HeavisideTheta

The Heaviside step function: 0 for x < 0, 1 for x > 0, unevaluated at x = 0.

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

HeavisideTheta(x)0 if x < 0, 1 if x > 0, and unevaluated (stays symbolic) at x = 0.

HeavisideTheta(x1, x2, …)the product of the individual steps -- 0 as soon as any argument is negative, unevaluated if none is negative but at least one is exactly 0.

Typed signatures

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

Domain: Elementary functions

Details

• Deliberately distinct from compute-engine's own native Heaviside, which this package does not touch: Heaviside(0) evaluates to 0 there, not Wolfram's convention. HeavisideTheta is a separate head so both conventions stay available.
• Also distinct from UnitStep (declared in unit-step.ts), which takes the value 1 at exactly 0 rather than staying unevaluated.
• D(HeavisideTheta(x), x) = DiracDelta(x).

Examples

Basic #

live ↗

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

live ↗

In$\mathrm{HeavisideTheta}(1)$Out$1$

Unevaluated at exactly 0 -- unlike UnitStep

live ↗

In$\mathrm{HeavisideTheta}(0)$Out$\mathrm{HeavisideTheta}(0)$

Scope #

Several arguments: the product of their steps

live ↗

In$\mathrm{HeavisideTheta}(1, 2)$Out$1$

Zero as soon as any argument is negative

live ↗

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

A matrix function of the step in terms of its eigenvalues

live ↗

In$\mathrm{MatrixFunction}(\mathrm{HeavisideTheta}, \bigl\lbrack\bigl\lbrack-1, -\frac{1}{2}\bigr\rbrack, \bigl\lbrack-\frac{1}{2}, \frac{1}{2}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\frac{1}{26}(\sqrt{13}(13^{1/2}-3)), -\frac{13^{1/2}}{13}\bigr\rbrack, \bigl\lbrack-\frac{13^{1/2}}{13}, \frac{1}{26}(\sqrt{13}(3+\sqrt{13}))\bigr\rbrack\bigr\rbrack$

With more than one argument the sign assumption does not resolve it

live ↗

In$\mathrm{Assuming}(a\gt0, \mathrm{Refine}(\mathrm{HeavisideTheta}(a, a, a, a)))$Out$\mathrm{HeavisideTheta}(a, a, a, a)$

Negative assumption leaves it symbolic

live ↗

In$\mathrm{Assuming}(a\lt0, \mathrm{Refine}(\mathrm{HeavisideTheta}(a, a, a, a)))$Out$\mathrm{HeavisideTheta}(a, a, a, a)$

Mixed-sign assumptions still leave it symbolic

live ↗

In$\mathrm{Assuming}(a\lt0∧b\gt0, \mathrm{Refine}(\mathrm{HeavisideTheta}(b, b, a)))$Out$\mathrm{HeavisideTheta}(b, b, a)$

Properties #

Derivative is DiracDelta

live ↗

In$\frac{\mathrm{d}\mathrm{HeavisideTheta}(x)}{\mathrm{d}x}$Out$\mathrm{DiracDelta}(x)$

live ↗

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

Known value · Wolfram MathematicalFunctionData, HeavisideTheta, FunctionalEquations

live ↗

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

Known value · Wolfram MathematicalFunctionData, HeavisideTheta, FunctionalEquations

About

Examples	12
Checked against	Wolfram Language 9 agree3 differ
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