UnitBox

The unit box function: 1 on the closed [-1/2, 1/2], 0 outside.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageUnitBox✓↗

UnitBox(x)1 for -1/2 <= x <= 1/2, 0 otherwise.

UnitBox(x1, x2, …)the product of the unit boxes -- 0 as soon as any argument is outside its own [-1/2, 1/2].

Typed signatures

Type	Package	Applies
(real, real*) -> number	analytic	always

Domain: Elementary functions

Details

• Defined everywhere, including the boundary -- unlike HeavisidePi, the same rectangular pulse shape but left unevaluated at x = ±1/2 instead of closed there. UnitBox and HeavisidePi differ exactly at the boundary.
• D(UnitBox(x), x) = Piecewise({{Indeterminate, x == 1/2 || x == -1/2}}, 0) -- 0 away from the jump, undefined exactly at the two points where UnitBox is discontinuous.

Examples

1 more kept as data, checked against the oracles

Basic #

Inside the box

live ↗

In$\mathrm{UnitBox}(0)$Out$1$

The boundary is closed

live ↗

In$\mathrm{UnitBox}(\frac{1}{2})$Out$1$

The boundary is closed

live ↗

In$\mathrm{UnitBox}(-\frac{1}{2})$Out$1$

UnitBox as a one-branch Piecewise

live ↗

In$\mathrm{PiecewiseExpand}(\mathrm{UnitBox}(x))$Out$\mathrm{Piecewise}(\bigl\lbrack\bigl\lbrack1, -1/2≤x≤1/2\bigr\rbrack\bigr\rbrack, 0)$

Scope #

Outside the box

live ↗

In$\mathrm{UnitBox}(1)$Out$0$

Two arguments -- the product of their unit boxes

live ↗

In$\mathrm{UnitBox}(0, 0)$Out$1$

An exact irrational operand: the boundary decision is certified, not a premature float

live ↗

In$\mathrm{UnitBox}(\frac{π}{4})$Out$0$

Known value · π/4 > 1/2

An exact irrational operand inside the box

live ↗

In$\mathrm{UnitBox}(\frac{\sqrt{2}}{4})$Out$1$

Known value · √2/4 ≈ 0.354 < 1/2

Zero outside the support

live ↗

In$\mathrm{UnitBox}(-1)$Out$0$

One inside $|x|<\frac12$

live ↗

In$\mathrm{UnitBox}(\frac{1}{4})$Out$1$

A matrix function with eigenvalues $\frac12$ and $\frac13$, giving the identity

live ↗

In$\mathrm{FullSimplify}(\mathrm{MatrixFunction}(\mathrm{UnitBox}, \bigl\lbrack\bigl\lbrack1/2, -1\bigr\rbrack, \bigl\lbrack0, 1/3\bigr\rbrack\bigr\rbrack))$Out$\bigl\lbrack\bigl\lbrack1, 0\bigr\rbrack, \bigl\lbrack0, 1\bigr\rbrack\bigr\rbrack$

The box is cut to what $0 < x < 2$ leaves open

live ↗

In$\mathrm{PiecewiseExpand}(\mathrm{UnitBox}(x), 0\lt x\lt2)$Out$\mathrm{Piecewise}(\bigl\lbrack\bigl\lbrack1, x≤1/2\bigr\rbrack\bigr\rbrack, 0)$

live ↗

In$\mathrm{Same}(\mathrm{UnitBox}^{\prime\prime}(x), \mathrm{UnitBox}^{\prime\prime\prime}(x))$Out$\mathrm{True}$

UnitBox is locally constant, so every derivative order is the same function

live ↗

In$\mathrm{UnitBox}^{\prime\prime\prime}(x)$Out$\mathrm{Piecewise}(\bigl\lbrack\bigl\lbrack\operatorname{Indeterminate}, x=1/2∨x=-1/2\bigr\rbrack\bigr\rbrack, 0)$

Properties #

0 away from the jump, undefined exactly at the boundary

live ↗

In$\frac{\mathrm{d}\mathrm{UnitBox}(x)}{\mathrm{d}x}$Out$\mathrm{Piecewise}(\bigl\lbrack\bigl\lbrack\operatorname{Indeterminate}, x=1/2∨x=-1/2\bigr\rbrack\bigr\rbrack, 0)$

Zero outside the box at 2

live ↗

In$\mathrm{UnitBox}(2)$Out$0$

Known value · Wolfram MathematicalFunctionData, UnitBox

About

Examples	17
Checked against	Wolfram Language 17 agree
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