
    &
    =
    \int \delta(x) b(x) \, dx
    \,.
  \end{aligned}
$$

=--

+-- {: .num_defn #FourierIntegralFormula}
###### Definition
**([[Fourier integral]] formula for step function)**


The Heaviside distribution $\Theta \in \mathcal{D}'(\mathbb{R})$ is equivalently the following [[limit of a sequence|limit]] of [[Fourier integrals]] (see at _[[Cauchy principal value]]_)

$$
  \begin{aligned}
    \Theta(x)
    & =
    \frac{1}{2\pi i} \int_{-\infty}^\infty \frac{e^{i \omega x}}{\omega - i 0^+}
    \\
    & \coloneqq
    \underset{ \epsilon \to 0^+}{\lim} 
    \frac{1}{2 \pi i}
    \int_{-\infty}^\infty \frac{e^{i \omega x}}{\omega - i \epsilon} d\omega
    \,,
  \end{aligned}
$$

where the limit is taken over [[sequences]] of [[positive numbers|positive]] [[real numbers]] $\epsilon \in (-\infty,0)$ tending to zero.

=--


+-- {: .proof}
###### Proof

We may think of the [[integrand]] $\frac{e^{i \omega x}}{\omega - i \epsilon}$ uniquely extended to a [[holomorphic function]] on the [[complex plane]] and consider computing the given real [[line integral]] for fixed $\epsilon$ as a [[contour integral]] in the [[complex plane]].

If $x \in (0,\infty)$ is [[positive number|positive]], then the exponent 

$$
  i \omega x = - Im(\omega) x + i Re(\omega) x
$$ 

<div style="float:right;margin:0 10px 10px 0;"><img src="https://ncatlab.org/nlab/files/ContoursForHeavisideFourierTransform.png" width="300">
</div>


has negative [[real part]] for _positive_ [[imaginary part]] of $\omega$. This means that the [[line integral]] equals the complex [[contour integral]] over a contour $C_+ \subset \mathbb{C}$ closing in the [[upper half plane]]. Since $i \epsilon$ has positive [[imaginary part]] by construction, this contour does encircle the [[pole]] of the [[integrand]] $\frac{e^{i \omega x}}{\omega - i \epsilon}$ at $\omega = i \epsilon$. Hence by the [[Cauchy integral formula]] in the case $x \gt 0$ one gets


$$
  \begin{aligned}
    \underset{\epsilon \to 0^+}{\lim}
    \frac{1}{2 \pi i}
    \int_{-\infty}^\infty \frac{e^{i \omega x}}{\omega - i \epsilon} d\omega
    & =
    \underset{\epsilon \to 0^+}{\lim}
    \frac{1}{2 \pi i}
    \oint_{C_+} \frac{e^{i \omega x}}{\omega - i \epsilon} d \omega
    \\
    &
    =
    \underset{\epsilon \to 0^+}{\lim}
    \left(e^{i \omega x}\vert_{\omega = i \epsilon}\right)
    \\
    & =
    \underset{\epsilon \to 0^+}{\lim}
    e^{- \epsilon x}
    \\
    & =
    e^0 = 1
  \end{aligned}
  \,.
$$

Conversely, for $x \lt 0$ the real part of the integrand decays as the _[[negative number|negative]]_ imaginary part increases, and hence in this case the given line integral equals the contour integral for a contour $C_- \subset \mathbb{C}$ closing in the lower half plane. Since the integrand has no pole in the lower half plane, in this case the [[Cauchy integral formula]] says that this integral is zero.

=--

+-- {: .num_remark}
###### Remark

The Fourier form of the step function in prop. \ref{FourierIntegralFormula} gives rise to the standard expression for the [[advanced propagator]], [[retarded propagator]] and [[Feynman propagator]] used in [[perturbative quantum field theory]]. See at _[[Feynman propagator]]_ for more.


=--

## Related concepts

* [[sign function]]

## References

* Wikipedia, _[Step function](https://en.wikipedia.org/wiki/Step_function)_

[[!redirects step functions]]

[[!redirects Heaviside function]]
[[!redirects Heaviside functions]]

[[!redirects Heaviside step function]]
[[!redirects Heaviside step functions]]

[[!redirects Heaviside distribution]]
[[!redirects Heaviside distributions]]

[[!redirects Heaviside step distribution]]
[[!redirects Heaviside step distributions]]