# UnitStep

UnitStep[x]

represents the unit step function, equal to 0 for and 1 for .

UnitStep[x1,x2,]

represents the multidimensional unit step function which is 1 only if none of the are negative.

# Details

• Some transformations are done automatically when UnitStep appears in a product of terms.
• UnitStep provides a convenient way to represent piecewise continuous functions.
• UnitStep has attribute Orderless.
• For exact numeric quantities, UnitStep internally uses numerical approximations to establish its result. This process can be affected by the setting of the global variable \$MaxExtraPrecision.
• UnitStep[] is 1.
• UnitStep automatically threads over lists. »

# Examples

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## Basic Examples(4)

Evaluate numerically:

Plot in one dimension:

Plot in two dimensions:

UnitStep is a piecewise function:

## Scope(34)

### Numerical Evaluation(6)

Evaluate numerically:

UnitStep always returns an exact result:

Evaluate efficiently at high precision:

UnitStep can deal with realvalued intervals:

Compute the elementwise values of an array using automatic threading:

Or compute the matrix UnitStep function using MatrixFunction:

Compute average-case statistical intervals using Around:

### Specific Values(4)

Value at zero:

Value at infinity:

Evaluate for symbolic parameters:

Find a value of x for which the UnitStep[x]=1:

### Visualization(4)

Plot the UnitStep function:

Visualize shifted UnitStep functions:

Visualize the composition of UnitStep with a periodic function:

Plot UnitStep in three dimensions:

### Function Properties(10)

Function domain of UnitStep:

It is restricted to real inputs:

Function range of UnitStep:

UnitStep has a jump discontinuity at the point :

UnitStep is not an analytic function:

It has both singularities and discontinuities:

UnitStep is nondecreasing:

UnitStep is not injective:

UnitStep is not surjective:

UnitStep is non-negative:

UnitStep is neither convex nor concave:

### Differentiation and Integration(6)

First derivative with respect to x:

All higher-order derivatives the same:

First derivative with respect to z:

Compute the indefinite integral using Integrate:

Verify the anti-derivative away from the singular point:

Definite integral:

Integral over an infinite domain:

### Integral Transforms(4)

Find the LaplaceTransform of UnitStep:

The convolution of UnitStep with itself:

## Applications(8)

Generate a square wave:

Compute a step response for a continuous-time system:

Using transform methods:

Compute a step response for a discrete-time system:

Using transform methods:

Solve the timeindependent Schrödinger equation with piecewise analytic potential:

This gives the probability of the random variable being in the interval :

Here is the resulting probability plotted:

Construct the Walsh function:

Define a BoseEinstein and a MaxwellBoltzmann distribution function with UnitStep and Exp:

Plot the distributions:

Find the representation of a mathematical expression with UnitStep in terms of FoxH:

## Properties & Relations(4)

The derivative of UnitStep is a piecewise function:

The derivative of HeavisideTheta is a distribution:

Expand into UnitStep of linear factors:

Convert into Piecewise:

Integrate over finite and infinite domains:

## Possible Issues(3)

Symbolic preprocessing of functions containing UnitStep can be timeconsuming:

Limit does not give UnitStep as a limit of smooth functions:

Differentiating Abs does not yield UnitStep:

Use RealAbs to get a derivative of absolute value on the reals:

But for the origin, where the derivative does not exist, this is equivalent to an expression in UnitStep:

Wolfram Research (1999), UnitStep, Wolfram Language function, https://reference.wolfram.com/language/ref/UnitStep.html (updated 2007).

#### Text

Wolfram Research (1999), UnitStep, Wolfram Language function, https://reference.wolfram.com/language/ref/UnitStep.html (updated 2007).

#### CMS

Wolfram Language. 1999. "UnitStep." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2007. https://reference.wolfram.com/language/ref/UnitStep.html.

#### APA

Wolfram Language. (1999). UnitStep. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/UnitStep.html

#### BibTeX

@misc{reference.wolfram_2024_unitstep, author="Wolfram Research", title="{UnitStep}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/UnitStep.html}", note=[Accessed: 15-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_unitstep, organization={Wolfram Research}, title={UnitStep}, year={2007}, url={https://reference.wolfram.com/language/ref/UnitStep.html}, note=[Accessed: 15-August-2024 ]}