nLab
step function

Contents

Idea

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

Specifically the function

Θ:x{0 | x<0 1 | x0 \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 Θ𝒟()\Theta \in \mathcal{D}'(\mathbb{R}) which sends a bump function bC c ()b \in C^\infty_c(\mathbb{R}) to its integral over the positive half-axis:

b(x)Θ(x)dx Θ,b 0 b(x)dx. \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 Θ\Theta is also called the Heaviside distribution.

The distributional derivative of the Heaviside distribution is the Dirac delta distribution (prop. 1 below).

Properties

Proposition

The distributional derivative of the Heaviside distribution Θ𝒟()\Theta \in \mathcal{D}'(\mathbb{R}) is the delta distribution δ𝒟()\delta \in \mathcal{D}'(\mathbb{R}):

Θ=δ. \partial \Theta = \delta \,.
Proof

For bC c ()b \in C^\infty_c(\mathbb{R}) any bump function we compute:

Θ(x)b(x)dx =Θ(x)b(x)dx = 0 b(x)dx =(b(x)| xb(0)) =b(0) =δ(x)b(x)dx. \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}
Definition

(Fourier integral formula for step function)

The Heaviside distribution Θ𝒟()\Theta \in \mathcal{D}'(\mathbb{R}) is equivalently the following limit of Fourier integrals (see at Cauchy principal value)

Θ(x) =12πi e iωxωi0 + limϵ0 +12πi e iωxωiϵdω, \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 real numbers ϵ(,0)\epsilon \in (-\infty,0) tending to zero.

Proof

We may think of the integrand e iωxωiϵ\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 ϵ\epsilon as a contour integral in the complex plane.

If x(0,)x \in (0,\infty) is positive, then the exponent

iωx=Im(ω)x+iRe(ω)x i \omega x = - Im(\omega) x + i Re(\omega) x

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

limϵ0 +12πi e iωxωiϵdω =limϵ0 +12πi C +e iωxωiϵdω =limϵ0 +(e iωx| ω=iϵ) =limϵ0 +e ϵx =e 0=1. \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<0x \lt 0 the real part of the integrand decays as the negative imaginary part increases, and hence in this case the given line integral equals the contour integral for a contour C 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.

Remark

The Fourier form of the step function in prop. 1 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.

References

Revised on November 20, 2017 10:12:34 by Urs Schreiber (94.220.49.59)